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Complete Book of Algebra and Geometry Grades 5-6 9780769643304 ISBN: 0769643302 Publisher: Carson-Dellosa Publishing, LLC Summary: The Complete Book of Algebra and Geometry offers children in grades 5-6 easy-to-understand lessons in higher math concepts, skills, and strategies. This best-selling, 352 page workbook teaches children how to understand algebraic and geometric languages and operations. Children complete a variety of activities that help them develop skills and then complete lessons that apply these skills and concepts to everyday sit...uations. Including a complete answer key this workbook features a user friendly format perfect for browsing, research, and review. Basic Skills Include: -Order of Operations -Numbers -Variables -Expressions -Integers -Powers -Exponents -Points -Lines -Rays -Angles -Area Over 4 million in print! The best-selling "Complete Book series" offers a full complement of instruction, activities, and information about a single topic or subject area. Containing over 30 titles and encompassing preschool to grade 8 this series helps children succeed in every subject area! Carson-Dellosa Publishing Staff is the author of Complete Book of Algebra and Geometry Grades 5-6, published under ISBN 9780769643304 and 0769643302. Fifty one Complete Book of Algebra and Geometry Grades 5-6 textbooks are available for sale on ValoreBooks.com, ten used from the cheapest price of $8.47, or buy new starting at $46.84
After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the König digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices that are part of the Perron–Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry. Using combinatorial and graph-theoretical tools, this book enables a solid understanding of the fundamentals of matrix theory and its application to scientific areas.
The Matrix Algebra Tutor: Learning by Example DVD Series teaches students about matrices and explains why they're useful in mathematics. This episode teaches students how to calculate matrix determinants, including what a matrix determinant is and why it is useful. Grades 9-College. 24 minutes on DVD.
Elementary Algebra for College Students [With CD-Fair 0131994573 Has heavy shelf & corner wear, but still a good reading copy. Includes CD-ROM Has moderate shelf and/or corner wear. Great used condition. Does not include CD-ROMVery Good 0131994573 Has some shelf wear, highlighting, underlining and/or writing. Great used condition. Textbook Only ANNOTATED INSTRUCTOR'S EDITION contains the COMPLETE STUDENT TEXT with some instructor comments or answers. May not include student CD or access code.Very Good 0131994570 FREE Used Good(6 013199457390 FREE Used Good(1 Copy): Good We ship everyday and offer PRIORITY SHIPPING. text book recycle ny, USA $1922.7084.4890.95 FREE eDeliverable Titles total 15 copies About the Book This dynamic new edition of this proven series adds cutting edge print and media resources. An emphasis on the practical applications of algebra motivates learners and encourages them to see algebra as an important part of their daily lives. The reader-friendly writing style uses short, clear sentences and easy-to-understand language, and the outstanding pedagogical program makes the material easy to follow and comprehend. KEY TOPICS Chapter topics cover real numbers, solving linear equations and inequalities, formulas and applications of algebra, exponents and polynomials, factoring, rational expressions and equations, graphing linear equations, systems of linear equations, roots and radicals, and quadratic equations. For the study of Algebra.
Elementary and Intermediate Algebra Worksheets for Classroom or Lab Practice Mathxl 12Mo Stu Cpn Business Mathematics MathXL Tutorials on CD for Elementary and Intermediate Algebra Pass the Test (Standalone) for Elementary and Intermediate Algebra Video Lectures on CD for Elementary and Intermediate Algebra Summary This study skills workbook, written by Alan Bass, expands upon George Woodburys study skills feature in the text, Building Your Study Strategy, and introduces new topics to help students be more successful in developmental math. Topics include: time management, note-taking, homework, and test preparation skills, overcoming math anxiety, among other topics. This no-nonsense approach to developing better math study skills provides students with the basic skills needed to be successful in developmental math.
Classes MTH 125: Everyday College Math This course is intended to further students' mathematical knowledge of concepts and applications they might encounter in everyday adult life. Students will read and understand college-level readings of mathematical topics. Topics will include three main subject areas: advanced consumer math and formulas (mortgage interest, compound interest, loans and credit cards), Logic and Sets (sets and operations, Venn Diagrams, basic logic) and statistics (probability, measures of center and spread, the normal curve
ISBN: 1429210737 / ISBN-13: 9781429210737 Calculus: Early Transcendentals What's the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of ...Show synopsisWhat's the ideal balance? How can you make sure students get both the computational skills they need and a deep understanding of the significance of what they are learning? With your teaching--supported by Rogawski's "Calculus Second Edition"--the most successful new calculus text in 25 years! Widely adopted in its first edition, Rogawski's "Calculus" "Calculus" success continues in a meticulously updated new edition. Revised in response to user feedback and classroom experiences, the new edition provides an even smoother teaching and learning experience.Hide synopsis Hide Calculus: Early Transcendentals This is a great book and it is very easy to understand. But the only downfall of it is I wish the answers to the selected answers showed each step to finding the solution because even though the problems are based on the same concept, some questions are harder. But it's an overall good back and totally worth it :)
Beginning And Intermediate Algebra An Integrated Approach 9780495117933 ISBN: 0495117935 Edition: 5 Pub Date: 2007 Publisher: Thomson Learning Summary: Easy to understand, filled with relevant applications, and focused on helping students develop problem-solving skills, BEGINNING AND INTERMEDIATE ALGEBRA is unparalleled in its ability to engage students in mathematics and prepare them for higher-level courses. Gustafson and Frisk's accessible style combines with drill problems, detailed examples, and careful explanations to help students overcome any mathematics anx...iety. Their proven five-step problem-solving strategy helps break each problem down into manageable segments: analyze the problem, form an equation, solve the equation, state the conclusion, and check the result. Examples and problems use real-life data to make the text more relevant to students and to show how mathematics is used in a wide variety of vocations. Plus, the text features plentiful real-world application problems that help build the strong mathematical foundation necessary for students to feel confident in applying their newly acquired skills in further mathematics courses, at home or on the job. Gustafson, R. David is the author of Beginning And Intermediate Algebra An Integrated Approach, published 2007 under ISBN 9780495117933 and 0495117935. Thirty one Beginning And Intermediate Algebra An Integrated Approach textbooks are available for sale on ValoreBooks.com, twenty eight used from the cheapest price of $1.00, or buy new starting at $34.24Marina Del Rey, CAShipping:StandardComments:Online Software included. New and Unread. Factory Sealed. All items guaranteed, and a portion of ... [more]Online Software included. New and Unread. Factory Sealed. All items guaranteed, and a portion of each sale supports social programs in Los Angeles. Ships from CA. [ [more
4 comments: As an algebra teacher, I can tell you that textbooks (and teachers) go with this approach because students cannot calculate the values of the polynomials even with calculators. Some of my fellow teachers say, "Oh, they'll always have calculators with them on their cell phones." My thinking is, "Oh, they'll always have their brains with them - so let's stuff that with some math." One of my state's GLOs is that students should be informed and ethical users of technology. Nothing about helping them have the ability to create any technology. The assessments for our students allow the use of calculators - another excuse for the silicon crutches. And only a freak might consider the possibility that some nasty solar flairs could leave us with severely diminished electronics capabilities for several years. Why would we want to prepare for that? Somewhere, someone else will make calculators for our kids if that ever happens. Why prepare them for a world that might not be exponentially snazzier than today? A note about tables. I have a jr son who is taking what purports to be a high-school level algebra course (he can get 1 year of credit for alg I when he gets to high school). It is trivial junk. One thing that is weird is how often they are given a table of (x, y) values and have to figure out, over and over again, whether the points fall on a line or a hyperbola. In the meantime they don't do much of anything that I consider algebra. I think algebra is about manipulating formulas and doing calculations to solve problems. For them it's tables and graphs, tables and graphs, tables and graphs,.... Their curriculum is an unholy alliance of Connected Math and a giant book published by Holt. The Connected Math is full of what I call fake word problems. There will be wordiness establishing some (irrelevant) context and then they graph some "data" points that will fall on a line, or a y = kx curve for positive x. Bleh
This course covers the following topics: factoring, algebraic fractions, radicals and rational exponents, complex numbers, quadratic equations, rational equations, linear equations and inequalities in two variables and their graphs, systems of linear equations and inequalities introduction to functions, and applications of the above topics. There is NO Lab Fee for Hybrid courses. Hybrid Sections require a MyMathLab access code. Use of MyMathLab in face-to-face sections is at the discretion of the instructor. Contact your instructor to determine if MyMathLab is required. For sections NOT requiring MyMathLab the textbook listed is required. Program Learning Outcomes: MATHEMATIC COURSE Global Learning Outcomes and Objectives: I. Critical Thinking: Students will evaluate the validity of their own and others' ideas through questioning, analyzing, and synthesizing results into the creative process. A. Evaluate information, text, numerical and/or graphical data for validity and reach conclusions that are supportable. B. Apply understanding and knowledge of mathematical concepts to devise and analyze solutions to problems. II. Scientific and Mathematical Literacy: Students will apply an understanding of mathematical, natural, and behavioral scientific principles and methods to solve abstract and practical problems. A. Engage in substantial mathematical problem solving. B. Apply knowledge and understanding of mathematical concepts through real world information. C. Acquire the skills necessary to communicate mathematical ideas and procedures using appropriate mathematical vocabulary and notation. III. Information Management: Students will use effective strategies to collect, verify, document and manage information from a variety of sources. A. Obtain information from the Web using traditional locator tools and assess the information. B. Use appropriate technology to address a variety of mathematical tasks and problems. At the end of the course the student will be able to set up, solve, and interpret intermediate level algebraic problems. Course Learning Outcomes: At the end of the course the student will be able to solve problems related to: factoring; algebraic fractions; radicals and rational exponents; complex numbers; quadric equations; rational equations; linear equations and inequalities in tow variables of their graphs, systems of linear equations and inequalities; introduction to functions; applications of the above topics. Methods of Evaluation: Evaluation of student progress towards achieving the stated learning outcomes and performance objectives is the responsibility of the instructor, within the polices of the college and the department. Detailed explanation is included in the expanded syllabus developed by
9780792357 of Braids (Mathematics and Its Applications (closed)) This book provides a comprehensive exposition of the theory of braids, beginning with the basic mathematical definitions and structures. Among the many topics explained in detail are: the braid group for various surfaces; the solution of the word problem for the braid group; braids in the context of knots and links (Alexander's theorem); Markov's theorem and its use in obtaining braid invariants; the connection between the Platonic solids (regular polyhedra) and braids; the use of braids in the solution of algebraic equations. Dirac's problem and special types of braids termed Mexican plaits are also discussed. Audience: Since the book relies on concepts and techniques from algebra and topology, the authors also provide a couple of appendices that cover the necessary material from these two branches of mathematics. Hence, the book is accessible not only to mathematicians but also to anybody who might have an interest in the theory of braids. In particular, as more and more applications of braid theory are found outside the realm of mathematics, this book is ideal for any physicist, chemist or biologist who would like to understand the mathematics of braids. With its use of numerous figures to explain clearly the mathematics, and exercises to solidify the understanding, this book may also be used as a textbook for a course on knots and braids, or as a supplementary textbook for a course on topology or algebra
I am trying to learn some calculus 3 and I understand HOW to do the problems but I just don't understand WHY I'm doing what I'm doing. So does anyone have any good recommendations on books that are really down to earth, and explain the concepts in terms that humans can understand. Here are the topics that I want to understand: 1 Answer 1 It is generally accepted that science provides a description of 'HOW' nature works, not "WHY" it works this way. Mathematics often provides 'higher' concepts which allow science to make these descriptions accurately. By 'higher', we mean 'generalized through the use of abstract reasoning'. To understand questions of 'WHY' usually involves the development and application of even higher concepts such as motivation, purpose and intention which are beyond the scope even of mathematics. If you want to understand "WHY" certain topics are taught in the mathematics curriculum, it might be useful to try and get a grasp on where some of these mathematical topics are used in practice. To this end, a book on Engineering Physics might be helpful, such as Serway's 'Physics for Scientists and Engineers'. The topics in part A are really fundamental to physics and engineering mechanics (both statics and dynamics). For example, surface area is important in calculating tension in mechanical components, as well as stresses and strain of engineering materials (eg: how many bolts of what size do you need to hold up a bridge?). Topics in part B are also fundamental to physics, especially hydrodynamics and electrodynamics. Although these are covered at a basic level in the previous text, however, a more specialized text on either topic would also be useful (eg: Hydrodynamics by Horace Lamb or Engineering Electrodynamics by William Hayt). Feynman's Lectures on Physics (parts 1 & 2) may also offer you some insights into the subtle connection between physics and mathematics by giving some practical applications of these mathematical topics. A fascinating discussion on the relationship between physics and mathematics can be found in Feynman's 'Character of Physical Law', which is also available as a series of six videos online at Microsoft Research under 'Project Tuva' (see Specifically, refer to Lecture 2: 'The Relation of Mathematics and Physics'.
Introduction You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry 10: Critical reflections on Hofstede 7: Hofstede's way of thinking about 4: What do you see? 3: Your own 2: Differences between national culture and organisational 1: Defining Overview Angles on a line and conditions), this content is made available under a No related items provided in this feed Introduction Partnership Sole tradersKey themes and
Preface This book covers calculus in two and three variables. It is suitable for a one-semester course, normally known as "Vector Calculus", "Multivariable Calculus", or simply "Calculus III". The prerequisites are the standard courses in single-variable calculus (a.k.a. Calculus I and II). I have tried to be somewhat rigorous about proving results. But while it is important for students to see full-blown proofs - since that is how mathematics works - too much rigor and emphasis on proofs can impede the flow of learning for the vast majority of the audience at this level. If I were to rate the level of rigor in the book on a scale of 1 to 10, with 1 being completely informal and 10 being completely rigorous, I would rate it as a 5. There are 420 exercises throughout the text, which in my experience are more than enough for a semester course in this subject. There are exercises at the end of each sec- tion, divided into three categories: A, B and C. The A exercises are mostly of a routine computational nature, the B exercises are slightly more involved, and the C exercises usu- ally require some effort or insight to solve. A crude way of describing A, B and C would be "Easy", "Moderate" and "Challenging", respectively. However, many of the B exercises are easy and not all the C exercises are difficult. There are a few exercises that require the student to write his or her own computer pro- gram to solve some numerical approximation problems (e.g. the Monte Carlo method for approximating multiple integrals, in Section 3.4). The code samples in the text are in the Java programming language, hopefully with enough comments so that the reader can figure out what is being done even without knowing Java. Those exercises do not mandate the use of Java, so students are free to implement the solutions using the language of their choice. While it would have been simple to use a scripting language like Python, and perhaps even easier with a functional programming language (such as Haskell or Scheme), Java was cho- sen due to its ubiquity, relatively clear syntax, and easy availability for multiple platforms. Answers and hints to most odd-numbered and some even-numbered exercises are pro- vided in Appendix A. Appendix B contains a proof of the right-hand rule for the cross prod- uct, which seems to have virtually disappeared from calculus texts over the last few decades. Appendix C contains a brief tutorial on Gnuplot for graphing functions of two variables. This book is released under the GNU Free Documentation License (GFDL), which allows others to not only copy and distribute the book but also to modify it. For more details, see the included copy of the GFDL. So that there is no ambiguity on this matter, anyone can make as many copies of this book as desired and distribute it as desired, without needing my permission. The PDF version will always be freely available to the public at no cost (go to Feel free to contact me at mcorral@schoolcraft.edu for iii iv Preface any questions on this or any other matter involving the book (e.g. comments, suggestions, corrections, etc). I welcome your input. Finally, I would like to thank my students in Math 240 for being the guinea pigs for the initial draft of this book, and for finding the numerous errors and typos it contained. January 2008 MICHAEL CORRAL 1 Vectors in Euclidean Space 1.1 Introduction In single-variable calculus, the functions that one encounters are functions of a variable (usually x or t) that varies over some subset of the real number line (which we denote by R). For such a function, say, y = f (x), the graph of the function f consists of the points (x, y) = (x, f (x)). These points lie in the Euclidean plane, which, in the Cartesian or rectangular coordinate system, consists of all ordered pairs of real numbers (a,b). We use the word "Euclidean" to denote a system in which all the usual rules of Euclidean geometry hold. We denote the Euclidean plane by R2 ; the "2" represents the number of dimensions of the plane. The Euclidean plane has two perpendicular coordinate axes: the x-axis and the y-axis. In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x, y or x, y, z, respectively). The graph of a function of two variables, say, z = f (x, y), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers (a,b, c). Since Euclidean space is 3-dimensional, we denote it by R3 . The graph of f consists of the points (x, y, z) = (x, y, f (x, y)). The 3-dimensional coordinate system of Euclidean space can be represented on a flat surface, such as this page or a black- board, only by giving the illusion of three dimensions, in the manner shown in Figure 1.1.1. Euclidean space has three mutually perpendicular coordinate axes (x, y and z), and three mutually perpendicular coordinate planes: the xy-plane, yz-plane and xz-plane (see Figure 1.1.2). x y z 0 P(a,b, c) a b c Figure 1.1.1 x y z 0 yz-plane xy-plane xz-plane Figure 1.1.2 1 2 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE The coordinate system shown in Figure 1.1.1 is known as a right-handed coordinate system, because it is possible, using the right hand, to point the index finger in the positive direction of the x-axis, the middle finger in the positive direction of the y-axis, and the thumb in the positive direction of the z-axis, as in Figure 1.1.3. x z y 0 Figure 1.1.3 Right-handed coordinate system An equivalent way of defining a right-handed system is if you can point your thumb up- wards in the positive z-axis direction while using the remaining four fingers to rotate the x-axis towards the y-axis. Doing the same thing with the left hand is what defines a left- handed coordinate system. Notice that switching the x- and y-axes in a right-handed system results in a left-handed system, and that rotating either type of system does not change its "handedness". Throughout the book we will use a right-handed system. For functions of three variables, the graphs exist in 4-dimensional space (i.e. R4 ), which we can not see in our 3-dimensional space, let alone simulate in 2-dimensional space. So we can only think of 4-dimensional space abstractly. For an entertaining discussion of this subject, see the book by ABBOTT.1 So far, we have discussed the position of an object in 2-dimensional or 3-dimensional space. But what about something such as the velocity of the object, or its acceleration? Or the gravitational force acting on the object? These phenomena all seem to involve motion and direction in some way. This is where the idea of a vector comes in. 1One thing you will learn is why a 4-dimensional creature would be able to reach inside an egg and remove the yolk without cracking the shell! 1.1 Introduction 3 You have already dealt with velocity and acceleration in single-variable calculus. For example, for motion along a straight line, if y = f (t) gives the displacement of an object after time t, then dy/dt = f ′ (t) is the velocity of the object at time t. The derivative f ′ (t) is just a number, which is positive if the object is moving in an agreed-upon "positive" direction, and negative if it moves in the opposite of that direction. So you can think of that number, which was called the velocity of the object, as having two components: a magnitude, indicated by a nonnegative number, preceded by a direction, indicated by a plus or minus symbol (representing motion in the positive direction or the negative direction, respectively), i.e. f ′ (t) = ±a for some number a ≥ 0. Then a is the magnitude of the velocity (normally called the speed of the object), and the ± represents the direction of the velocity (though the + is usually omitted for the positive direction). For motion along a straight line, i.e. in a 1-dimensional space, the velocities are also con- tained in that 1-dimensional space, since they are just numbers. For general motion along a curve in 2- or 3-dimensional space, however, velocity will need to be represented by a multi- dimensional object which should have both a magnitude and a direction. A geometric object which has those features is an arrow, which in elementary geometry is called a "directed line segment". This is the motivation for how we will define a vector. Definition 1.1. A (nonzero) vector is a directed line segment drawn from a point P (called its initial point) to a point Q (called its terminal point), with P and Q being distinct points. The vector is denoted by −−→ PQ. Its magnitude is the length of the line segment, denoted by −−→ PQ , and its direction is the same as that of the directed line segment. The zero vector is just a point, and it is denoted by 0. To indicate the direction of a vector, we draw an arrow from its initial point to its terminal point. We will often denote a vector by a single bold-faced letter (e.g. v) and use the terms "magnitude" and "length" interchangeably. Note that our definition could apply to systems with any number of dimensions (see Figure 1.1.4 (a)-(c)). 0 xP QRS −−→ PQ −−→ RS (a) One dimension x y 0 P Q R S −−→ PQ −−→ RS v (b) Two dimensions x y z 0 P Q R S −−→PQ −−→ R S v (c) Three dimensions Figure 1.1.4 Vectors in different dimensions 4 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE A few things need to be noted about the zero vector. Our motivation for what a vector is included the notions of magnitude and direction. What is the magnitude of the zero vector? We define it to be zero, i.e. 0 = 0. This agrees with the definition of the zero vector as just a point, which has zero length. What about the direction of the zero vector? A single point really has no well-defined direction. Notice that we were careful to only define the direction of a nonzero vector, which is well-defined since the initial and terminal points are distinct. Not everyone agrees on the direction of the zero vector. Some contend that the zero vector has arbitrary direction (i.e. can take any direction), some say that it has indeterminate direction (i.e. the direction can not be determined), while others say that it has no direction. Our definition of the zero vector, however, does not require it to have a direction, and we will leave it at that.2 Now that we know what a vector is, we need a way of determining when two vectors are equal. This leads us to the following definition. Definition 1.2. Two nonzero vectors are equal if they have the same magnitude and the same direction. Any vector with zero magnitude is equal to the zero vector. By this definition, vectors with the same magnitude and direction but with different initial points would be equal. For example, in Figure 1.1.5 the vectors u, v and w all have the same magnitude 5 (by the Pythagorean Theorem). And we see that u and w are parallel, since they lie on lines having the same slope 1 2 , and they point in the same direction. So u = w, even though they have different initial points. We also see that v is parallel to u but points in the opposite direction. So u = v. 1 2 3 4 1 2 3 4 x y 0 u v w Figure 1.1.5 So we can see that there are an infinite number of vectors for a given magnitude and direction, those vectors all being equal and differing only by their initial and terminal points. Is there a single vector which we can choose to represent all those equal vectors? The answer is yes, and is suggested by the vector w in Figure 1.1.5. 2In the subject of linear algebra there is a more abstract way of defining a vector where the concept of "direction" is not really used. See ANTON and RORRES. 1.1 Introduction 5 Unless otherwise indicated, when speaking of "the vector" with a given magnitude and direction, we will mean the one whose initial point is at the origin of the coordinate system. Thinking of vectors as starting from the origin provides a way of dealing with vectors in a standard way, since every coordinate system has an origin. But there will be times when it is convenient to consider a different initial point for a vector (for example, when adding vectors, which we will do in the next section). Another advantage of using the origin as the initial point is that it provides an easy cor- respondence between a vector and its terminal point. Example 1.1. Let v be the vector in R3 whose initial point is at the origin and whose ter- minal point is (3,4,5). Though the point (3,4,5) and the vector v are different objects, it is convenient to write v = (3,4,5). When doing this, it is understood that the initial point of v is at the origin (0,0,0) and the terminal point is (3,4,5). x y z 0 P(3,4,5) (a) The point (3,4,5) x y z 0 v = (3,4,5) (b) The vector (3,4,5) Figure 1.1.6 Correspondence between points and vectors Unless otherwise stated, when we refer to vectors as v = (a,b) in R2 or v = (a,b, c) in R3 , we mean vectors in Cartesian coordinates starting at the origin. Also, we will write the zero vector 0 in R2 and R3 as (0,0) and (0,0,0), respectively. The point-vector correspondence provides an easy way to check if two vectors are equal, without having to determine their magnitude and direction. Similar to seeing if two points are the same, you are now seeing if the terminal points of vectors starting at the origin are the same. For each vector, find the (unique!) vector it equals whose initial point is the origin. Then compare the coordinates of the terminal points of these "new" vectors: if those coordinates are the same, then the original vectors are equal. To get the "new" vectors starting at the origin, you translate each vector to start at the origin by subtracting the coordinates of the original initial point from the original terminal point. The resulting point will be the terminal point of the "new" vector whose initial point is the origin. Do this for each original vector then compare. 1.2 Vector Algebra 9 1.2 Vector Algebra Now that we know what vectors are, we can start to perform some of the usual algebraic operations on them (e.g. addition, subtraction). Before doing that, we will introduce the notion of a scalar. Definition 1.3. A scalar is a quantity that can be represented by a single number. For our purposes, scalars will always be real numbers.3 Examples of scalar quantities are mass, electric charge, and speed (not velocity).4 We can now define scalar multiplication of a vector. Definition 1.4. For a scalar k and a nonzero vector v, the scalar multiple of v by k, denoted by kv, is the vector whose magnitude is \|k\| v , points in the same direction as v if k > 0, points in the opposite direction as v if k < 0, and is the zero vector 0 if k = 0. For the zero vector 0, we define k0 = 0 for any scalar k. Two vectors v and w are parallel (denoted by v ∥ w) if one is a scalar multiple of the other. You can think of scalar multiplication of a vector as stretching or shrinking the vector, and as flipping the vector in the opposite direction if the scalar is a negative number (see Figure 1.2.1). v 2v 3v 0.5v −v −2v Figure 1.2.1 Recall that translating a nonzero vector means that the initial point of the vector is changed but the magnitude and direction are preserved. We are now ready to define the sum of two vectors. Definition 1.5. The sum of vectors v and w, denoted by v+w, is obtained by translating w so that its initial point is at the terminal point of v; the initial point of v+w is the initial point of v, and its terminal point is the new terminal point of w. 3The term scalar was invented by 19th century Irish mathematician, physicist and astronomer William Rowan Hamilton, to convey the sense of something that could be represented by a point on a scale or graduated ruler. The word vector comes from Latin, where it means "carrier". 4An alternate definition of scalars and vectors, used in physics, is that under certain types of coordinate trans- formations (e.g. rotations), a quantity that is not affected is a scalar, while a quantity that is affected (in a certain way) is a vector. See MARION for details. 10 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Intuitively, adding w to v means tacking on w to the end of v (see Figure 1.2.2). v w (a) Vectors v and w v w (b) Translate w to the end of v v w v+w (c) The sum v+w Figure 1.2.2 Adding vectors v and w Notice that our definition is valid for the zero vector (which is just a point, and hence can be translated), and so we see that v+0 = v = 0+v for any vector v. In particular, 0+0 = 0. Also, it is easy to see that v + (−v) = 0, as we would expect. In general, since the scalar multiple −v = −1v is a well-defined vector, we can define vector subtraction as follows: v−w = v+(−w). See Figure 1.2.3. v w (a) Vectors v and w v −w (b) Translate −w to the end of v v −w v−w (c) The difference v−w Figure 1.2.3 Subtracting vectors v and w Figure 1.2.4 shows the use of "geometric proofs" of various laws of vector algebra, that is, it uses laws from elementary geometry to prove statements about vectors. For example, (a) shows that v+w = w+v for any vectors v, w. And (c) shows how you can think of v−w as the vector that is tacked on to the end of w to add up to v. v v w ww+v v+w (a) Add vectors −w w v−w v−wv (b) Subtract vectors v w v+w v−w (c) Combined add/subtract Figure 1.2.4 "Geometric" vector algebra Notice that we have temporarily abandoned the practice of starting vectors at the origin. In fact, we have not even mentioned coordinates in this section so far. Since we will deal mostly with Cartesian coordinates in this book, the following two theorems are useful for performing vector algebra on vectors in R2 and R3 starting at the origin. 1.2 Vector Algebra 11 Theorem 1.3. Let v = (v1,v2), w = (w1,w2) be vectors in R2 , and let k be a scalar. Then (a) kv = (kv1,kv2) (b) v + w = (v1 + w1,v2 + w2) Proof: (a) Without loss of generality, we assume that v1,v2 > 0 (the other possibilities are handled in a similar manner). If k = 0 then kv = 0v = 0 = (0,0) = (0v1,0v2) = (kv1,kv2), which is what we needed to show. If k = 0, then (kv1,kv2) lies on a line with slope kv2 kv1 = v2 v1 , which is the same as the slope of the line on which v (and hence kv) lies, and (kv1,kv2) points in the same direction on that line as kv. Also, by formula (1.3) the magnitude of (kv1,kv2) is (kv1)2 +(kv2)2 = k2v2 1 + k2v2 2 = k2(v2 1 + v2 2 ) = \|k\| v2 1 + v2 2 = \|k\| v . So kv and (kv1,kv2) have the same magnitude and direction. This proves (a). x y 0 w2 v2 w1 v1 v1 + w1 v2 + w2 w2 w1 v v w w v+w Figure 1.2.5 (b) Without loss of generality, we assume that v1,v2,w1,w2 > 0 (the other possibilities are han- dled in a similar manner). From Figure 1.2.5, we see that when translating w to start at the end of v, the new terminal point of w is (v1 +w1,v2 +w2), so by the definition of v+w this must be the ter- minal point of v+w. This proves (b). QED Theorem 1.4. Let v = (v1,v2,v3), w = (w1,w2,w3) be vectors in R3 , let k be a scalar. Then (a) kv = (kv1,kv2,kv3) (b) v + w = (v1 + w1,v2 + w2,v3 + w3) The following theorem summarizes the basic laws of vector algebra. Theorem 1.5. For any vectors u, v, w, and scalars k,l, we have (a) v+w = w+v Commutative Law (b) u+(v+w) = (u+v)+w Associative Law (c) v+0 = v = 0+v Additive Identity (d) v+(−v) = 0 Additive Inverse (e) k(lv) = (kl)v Associative Law (f) k(v+w) = kv+ kw Distributive Law (g) (k + l)v = kv+ lv Distributive Law Proof: (a) We already presented a geometric proof of this in Figure 1.2.4(a). (b) To illustrate the difference between analytic proofs and geometric proofs in vector alge- bra, we will present both types here. For the analytic proof, we will use vectors in R3 (the proof for R2 is similar). 1.3 Dot Product 15 1.3 Dot Product You may have noticed that while we did define multiplication of a vector by a scalar in the previous section on vector algebra, we did not define multiplication of a vector by a vector. We will now see one type of multiplication of vectors, called the dot product. Definition 1.6. Let v = (v1,v2,v3) and w = (w1,w2,w3) be vectors in R3 . The dot product of v and w, denoted by v···w, is given by: v···w = v1w1 + v2w2 + v3w3 (1.6) Similarly, for vectors v = (v1,v2) and w = (w1,w2) in R2 , the dot product is: v···w = v1w1 + v2w2 (1.7) Notice that the dot product of two vectors is a scalar, not a vector. So the associative law that holds for multiplication of numbers and for addition of vectors (see Theorem 1.5(b),(e)), does not hold for the dot product of vectors. Why? Because for vectors u, v, w, the dot product u···v is a scalar, and so (u···v)···w is not defined since the left side of that dot product (the part in parentheses) is a scalar and not a vector. For vectors v = v1 i+v2 j+v3 k and w = w1 i+w2 j+w3 k in component form, the dot product is still v···w = v1w1 + v2w2 + v3w3. Also notice that we defined the dot product in an analytic way, i.e. by referencing vector coordinates. There is a geometric way of defining the dot product, which we will now develop as a consequence of the analytic definition. Definition 1.7. The angle between two nonzero vectors with the same initial point is the smallest angle between them. We do not define the angle between the zero vector and any other vector. Any two nonzero vectors with the same initial point have two angles between them: θ and 360◦ −θ. We will always choose the smallest nonnegative angle θ between them, so that 0◦ ≤ θ ≤ 180◦ . See Figure 1.3.1. θ 360◦ −θ (a) 0◦ < θ < 180◦ θ 360◦ −θ (b) θ = 180◦ θ 360◦ −θ (c) θ = 0◦ Figure 1.3.1 Angle between vectors We can now take a more geometric view of the dot product by establishing a relationship between the dot product of two vectors and the angle between them. 18 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Using Theorem 1.9, we see that if u···v = 0 and u···w = 0, then u···(kv+lw) = k(u···v)+l(u···w) = k(0)+ l(0) = 0 for all scalars k,l. Thus, we have the following fact: If u ⊥ v and u ⊥ w, then u ⊥ (kv+ lw) for all scalars k,l. For vectors v and w, the collection of all scalar combinations kv + lw is called the span of v and w. If nonzero vectors v and w are parallel, then their span is a line; if they are not parallel, then their span is a plane. So what we showed above is that a vector which is perpendicular to two other vectors is also perpendicular to their span. The dot product can be used to derive properties of the magnitudes of vectors, the most important of which is the Triangle Inequality, as given in the following theorem: Theorem 1.10. For any vectors v, w, we have (a) v 2 = v···v (b) v+w ≤ v + w Triangle Inequality (c) v−w ≥ v − w Proof: (a) Left as an exercise for the reader. (b) By part (a) and Theorem 1.9, we have v+w 2 = (v+w)···(v+w) = v···v+v···w+w···v+w···w = v 2 +2(v···w)+ w 2 , so since a ≤ \|a\| for any real number a, we have ≤ v 2 +2\|v···w\|+ w 2 , so by Theorem 1.9(f) we have ≤ v 2 +2 v w + w 2 = ( v + w )2 and so v+w ≤ v + w after taking square roots of both sides, which proves (b). (c) Since v = w+(v−w), then v = w+(v−w) ≤ w + v−w by the Triangle Inequality, so subtracting w from both sides gives v − w ≤ v−w . QED v w v+w Figure 1.3.4 The Triangle Inequality gets its name from the fact that in any triangle, no one side is longer than the sum of the lengths of the other two sides (see Figure 1.3.4). Another way of saying this is with the familiar statement "the shortest distance between two points is a straight line." Exercises A 1. Let v = (5,1,−2) and w = (4,−4,3). Calculate v···w. 2. Let v = −3i−2j−k and w = 6i+4j+2k. Calculate v···w. For Exercises 3-8, find the angle θ between the vectors v and w. 20 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE 1.4 Cross Product In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vectors that does result in another vector. This product, called the cross product, is only defined for vectors in R3 . The definition may appear strange and lacking motivation, but we will see the geometric basis for it shortly. Definition 1.8. Let v = (v1,v2,v3) and w = (w1,w2,w3) be vectors in R3 . The cross product of v and w, denoted by v×××w, is the vector in R3 given by: v×××w = (v2w3 − v3w2,v3w1 − v1w3,v1w2 − v2w1) (1.10) 1 1 1 x y z 0i j k = i×××j Figure 1.4.1 Example 1.7. Find i×××j. Solution: Since i = (1,0,0) and j = (0,1,0), then i×××j = ((0)(0)−(0)(1),(0)(0)−(1)(0),(1)(1)−(0)(0)) = (0,0,1) = k Similarly it can be shown that j×××k = i and k×××i = j. In the above example, the cross product of the given vectors was perpendicular to both those vectors. It turns out that this will always be the case. Theorem 1.11. If the cross product v×××w of two nonzero vectors v and w is also a nonzero vector, then it is perpendicular to both v and w. Proof: We will show that (v×××w)···v = 0: (v×××w)···v = (v2w3 − v3w2,v3w1 − v1w3,v1w2 − v2w1)···(v1,v2,v3) = v2w3v1 − v3w2v1 + v3w1v2 − v1w3v2 + v1w2v3 − v2w1v3 = v1v2w3 − v1v2w3 + w1v2v3 − w1v2v3 + v1w2v3 − v1w2v3 = 0 , after rearranging the terms. ∴ v×××w ⊥ v by Corollary 1.7. The proof that v×××w ⊥ w is similar. QED As a consequence of the above theorem and Theorem 1.9, we have the following: Corollary 1.12. If the cross product v×××w of two nonzero vectors v and w is also a nonzero vector, then it is perpendicular to the span of v and w. 22 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE If θ is the angle between nonzero vectors v and w in R3 , then v×××w = v w sinθ (1.11) It may seem strange to bother with the above formula, when the magnitude of the cross product can be calculated directly, like for any other vector. The formula is more useful for its applications in geometry, as in the following example. Example 1.8. Let △PQR and PQRS be a triangle and parallelogram, respectively, as shown in Figure 1.4.3. b h h θ θ P P Q QR R S S v w Figure 1.4.3 Think of the triangle as existing in R3 , and identify the sides QR and QP with vectors v and w, respectively, in R3 . Let θ be the angle between v and w. The area APQR of △PQR is 1 2 bh, where b is the base of the triangle and h is the height. So we see that b = v and h = w sinθ APQR = 1 2 v w sinθ = 1 2 v×××w So since the area APQRS of the parallelogram PQRS is twice the area of the triangle △PQR, then APQRS = v w sinθ By the discussion in Example 1.8, we have proved the following theorem: Theorem 1.13. Area of triangles and parallelograms (a) The area A of a triangle with adjacent sides v, w (as vectors in R3 ) is: A = 1 2 v×××w (b) The area A of a parallelogram with adjacent sides v, w (as vectors in R3 ) is: A = v×××w 1.4 Cross Product 23 It may seem at first glance that since the formulas derived in Example 1.8 were for the adjacent sides QP and QR only, then the more general statements in Theorem 1.13 that the formulas hold for any adjacent sides are not justified. We would get a different formula for the area if we had picked PQ and PR as the adjacent sides, but it can be shown (see Exercise 26) that the different formulas would yield the same value, so the choice of adjacent sides indeed does not matter, and Theorem 1.13 is valid. Theorem 1.13 makes it simpler to calculate the area of a triangle in 3-dimensional space than by using traditional geometric methods. Example 1.9. Calculate the area of the triangle △PQR, where P = (2,4,−7), Q = (3,7,18), and R = (−5,12,8). y z x 0 v w R(−5,12,8) Q(3,7,18) P(2,4,−7) Figure 1.4.4 Solution: Let v = −−→ PQ and w = −−→ PR, as in Figure 1.4.4. Then v = (3,7,18)−(2,4,−7) = (1,3,25) and w = (−5,12,8)−(2,4,−7) = (−7,8,15), so the area A of the triangle △PQR is A = 1 2 v×××w = 1 2 (1,3,25)×××(−7,8,15) = 1 2 ((3)(15)−(25)(8),(25)(−7)−(1)(15),(1)(8)−(3)(−7)) = 1 2 (−155,−190,29) = 1 2 (−155)2 +(−190)2 +292 = 1 2 60966 A ≈ 123.46 Example 1.10. Calculate the area of the parallelogram PQRS, where P = (1,1), Q = (2,3), R = (5,4), and S = (4,2). x y 0 1 2 3 4 1 2 3 4 5 P Q R S v w Figure 1.4.5 Solution: Let v = −−→ SP and w = −−→ SR, as in Figure 1.4.5. Then v = (1,1) − (4,2) = (−3,−1) and w = (5,4) − (4,2) = (1,2). But these are vectors in R2 , and the cross product is only defined for vectors in R3 . However, R2 can be thought of as the subset of R3 such that the z-coordinate is always 0. So we can write v = (−3,−1,0) and w = (1,2,0). Then the area A of PQRS is A = v×××w = (−3,−1,0)×××(1,2,0) = ((−1)(0)−(0)(2),(0)(1)−(−3)(0),(−3)(2)−(−1)(1)) = (0,0,−5) A = 5 24 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE The following theorem summarizes the basic properties of the cross product. Theorem 1.14. For any vectors u, v, w in R3 , and scalar k, we have (a) v×××w = −w×××v Anticommutative Law (b) u×××(v+w) = u×××v+u×××w Distributive Law (c) (u+v)×××w = u×××w+v×××w Distributive Law (d) (kv)×××w = v×××(kw) = k(v×××w) Associative Law (e) v×××0 = 0 = 0×××v (f) v×××v = 0 (g) v×××w = 0 if and only if v ∥ w Proof: The proofs of properties (b)-(f) are straightforward. We will prove parts (a) and (g) and leave the rest to the reader as exercises. x y z 0 v w v×××w w×××v Figure 1.4.6 (a) By the definition of the cross product and scalar multipli- cation, we have: v×××w = (v2w3 − v3w2,v3w1 − v1w3,v1w2 − v2w1) = −(v3w2 − v2w3,v1w3 − v3w1,v2w1 − v1w2) = −(w2v3 − w3v2,w3v1 − w1v3,w1v2 − w2v1) = −w×××v Note that this says that v×××w and w×××v have the same mag- nitude but opposite direction (see Figure 1.4.6). (g) If either v or w is 0 then v×××w = 0 by part (e), and either v = 0 = 0w or w = 0 = 0v, so v and w are scalar multiples, i.e. they are parallel. If both v and w are nonzero, and θ is the angle between them, then by formula (1.11), v×××w = 0 if and only if v w sinθ = 0, which is true if and only if sinθ = 0 (since v > 0 and w > 0). So since 0◦ ≤ θ ≤ 180◦ , then sinθ = 0 if and only if θ = 0◦ or 180◦ . But the angle between v and w is 0◦ or 180◦ if and only if v ∥ w. QED Example 1.11. Adding to Example 1.7, we have i×××j = k j×××k = i k×××i = j j×××i = −k k×××j = −i i×××k = −j i×××i = j×××j = k×××k = 0 Recall from geometry that a parallelepiped is a 3-dimensional solid with 6 faces, all of which are parallelograms.6 6An equivalent definition of a parallelepiped is: the collection of all scalar combinations k1v1 + k2v2 + k3v3 of some vectors v1, v2, v3 in R3, where 0 ≤ k1,k2,k3 ≤ 1. 1.4 Cross Product 25 Example 1.12. Volume of a parallelepiped: Let the vectors u, v, w in R3 represent adjacent sides of a parallelepiped P, with u, v, w forming a right-handed system, as in Figure 1.4.7. Show that the volume of P is the scalar triple product u···(v×××w). h θ u w v v×××w Figure 1.4.7 Parallelepiped P Solution: Recall that the volume vol(P) of a par- allelepiped P is the area A of the base parallel- ogram times the height h. By Theorem 1.13(b), the area A of the base parallelogram is v×××w . And we can see that since v×××w is perpendicular to the base parallelogram determined by v and w, then the height h is u cosθ, where θ is the angle between u and v×××w. By Theorem 1.6 we know that cosθ = u···(v×××w) u v×××w . Hence, vol(P) = A h = v×××w u u···(v×××w) u v×××w = u···(v×××w) In Example 1.12 the height h of the parallelepiped is u cosθ, and not − u cosθ, be- cause the vector u is on the same side of the base parallelogram's plane as the vector v×××w (so that cosθ > 0). Since the volume is the same no matter which base and height we use, then repeating the same steps using the base determined by u and v (since w is on the same side of that base's plane as u ××× v), the volume is w ··· (u ××× v). Repeating this with the base determined by w and u, we have the following result: For any vectors u, v, w in R3 , u···(v×××w) = w···(u×××v) = v···(w×××u) (1.12) (Note that the equalities hold trivially if any of the vectors are 0.) Since v×××w = −w×××v for any vectors v, w in R3 , then picking the wrong order for the three adjacent sides in the scalar triple product in formula (1.12) will give you the negative of the volume of the parallelepiped. So taking the absolute value of the scalar triple product for any order of the three adjacent sides will always give the volume: Theorem 1.15. If vectors u, v, w in R3 represent any three adjacent sides of a paral- lelepiped, then the volume of the parallelepiped is \|u···(v×××w)\|. Another type of triple product is the vector triple product u ××× (v ××× w). The proof of the following theorem is left as an exercise for the reader: 26 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Theorem 1.16. For any vectors u, v, w in R3 , u×××(v×××w) = (u···w)v−(u···v)w (1.13) An examination of the formula in Theorem 1.16 gives some idea of the geometry of the vector triple product. By the right side of formula (1.13), we see that u×××(v×××w) is a scalar combination of v and w, and hence lies in the plane containing v and w (i.e. u×××(v×××w), v and w are coplanar). This makes sense since, by Theorem 1.11, u×××(v×××w) is perpendicular to both u and v×××w. In particular, being perpendicular to v×××w means that u×××(v×××w) lies in the plane containing v and w, since that plane is itself perpendicular to v×××w. But then how is u×××(v×××w) also perpendicular to u, which could be any vector? The following example may help to see how this works. Example 1.13. Find u×××(v×××w) for u = (1,2,4), v = (2,2,0), w = (1,3,0). Solution: Since u···v = 6 and u···w = 7, then u×××(v×××w) = (u···w)v−(u···v)w = 7(2,2,0)−6(1,3,0) = (14,14,0)−(6,18,0) = (8,−4,0) Note that v and w lie in the xy-plane, and that u ××× (v ××× w) also lies in that plane. Also, u×××(v×××w) is perpendicular to both u and v×××w = (0,0,4) (see Figure 1.4.8). y z x 0 u v w v ××× w u ××× (v ××× w) Figure 1.4.8 For vectors v = v1 i+v2 j+v3 k and w = w1 i+w2 j+w3 k in component form, the cross product is written as: v×××w = (v2w3 −v3w2)i+(v3w1 −v1w3)j+(v1w2 −v2w1)k. It is often easier to use the component form for the cross product, because it can be represented as a determinant. We will not go too deeply into the theory of determinants7 ; we will just cover what is essential for our purposes. 7See ANTON and RORRES for a fuller development. 1.4 Cross Product 27 A 2×××2 matrix is an array of two rows and two columns of scalars, written as a b c d or a b c d where a,b, c,d are scalars. The determinant of such a matrix, written as a b c d or det a b c d , is the scalar defined by the following formula: a b c d = ad − bc It may help to remember this formula as being the product of the scalars on the downward diagonal minus the product of the scalars on the upward diagonal. Example 1.14. 1 2 3 4 = (1)(4)−(2)(3) = 4−6 = −2 A 3×××3 matrix is an array of three rows and three columns of scalars, written as   a1 a2 a3 b1 b2 b3 c1 c2 c3   or   a1 a2 a3 b1 b2 b3 c1 c2 c3  , and its determinant is given by the formula: a1 a2 a3 b1 b2 b3 c1 c2 c3 = a1 b2 b3 c2 c3 − a2 b1 b3 c1 c3 + a3 b1 b2 c1 c2 (1.14) One way to remember the above formula is the following: multiply each scalar in the first row by the determinant of the 2×2 matrix that remains after removing the row and column that contain that scalar, then sum those products up, putting alternating plus and minus signs in front of each (starting with a plus). Example 1.15. 1 0 2 4 −1 3 1 0 2 = 1 −1 3 0 2 − 0 4 3 1 2 + 2 4 −1 1 0 = 1(−2−0)−0(8−3)+2(0+1) = 0 1.5 Lines and Planes 31 1.5 Lines and Planes Now that we know how to perform some operations on vectors, we can start to deal with some familiar geometric objects, like lines and planes, in the language of vectors. The reason for doing this is simple: using vectors makes it easier to study objects in 3-dimensional Euclidean space. We will first consider lines. Line through a point, parallel to a vector Let P = (x0, y0, z0) be a point in R3 , let v = (a,b, c) be a nonzero vector, and let L be the line through P which is parallel to v (see Figure 1.5.1). x y z 0 L t > 0 t < 0 P(x0, y0, z0) r v tv r+ tv r+ tv Figure 1.5.1 Let r = (x0, y0, z0) be the vector pointing from the origin to P. Since multiplying the vector v by a scalar t lengthens or shrinks v while preserving its direction if t > 0, and reversing its direction if t < 0, then we see from Figure 1.5.1 that every point on the line L can be obtained by adding the vector tv to the vector r for some scalar t. That is, as t varies over all real numbers, the vector r+ tv will point to every point on L. We can summarize the vector representation of L as follows: For a point P = (x0, y0, z0) and nonzero vector v in R3 , the line L through P parallel to v is given by r+ tv, for −∞ < t < ∞ (1.16) where r = (x0, y0, z0) is the vector pointing to P. Note that we used the correspondence between a vector and its terminal point. Since v = (a,b, c), then the terminal point of the vector r+ tv is (x0 +at, y0 +bt, z0 + ct). We then get the parametric representation of L with the parameter t: For a point P = (x0, y0, z0) and nonzero vector v = (a,b, c) in R3 , the line L through P parallel to v consists of all points (x, y, z) given by x = x0 + at, y = y0 + bt, z = z0 + ct, for −∞ < t < ∞ (1.17) Note that in both representations we get the point P on L by letting t = 0. 32 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE In formula (1.17), if a = 0, then we can solve for the parameter t: t = (x−x0)/a. We can also solve for t in terms of y and in terms of z if neither b nor c, respectively, is zero: t = (y− y0)/b and t = (z−z0)/c. These three values all equal the same value t, so we can write the following system of equalities, called the symmetric representation of L: For a point P = (x0, y0, z0) and vector v = (a,b, c) in R3 with a, b and c all nonzero, the line L through P parallel to v consists of all points (x, y, z) given by the equations x− x0 a = y− y0 b = z − z0 c (1.18) x y z 0 x = x0 x0 L Figure 1.5.2 What if, say, a = 0 in the above scenario? We can not divide by zero, but we do know that x = x0 +at, and so x = x0 +0t = x0. Then the symmetric representation of L would be: x = x0, y− y0 b = z − z0 c (1.19) Note that this says that the line L lies in the plane x = x0, which is parallel to the yz-plane (see Figure 1.5.2). Similar equations can be derived for the cases when b = 0 or c = 0. You may have noticed that the vector representation of L in formula (1.16) is more compact than the parametric and symmetric formulas. That is an advantage of using vector notation. Technically, though, the vector representation gives us the vectors whose terminal points make up the line L, not just L itself. So you have to remember to identify the vectors r+ tv with their terminal points. On the other hand, the parametric representation always gives just the points on L and nothing else. Example 1.19. Write the line L through the point P = (2,3,5) and parallel to the vector v = (4,−1,6), in the following forms: (a) vector, (b) parametric, (c) symmetric. Lastly: (d) find two points on L distinct from P. Solution: (a) Let r = (2,3,5). Then by formula (1.16), L is given by: r+ tv = (2,3,5)+ t(4,−1,6), for −∞ < t < ∞ (b) L consists of the points (x, y, z) such that x = 2+4t, y = 3− t, z = 5+6t, for −∞ < t < ∞ (c) L consists of the points (x, y, z) such that x−2 4 = y−3 −1 = z −5 6 (d) Letting t = 1 and t = 2 in part(b) yields the points (6,2,11) and (10,1,17) on L. 1.5 Lines and Planes 33 Line through two points x y z 0 L P1(x1, y1, z1) P2(x2, y2, z2) r1 r2 r2 −r1 r1 + t(r2 −r1) Figure 1.5.3 Let P1 = (x1, y1, z1) and P2 = (x2, y2, z2) be distinct points in R3 , and let L be the line through P1 and P2. Let r1 = (x1, y1, z1) and r2 = (x2, y2, z2) be the vectors pointing to P1 and P2, respectively. Then as we can see from Figure 1.5.3, r2 −r1 is the vector from P1 to P2. So if we multiply the vector r2 − r1 by a scalar t and add it to the vector r1, we will get the entire line L as t varies over all real numbers. The following is a summary of the vector, para- metric, and symmetric forms for the line L: Let P1 = (x1, y1, z1), P2 = (x2, y2, z2) be distinct points in R3 , and let r1 = (x1, y1, z1), r2 = (x2, y2, z2). Then the line L through P1 and P2 has the following representations: Vector: r1 + t(r2 −r1) , for −∞ < t < ∞ (1.20) Parametric: x = x1 +(x2 − x1)t, y = y1 +(y2 − y1)t, z = z1 +(z2 − z1)t, for −∞ < t < ∞ (1.21) Symmetric: x− x1 x2 − x1 = y− y1 y2 − y1 = z − z1 z2 − z1 (if x1 = x2, y1 = y2, and z1 = z2) (1.22) Example 1.20. Write the line L through the points P1 = (−3,1,−4) and P2 = (4,4,−6) in parametric form. Solution: By formula (1.21), L consists of the points (x, y, z) such that x = −3+7t, y = 1+3t, z = −4−2t, for −∞ < t < ∞ Distance between a point and a line θ L v w d Q P Figure 1.5.4 Let L be a line in R3 in vector form as r + tv (for −∞ < t < ∞), and let P be a point not on L. The distance d from P to L is the length of the line segment from P to L which is perpendicular to L (see Figure 1.5.4). Pick a point Q on L, and let w be the vector from Q to P. If θ is the angle between w and v, then d = w sinθ. So since v×××w = v w sinθ and v = 0, then: d = v×××w v (1.23) 34 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Example 1.21. Find the distance d from the point P = (1,1,1) to the line L in Example 1.20. Solution: From Example 1.20, we see that we can represent L in vector form as: r+ tv, for r = (−3,1,−4) and v = (7,3,−2). Since the point Q = (−3,1,−4) is on L, then for w = −−→ QP = (1,1,1)−(−3,1,−4) = (4,0,5), we have: v×××w = i j k 7 3 −2 4 0 5 = 3 −2 0 5 i − 7 −2 4 5 j + 7 3 4 0 k = 15i−43j−12k , so d = v×××w v = 15i−43j−12k (7,3,−2) = 152 +(−43)2 +(−12)2 72 +32 +(−2)2 = 2218 62 = 5.98 It is clear that two lines L1 and L2, represented in vector form as r1 + sv1 and r2 + tv2, respectively, are parallel (denoted as L1 ∥ L2) if v1 and v2 are parallel. Also, L1 and L2 are perpendicular (denoted as L1 ⊥ L2) if v1 and v2 are perpendicular. x y z 0 L1 L2 Figure 1.5.5 In 2-dimensional space, two lines are either identical, parallel, or they intersect. In 3-dimensional space, there is an additional possibility: two lines can be skew, that is, they do not intersect but they are not parallel. However, even though they are not parallel, skew lines are on parallel planes (see Figure 1.5.5). To determine whether two lines in R3 intersect, it is often easier to use the parametric representation of the lines. In this case, you should use dif- ferent parameter variables (usually s and t) for the lines, since the values of the parameters may not be the same at the point of intersection. Setting the two (x, y, z) triples equal will result in a system of 3 equations in 2 unknowns (s and t). Example 1.22. Find the point of intersection (if any) of the following lines: x+1 3 = y−2 2 = z −1 −1 and x+3 = y−8 −3 = z +3 2 Solution: First we write the lines in parametric form, with parameters s and t: x = −1+3s, y = 2+2s, z = 1− s and x = −3+ t, y = 8−3t, z = −3+2t The lines intersect when (−1+3s,2+2s,1− s) = (−3+ t,8−3t,−3+2t) for some s, t: −1+3s = −3+ t : ⇒ t = 2+3s 2+2s = 8−3t : ⇒ 2+2s = 8−3(2+3s) = 2−9s ⇒ 2s = −9s ⇒ s = 0 ⇒ t = 2+3(0) = 2 1− s = −3+2t : 1−0 = −3+2(2) ⇒ 1 = 1 (Note that we had to check this.) Letting s = 0 in the equations for the first line, or letting t = 2 in the equations for the second line, gives the point of intersection (−1,2,1). 1.5 Lines and Planes 35 We will now consider planes in 3-dimensional Euclidean space. Plane through a point, perpendicular to a vector Let P be a plane in R3 , and suppose it contains a point P0 = (x0, y0, z0). Let n = (a,b, c) be a nonzero vector which is perpendicular to the plane P. Such a vector is called a normal vector (or just a normal) to the plane. Now let (x, y, z) be any point in the plane P. Then the vector r = (x−x0, y− y0, z− z0) lies in the plane P (see Figure 1.5.6). So if r = 0, then r ⊥ n and hence n···r = 0. And if r = 0 then we still have n···r = 0. (x0, y0, z0)(x, y, z) n r Figure 1.5.6 The plane P Conversely, if (x, y, z) is any point in R3 such that r = (x− x0, y− y0, z − z0) = 0 and n···r = 0, then r ⊥ n and so (x, y, z) lies in P. This proves the following theorem: Theorem 1.18. Let P be a plane in R3 , let (x0, y0, z0) be a point in P, and let n = (a,b, c) be a nonzero vector which is perpendicular to P. Then P consists of the points (x, y, z) satisfying the vector equation: n···r = 0 (1.24) where r = (x− x0, y− y0, z − z0), or equivalently: a(x− x0)+ b(y− y0)+ c(z − z0) = 0 (1.25) The above equation is called the point-normal form of the plane P. Example 1.23. Find the equation of the plane P containing the point (−3,1,3) and perpen- dicular to the vector n = (2,4,8). Solution: By formula (1.25), the plane P consists of all points (x, y, z) such that: 2(x+3)+4(y−1)+8(z −3) = 0 If we multiply out the terms in formula (1.25) and combine the constant terms, we get an equation of the plane in normal form: ax+ by+ cz + d = 0 (1.26) For example, the normal form of the plane in Example 1.23 is 2x+4y+8z −22 = 0. 36 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Plane containing three noncollinear points In 2-dimensional and 3-dimensional space, two points determine a line. Two points do not determine a plane in R3 . In fact, three collinear points (i.e. all on the same line) do not determine a plane; an infinite number of planes would contain the line on which those three points lie. However, three noncollinear points do determine a plane. For if Q, R and S are noncollinear points in R3 , then −−→ QR and −−→ QS are nonzero vectors which are not parallel (by noncollinearity), and so their cross product −−→ QR ××× −−→ QS is perpendicular to both −−→ QR and −−→ QS. So −−→ QR and −−→ QS (and hence Q, R and S) lie in the plane through the point Q with normal vector n = −−→ QR ××× −−→ QS (see Figure 1.5.7). Q R S n = −−→ QR××× −−→ QS −−→ QR −−→ QS Figure 1.5.7 Noncollinear points Q, R, S Example 1.24. Find the equation of the plane P containing the points (2,1,3), (1,−1,2) and (3,2,1). Solution: Let Q = (2,1,3), R = (1,−1,2) and S = (3,2,1). Then for the vectors −−→ QR = (−1,−2,−1) and −−→ QS = (1,1,−2), the plane P has a normal vector n = −−→ QR ××× −−→ QS = (−1,−2,−1)×××(1,1,−2) = (5,−3,1) So using formula (1.25) with the point Q (we could also use R or S), the plane P consists of all points (x, y, z) such that: 5(x−2)−3(y−1)+(z −3) = 0 or in normal form, 5x−3y+ z −10 = 0 We mentioned earlier that skew lines in R3 lie on separate, parallel planes. So two skew lines do not determine a plane. But two (nonidentical) lines which either intersect or are parallel do determine a plane. In both cases, to find the equation of the plane that contains those two lines, simply pick from the two lines a total of three noncollinear points (i.e. one point from one line and two points from the other), then use the technique above, as in Example 1.24, to write the equation. We will leave examples of this as exercises for the reader. 1.5 Lines and Planes 37 Distance between a point and a plane The distance between a point in R3 and a plane is the length of the line segment from that point to the plane which is perpendicular to the plane. The following theorem gives a formula for that distance. Theorem 1.19. Let Q = (x0, y0, z0) be a point in R3 , and let P be a plane with normal form ax+ by+ cz + d = 0 that does not contain Q. Then the distance D from Q to P is: D = \|ax0 + by0 + cz0 + d\| a2 + b2 + c2 (1.27) Proof: Let R = (x, y, z) be any point in the plane P (so that ax + by + cz + d = 0) and let r = −−→ RQ = (x0 − x, y0 − y, z0 − z). Then r = 0 since Q does not lie in P. From the normal form equation for P, we know that n = (a,b, c) is a normal vector for P. Now, any plane divides R3 into two disjoint parts. Assume that n points toward the side of P where the point Q is located. Place n so that its initial point is at R, and let θ be the angle between r and n. Then 0◦ < θ < 90◦ , so cosθ > 0. Thus, the distance D is cosθ r = \|cosθ\| r (see Figure 1.5.8). Q R n r D θ D P Figure 1.5.8 By Theorem 1.6 in Section 1.3, we know that cosθ = n···r n r , so D = \|cosθ\| r = n···r n r r = n···r n = \|a(x0 − x)+ b(y0 − y)+ c(z0 − z)\| a2 + b2 + c2 = \|ax0 + by0 + cz0 −(ax+ by+ cz)\| a2 + b2 + c2 = \|ax0 + by0 + cz0 −(−d)\| a2 + b2 + c2 = \|ax0 + by0 + cz0 + d\| a2 + b2 + c2 If n points away from the side of P where the point Q is located, then 90◦ < θ < 180◦ and so cosθ < 0. The distance D is then \|cosθ\| r , and thus repeating the same argument as above still gives the same result. QED Example 1.25. Find the distance D from (2,4,−5) to the plane from Example 1.24. Solution: Recall that the plane is given by 5x−3y+ z −10 = 0. So D = \|5(2)−3(4)+1(−5)−10\| 52 +(−3)2 +12 = \|−17\| 35 = 17 35 ≈ 2.87 38 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Line of intersection of two planes L Figure 1.5.9 Note that two planes are parallel if they have normal vectors that are parallel, and the planes are perpendicular if their normal vectors are perpendicular. If two planes do intersect, they do so in a line (see Figure 1.5.9). Suppose that two planes P1 and P2 with normal vectors n1 and n2, respectively, intersect in a line L. Since n1 ××× n2 ⊥ n1, then n1 ××× n2 is parallel to the plane P1. Likewise, n1 ××× n2 ⊥ n2 means that n1 ×××n2 is also parallel to P2. Thus, n1 ×××n2 is parallel to the intersection of P1 and P2, i.e. n1 ×××n2 is parallel to L. Thus, we can write L in the following vector form: L : r+ t(n1 ×××n2) , for −∞ < t < ∞ (1.28) where r is any vector pointing to a point belonging to both planes. To find a point in both planes, find a common solution (x, y, z) to the two normal form equations of the planes. This can often be made easier by setting one of the coordinate variables to zero, which leaves you to solve two equations in just two unknowns. Example 1.26. Find the line of intersection L of the planes 5x−3y+ z−10 = 0 and 2x+4y− z +3 = 0. Solution: The plane 5x −3y+ z −10 = 0 has normal vector n1 = (5,−3,1) and the plane 2x + 4y− z+3 = 0 has normal vector n2 = (2,4,−1). Since n1 and n2 are not scalar multiples, then the two planes are not parallel and hence will intersect. A point (x, y, z) on both planes will satisfy the following system of two equations in three unknowns: 5x−3y+ z −10 = 0 2x+4y− z + 3 = 0 Set x = 0 (why is that a good choice?). Then the above equations are reduced to: −3y+ z −10 = 0 4y− z + 3 = 0 The second equation gives z = 4y + 3, substituting that into the first equation gives y = 7. Then z = 31, and so the point (0,7,31) is on L. Since n1 ×××n2 = (−1,7,26), then L is given by: r+ t(n1 ×××n2) = (0,7,31)+ t(−1,7,26), for −∞ < t < ∞ or in parametric form: x = −t, y = 7+7t, z = 31+26t, for −∞ < t < ∞ 40 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE 1.6 Surfaces In the previous section we discussed planes in Euclidean space. A plane is an example of a surface, which we will define informally8 as the solution set of the equation F(x, y, z) = 0 in R3 , for some real-valued function F. For example, a plane given by ax + by + cz + d = 0 is the solution set of F(x, y, z) = 0 for the function F(x, y, z) = ax + by + cz + d. Surfaces are 2-dimensional. The plane is the simplest surface, since it is "flat". In this section we will look at some surfaces that are more complex, the most important of which are the sphere and the cylinder. Definition 1.9. A sphere S is the set of all points (x, y, z) in R3 which are a fixed distance r (called the radius) from a fixed point P0 = (x0, y0, z0) (called the center of the sphere): S = {(x, y, z) : (x− x0)2 +(y− y0)2 +(z − z0)2 = r2 } (1.29) Using vector notation, this can be written in the equivalent form: S = {x : x−x0 = r} (1.30) where x = (x, y, z) and x0 = (x0, y0, z0) are vectors. Figure 1.6.1 illustrates the vectorial approach to spheres. y z x 0 x = r x (a) radius r, center (0,0,0) y z x 0 x−x0 = r x x0 x−x0 (x0, y0, z0) (b) radius r, center (x0, y0, z0) Figure 1.6.1 Spheres in R3 Note in Figure 1.6.1(a) that the intersection of the sphere with the xy-plane is a circle of radius r (i.e. a great circle, given by x2 + y2 = r2 as a subset of R2 ). Similarly for the intersections with the xz-plane and the yz-plane. In general, a plane intersects a sphere either at a single point or in a circle. 8See O'NEILL for a deeper and more rigorous discussion of surfaces. 42 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE If two spheres intersect, they do so either at a single point or in a circle. Example 1.30. Find the intersection (if any) of the spheres x2 + y2 +z2 = 25 and x2 + y2 +(z− 2)2 = 16. Solution: For any point (x, y, z) on both spheres, we see that x2 + y2 + z2 = 25 ⇒ x2 + y2 = 25− z2 , and x2 + y2 +(z −2)2 = 16 ⇒ x2 + y2 = 16−(z −2)2 , so 16−(z −2)2 = 25− z2 ⇒ 4z −4 = 9 ⇒ z = 13/4 ⇒ x2 + y2 = 25−(13/4)2 = 231/16 ∴ The intersection is the circle x2 + y2 = 231 16 of radius 231 4 ≈ 3.8 centered at (0,0, 13 4 ). The cylinders that we will consider are right circular cylinders. These are cylinders ob- tained by moving a line L along a circle C in R3 in a way so that L is always perpendicular to the plane containing C. We will only consider the cases where the plane containing C is parallel to one of the three coordinate planes (see Figure 1.6.3). y z x 0 r (a) x2 + y2 = r2, any z y z x 0 r (b) x2 + z2 = r2, any y y z x 0 r (c) y2 + z2 = r2, any x Figure 1.6.3 Cylinders in R3 For example, the equation of a cylinder whose base circle C lies in the xy-plane and is centered at (a,b,0) and has radius r is (x− a)2 +(y− b)2 = r2 , (1.32) where the value of the z coordinate is unrestricted. Similar equations can be written when the base circle lies in one of the other coordinate planes. A plane intersects a right circular cylinder in a circle, ellipse, or one or two lines, depending on whether that plane is parallel, oblique9 , or perpendicular, respectively, to the plane containing C. The intersection of a surface with a plane is called the trace of the surface. 9i.e. at an angle strictly between 0◦ and 90◦. 1.6 Surfaces 43 The equations of spheres and cylinders are examples of second-degree equations in R3 , i.e. equations of the form Ax2 +By2 +Cz2 + Dxy+ Exz + F yz +Gx+ H y+ Iz + J = 0 (1.33) for some constants A, B, ..., J. If the above equation is not that of a sphere, cylinder, plane, line or point, then the resulting surface is called a quadric surface. y z x 0 a b c Figure 1.6.4 Ellipsoid One type of quadric surface is the ellipsoid, given by an equation of the form: x2 a2 + y2 b2 + z2 c2 = 1 (1.34) In the case where a = b = c, this is just a sphere. In general, an ellipsoid is egg-shaped (think of an ellipse rotated around its major axis). Its traces in the coordinate planes are ellipses. Two other types of quadric surfaces are the hyperboloid of one sheet, given by an equation of the form: x2 a2 + y2 b2 − z2 c2 = 1 (1.35) and the hyperboloid of two sheets, whose equation has the form: x2 a2 − y2 b2 − z2 c2 = 1 (1.36) y z x 0 Figure 1.6.5 Hyperboloid of one sheet y z x 0 Figure 1.6.6 Hyperboloid of two sheets 44 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE For the hyperboloid of one sheet, the trace in any plane parallel to the xy-plane is an ellipse. The traces in the planes parallel to the xz- or yz-planes are hyperbolas (see Figure 1.6.5), except for the special cases x = ±a and y = ±b; in those planes the traces are pairs of intersecting lines (see Exercise 8). For the hyperboloid of two sheets, the trace in any plane parallel to the xy- or xz-plane is a hyperbola (see Figure 1.6.6). There is no trace in the yz-plane. In any plane parallel to the yz-plane for which \|x\| > \|a\|, the trace is an ellipse. y z x 0 Figure 1.6.7 Paraboloid The elliptic paraboloid is another type of quadric surface, whose equation has the form: x2 a2 + y2 b2 = z c (1.37) The traces in planes parallel to the xy-plane are ellipses, though in the xy-plane itself the trace is a single point. The traces in planes parallel to the xz- or yz-planes are parabolas. Figure 1.6.7 shows the case where c > 0. When c < 0 the surface is turned downward. In the case where a = b, the surface is called a paraboloid of revolution, which is often used as a reflecting sur- face, e.g. in vehicle headlights.10 A more complicated quadric surface is the hyperbolic paraboloid, given by: x2 a2 − y2 b2 = z c (1.38) -10 -5 0 5 10 -10 -5 0 5 10 -100 -50 0 50 100 z x y z Figure 1.6.8 Hyperbolic paraboloid 10 For a discussion of this see pp. 157-158 in HECHT. 1.6 Surfaces 45 The hyperbolic paraboloid can be tricky to draw; using graphing software on a computer can make it easier. For example, Figure 1.6.8 was created using the free Gnuplot package (see Appendix C). It shows the graph of the hyperbolic paraboloid z = y2 − x2 , which is the special case where a = b = 1 and c = −1 in equation (1.38). The mesh lines on the surface are the traces in planes parallel to the coordinate planes. So we see that the traces in planes parallel to the xz-plane are parabolas pointing upward, while the traces in planes parallel to the yz-plane are parabolas pointing downward. Also, notice that the traces in planes parallel to the xy-plane are hyperbolas, though in the xy-plane itself the trace is a pair of intersecting lines through the origin. This is true in general when c < 0 in equation (1.38). When c > 0, the surface would be similar to that in Figure 1.6.8, only rotated 90◦ around the z-axis and the nature of the traces in planes parallel to the xz- or yz-planes would be reversed. y z x 0 Figure 1.6.9 Elliptic cone The last type of quadric surface that we will consider is the elliptic cone, which has an equation of the form: x2 a2 + y2 b2 − z2 c2 = 0 (1.39) The traces in planes parallel to the xy-plane are ellipses, ex- cept in the xy-plane itself where the trace is a single point. The traces in planes parallel to the xz- or yz-planes are hyper- bolas, except in the xz- and yz-planes themselves where the traces are pairs of intersecting lines. Notice that every point on the elliptic cone is on a line which lies entirely on the surface; in Figure 1.6.9 these lines all go through the origin. This makes the elliptic cone an example of a ruled surface. The cylinder is also a ruled surface. What may not be as obvious is that both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces. In fact, on both surfaces there are two lines through each point on the surface (see Exercises 11-12). Such surfaces are called doubly ruled surfaces, and the pairs of lines are called a regulus. It is clear that for each of the six types of quadric surfaces that we discussed, the surface can be translated away from the origin (e.g. by replacing x2 by (x−x0)2 in its equation). It can be proved11 that every quadric surface can be translated and/or rotated so that its equation matches one of the six types that we described. For example, z = 2xy is a case of equation (1.33) with "mixed" variables, e.g. with D = 0 so that we get an xy term. This equation does not match any of the types we considered. However, by rotating the x- and y-axes by 45◦ in the xy-plane by means of the coordinate transformation x = (x′ −y′ )/ 2, y = (x′ +y′ )/ 2, z = z′ , then z = 2xy becomes the hyperbolic paraboloid z′ = (x′ )2 − (y′ )2 in the (x′ , y′ , z′ ) coordinate system. That is, z = 2xy is a hyperbolic paraboloid as in equation (1.38), but rotated 45◦ in the xy-plane. 11See Ch. 7 in POGORELOV. 46 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Exercises A For Exercises 1-4, determine if the given equation describes a sphere. If so, find its radius and center. 1. x2 + y2 + z2 −4x−6y−10z +37 = 0 2. x2 + y2 + z2 +2x−2y−8z +19 = 0 3. 2x2 +2y2 +2z2 +4x+4y+4z −44 = 0 4. x2 + y2 − z2 +12x+2y−4z +32 = 0 5. Find the point(s) of intersection of the sphere (x −3)2 +(y+1)2 +(z −3)2 = 9 and the line x = −1+2t, y = −2−3t, z = 3+ t. B 6. Find the intersection of the spheres x2 + y2 + z2 = 9 and (x−4)2 +(y+2)2 +(z −4)2 = 9. 7. Find the intersection of the sphere x2 + y2 + z2 = 9 and the cylinder x2 + y2 = 4. 8. Find the trace of the hyperboloid of one sheet x2 a2 + y2 b2 − z2 c2 = 1 in the plane x = a, and the trace in the plane y = b. 9. Find the trace of the hyperbolic paraboloid x2 a2 − y2 b2 = z c in the xy-plane. C 10. It can be shown that any four noncoplanar points (i.e. points that do not lie in the same plane) determine a sphere.12 Find the equation of the sphere that passes through the points (0,0,0), (0,0,2), (1,−4,3) and (0,−1,3). (Hint: Equation (1.31)) 11. Show that the hyperboloid of one sheet is a doubly ruled surface, i.e. each point on the surface is on two lines lying entirely on the surface. (Hint: Write equation (1.35) as x2 a2 − z2 c2 = 1− y2 b2 , factor each side. Recall that two planes intersect in a line.) 12. Show that the hyperbolic paraboloid is a doubly ruled surface. (Hint: Exercise 11) y z x 0 (0,0,2) (x, y,0) (a,b, c) 1 S Figure 1.6.10 13. Let S be the sphere with radius 1 centered at (0,0,1), and let S∗ be S without the "north pole" point (0,0,2). Let (a,b, c) be an arbitrary point on S∗ . Then the line passing through (0,0,2) and (a,b, c) intersects the xy-plane at some point (x, y,0), as in Figure 1.6.10. Find this point (x, y,0) in terms of a, b and c. (Note: Every point in the xy-plane can be matched with a point on S∗ , and vice versa, in this manner. This method is called stereographic projection, which essentially identifies all of R2 with a "punctured" sphere.) 12See WELCHONS and KRICKENBERGER, p. 160, for a proof. 1.7 Curvilinear Coordinates 47 1.7 Curvilinear Coordinates x y z 0 (x, y, z) x y z Figure 1.7.1 The Cartesian coordinates of a point (x, y, z) are determined by following straight paths starting from the origin: first along the x-axis, then parallel to the y-axis, then parallel to the z-axis, as in Figure 1.7.1. In curvilinear coordinate systems, these paths can be curved. The two types of curvilinear coordinates which we will consider are cylindrical and spherical coordinates. Instead of ref- erencing a point in terms of sides of a rectangular parallelepiped, as with Cartesian coordinates, we will think of the point as ly- ing on a cylinder or sphere. Cylindrical coordinates are often used when there is symmetry around the z-axis; spherical coordinates are useful when there is symmetry about the origin. Let P = (x, y, z) be a point in Cartesian coordinates in R3 , and let P0 = (x, y,0) be the projection of P upon the xy-plane. Treating (x, y) as a point in R2 , let (r,θ) be its polar coordinates (see Figure 1.7.2). Let ρ be the length of the line segment from the origin to P, and let φ be the angle between that line segment and the positive z-axis (see Figure 1.7.3). φ is called the zenith angle. Then the cylindrical coordinates (r,θ, z) and the spherical coordinates (ρ,θ,φ) of P(x, y, z) are defined as follows:13 x y z 0 P(x, y, z) P0(x, y,0) θx y z r Figure 1.7.2 Cylindrical coordinates Cylindrical coordinates (r,θ, z): x = rcosθ r = x2 + y2 y = rsinθ θ = tan−1 y x z = z z = z where 0 ≤ θ ≤ π if y ≥ 0 and π < θ < 2π if y < 0 x y z 0 P(x, y, z) P0(x, y,0) θx y z ρ φ Figure 1.7.3 Spherical coordinates Spherical coordinates (ρ,θ,φ): x = ρ sinφ cosθ ρ = x2 + y2 + z2 y = ρ sinφ sinθ θ = tan−1 y x z = ρ cosφ φ = cos−1 z x2+y2+z2 where 0 ≤ θ ≤ π if y ≥ 0 and π < θ < 2π if y < 0 Both θ and φ are measured in radians. Note that r ≥ 0, 0 ≤ θ < 2π, ρ ≥ 0 and 0 ≤ φ ≤ π. Also, θ is undefined when (x, y) = (0,0), and φ is undefined when (x, y, z) = (0,0,0). 13This "standard" definition of spherical coordinates used by mathematicians results in a left-handed system. For this reason, physicists usually switch the definitions of θ and φ to make (ρ,θ,φ) a right-handed system. 1.7 Curvilinear Coordinates 49 Sometimes the equation of a surface in Cartesian coordinates can be transformed into a simpler equation in some other coordinate system, as in the following example. Example 1.32. Write the equation of the cylinder x2 + y2 = 4 in cylindrical coordinates. Solution: Since r = x2 + y2, then the equation in cylindrical coordinates is r = 2. Using spherical coordinates to write the equation of a sphere does not necessarily make the equation simpler, if the sphere is not centered at the origin. Example 1.33. Write the equation (x−2)2 +(y−1)2 + z2 = 9 in spherical coordinates. Solution: Multiplying the equation out gives x2 + y2 + z2 −4x−2y+5 = 9 , so we get ρ2 −4ρ sinφ cosθ −2ρ sinφ sinθ −4 = 0 , or ρ2 −2sinφ(2cosθ −sinθ)ρ −4 = 0 after combining terms. Note that this actually makes it more difficult to figure out what the surface is, as opposed to the Cartesian equation where you could immediately identify the surface as a sphere of radius 3 centered at (2,1,0). Example 1.34. Describe the surface given by θ = z in cylindrical coordinates. Solution: This surface is called a helicoid. As the (vertical) z coordinate increases, so does the angle θ, while the radius r is unrestricted. So this sweeps out a (ruled!) surface shaped like a spiral staircase, where the spiral has an infinite radius. Figure 1.7.6 shows a section of this surface restricted to 0 ≤ z ≤ 4π and 0 ≤ r ≤ 2. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 2 4 6 8 10 12 14 z x y z Figure 1.7.6 Helicoid θ = z 50 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Exercises A For Exercises 1-4, find the (a) cylindrical and (b) spherical coordinates of the point whose Cartesian coordinates are given. 1. (2,2 3,−1) 2. (−5,5,6) 3. ( 21,− 7,0) 4. (0, 2,2) For Exercises 5-7, write the given equation in (a) cylindrical and (b) spherical coordinates. 5. x2 + y2 + z2 = 25 6. x2 + y2 = 2y 7. x2 + y2 +9z2 = 36 B 8. Describe the intersection of the surfaces whose equations in spherical coordinates are θ = π 2 and φ = π 4 . 9. Show that for a = 0, the equation ρ = 2asinφ cosθ in spherical coordinates describes a sphere centered at (a,0,0) with radius \|a\|. C 10. Let P = (a,θ,φ) be a point in spherical coordinates, with a > 0 and 0 < φ < π. Then P lies on the sphere ρ = a. Since 0 < φ < π, the line segment from the origin to P can be extended to intersect the cylinder given by r = a (in cylindrical coordinates). Find the cylindrical coordinates of that point of intersection. 11. Let P1 and P2 be points whose spherical coordinates are (ρ1,θ1,φ1) and (ρ2,θ2,φ2), respec- tively. Let v1 be the vector from the origin to P1, and let v2 be the vector from the origin to P2. For the angle γ between v1 and v2, show that cosγ = cosφ1 cosφ2 +sinφ1 sinφ2 cos(θ2 −θ1 ). This formula is used in electrodynamics to prove the addition theorem for spherical har- monics, which provides a general expression for the electrostatic potential at a point due to a unit charge. See pp. 100-102 in JACKSON. 12. Show that the distance d between the points P1 and P2 with cylindrical coordinates (r1,θ1, z1) and (r2,θ2, z2), respectively, is d = r2 1 + r2 2 −2r1 r2 cos(θ2 −θ1 )+(z2 − z1)2 . 13. Show that the distance d between the points P1 and P2 with spherical coordinates (ρ1,θ1,φ1) and (ρ2,θ2,φ2), respectively, is d = ρ2 1 +ρ2 2 −2ρ1 ρ2[sinφ1 sinφ2 cos(θ2 −θ1 )+cosφ1 cosφ2]. 1.8 Vector-Valued Functions 51 1.8 Vector-Valued Functions Now that we are familiar with vectors and their operations, we can begin discussing func- tions whose values are vectors. Definition 1.10. A vector-valued function of a real variable is a rule that associates a vector f(t) with a real number t, where t is in some subset D of R1 (called the domain of f). We write f : D → R3 to denote that f is a mapping of D into R3 . For example, f(t) = ti+ t2 j+ t3 k is a vector-valued function in R3 , defined for all real num- bers t. We would write f : R → R3 . At t = 1 the value of the function is the vector i + j + k, which in Cartesian coordinates has the terminal point (1,1,1). A vector-valued function of a real variable can be written in component form as f(t) = f1(t)i+ f2(t)j+ f3(t)k or in the form f(t) = (f1(t), f2(t), f3(t)) for some real-valued functions f1(t), f2(t), f3(t), called the component functions of f. The first form is often used when emphasizing that f(t) is a vector, and the second form is useful when considering just the terminal points of the vectors. By identifying vectors with their terminal points, a curve in space can be written as a vector-valued function. y z x 0 f(0) f(2π) Figure 1.8.1 Example 1.35. Define f : R → R3 by f(t) = (cost,sint,t). This is the equation of a helix (see Figure 1.8.1). As the value of t increases, the terminal points of f(t) trace out a curve spiraling upward. For each t, the x- and y-coordinates of f(t) are x = cost and y = sint, so x2 + y2 = cos2 t+sin2 t = 1. Thus, the curve lies on the surface of the right circular cylinder x2 + y2 = 1. It may help to think of vector-valued functions of a real variable in R3 as a generalization of the parametric functions in R2 which you learned about in single-variable calculus. Much of the theory of real-valued functions of a single real variable can be applied to vector-valued functions of a real variable. Since each of the three component functions are real-valued, it will sometimes be the case that results from single-variable calculus can simply be applied to each of the component functions to yield a similar result for the vector-valued function. However, there are times when such generalizations do not hold (see Exercise 13). The concept of a limit, though, can be extended naturally to vector-valued functions, as in the following definition. 52 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Definition 1.11. Let f(t) be a vector-valued function, let a be a real number and let c be a vector. Then we say that the limit of f(t) as t approaches a equals c, written as lim t→a f(t) = c, if lim t→a f(t)−c = 0. If f(t) = (f1(t), f2(t), f3(t)), then lim t→a f(t) = lim t→a f1(t),lim t→a f2(t),lim t→a f3(t) provided that all three limits on the right side exist. The above definition shows that continuity and the derivative of vector-valued functions can also be defined in terms of its component functions. Definition 1.12. Let f(t) = (f1(t), f2(t), f3(t)) be a vector-valued function, and let a be a real number in its domain. Then f(t) is continuous at a if lim t→a f(t) = f(a). Equivalently, f(t) is continuous at a if and only if f1(t), f2(t), and f3(t) are continuous at a. The derivative of f(t) at a, denoted by f′ (a) or df dt (a), is the limit f′ (a) = lim h→0 f(a+ h)−f(a) h if that limit exists. Equivalently, f′ (a) = (f1 ′ (a), f2 ′ (a), f3 ′ (a)), if the component derivatives exist. We say that f(t) is differentiable at a if f′ (a) exists. Recall that the derivative of a real-valued function of a single variable is a real number, representing the slope of the tangent line to the graph of the function at a point. Similarly, the derivative of a vector-valued function is a tangent vector to the curve in space which the function represents, and it lies on the tangent line to the curve (see Figure 1.8.2). y z x 0 L f(t) f′ (a) f(a) f(a+ h) f(a+ h)− f(a) Figure 1.8.2 Tangent vector f′ (a) and tangent line L = f(a)+ sf′ (a) Example 1.36. Let f(t) = (cost,sint,t). Then f′ (t) = (−sint,cost,1) for all t. The tangent line L to the curve at f(2π) = (1,0,2π) is L = f(2π)+ sf′ (2π) = (1,0,2π)+ s(0,1,1), or in parametric form: x = 1, y = s, z = 2π+ s for −∞ < s < ∞. 1.8 Vector-Valued Functions 55 Just as in single-variable calculus, higher-order derivatives of vector-valued functions are obtained by repeatedly differentiating the (first) derivative of the function: f′′ (t) = d dt f′ (t) , f′′′ (t) = d dt f′′ (t) , ... , dn f dtn = d dt dn−1 f dtn−1 (for n = 2,3,4,...) We can use vector-valued functions to represent physical quantities, such as velocity, ac- celeration, force, momentum, etc. For example, let the real variable t represent time elapsed from some initial time (t = 0), and suppose that an object of constant mass m is subjected to some force so that it moves in space, with its position (x, y, z) at time t a function of t. That is, x = x(t), y = y(t), z = z(t) for some real-valued functions x(t), y(t), z(t). Call r(t) = (x(t), y(t), z(t)) the position vector of the object. We can define various physical quan- tities associated with the object as follows:14 position: r(t) = (x(t), y(t), z(t)) velocity: v(t) = ˙r(t) = r′ (t) = dr dt = (x′ (t), y′ (t), z′ (t)) acceleration: a(t) = ˙v(t) = v′ (t) = dv dt = ¨r(t) = r′′ (t) = d2 r dt2 = (x′′ (t), y′′ (t), z′′ (t)) momentum: p(t) = mv(t) force: F(t) = ˙p(t) = p′ (t) = dp dt (Newton's Second Law of Motion) The magnitude v(t) of the velocity vector is called the speed of the object. Note that since the mass m is a constant, the force equation becomes the familiar F(t) = ma(t). Example 1.39. Let r(t) = (5cost,3sint,4sint) be the position vector of an object at time t ≥ 0. Find its (a) velocity and (b) acceleration vectors. Solution: (a) v(t) = ˙r(t) = (−5sint,3cost,4cost) (b) a(t) = ˙v(t) = (−5cost,−3sint,−4sint) Note that r(t) = 25cos2 t+25sin2 t = 5 for all t, so by Example 1.37 we know that r(t)··· ˙r(t) = 0 for all t (which we can verify from part (a)). In fact, v(t) = 5 for all t also. And not only does r(t) lie on the sphere of radius 5 centered at the origin, but perhaps not so obvious is that it lies completely within a circle of radius 5 centered at the origin. Also, note that a(t) = −r(t). It turns out (see Exercise 16) that whenever an object moves in a circle with constant speed, the acceleration vector will point in the opposite direction of the position vector (i.e. towards the center of the circle). 14We will often use the older dot notation for derivatives when physics is involved. 56 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Recall from Section 1.5 that if r1, r2 are position vectors to distinct points then r1 +t(r2 −r1) represents a line through those two points as t varies over all real numbers. That vector sum can be written as (1 − t)r1 + tr2. So the function l(t) = (1 − t)r1 + tr2 is a line through the terminal points of r1 and r2, and when t is restricted to the interval [0,1] it is the line segment between the points, with l(0) = r1 and l(1) = r2. In general, a function of the form f(t) = (a1 t+b1,a2 t+b2,a3 t+b3) represents a line in R3 . A function of the form f(t) = (a1 t2 + b1 t+ c1,a2 t2 + b2 t+ c2,a3 t2 + b3 t+ c3) represents a (possibly degenerate) parabola in R3 . Example 1.40. Bézier curves are used in Computer Aided Design (CAD) to approximate the shape of a polygonal path in space (called the Bézier polygon or control polygon). For instance, given three points (or position vectors) b0, b1, b2 in R3 , define b1 0(t) = (1− t)b0 + tb1 b1 1(t) = (1− t)b1 + tb2 b2 0(t) = (1− t)b1 0(t)+ tb1 1(t) = (1− t)2 b0 +2t(1− t)b1 + t2 b2 for all real t. For t in the interval [0,1], we see that b1 0(t) is the line segment between b0 and b1, and b1 1(t) is the line segment between b1 and b2. The function b2 0(t) is the Bézier curve for the points b0, b1, b2. Note from the last formula that the curve is a parabola that goes through b0 (when t = 0) and b2 (when t = 1). As an example, let b0 = (0,0,0), b1 = (1,2,3), and b2 = (4,5,2). Then the explicit formula for the Bézier curve is b2 0(t) = (2t+2t2 ,4t+ t2 ,6t−4t2 ), as shown in Figure 1.8.4, where the line segments are b1 0(t) and b1 1(t), and the curve is b2 0(t). 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 z x y z (0,0,0) (1,2,3) (4,5,2) Figure 1.8.4 Bézier curve approximation for three points 1.8 Vector-Valued Functions 57 In general, the polygonal path determined by n ≥ 3 noncollinear points in R3 can be used to define the Bézier curve recursively by a process called repeated linear interpolation. This curve will be a vector-valued function whose components are polynomials of degree n − 1, and its formula is given by de Casteljau's algorithm.15 In the exercises, the reader will be given the algorithm for the case of n = 4 points and asked to write the explicit formula for the Bézier curve for the four points shown in Figure 1.8.5. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 0 0.5 1 1.5 2 z x y z (0,0,0) (0,1,1) (2,3,0) (4,5,2) Figure 1.8.5 Bézier curve approximation for four points Exercises A For Exercises 1-4, calculate f′ (t) and find the tangent line at f(0). 1. f(t) = (t+1,t2 +1,t3 +1) 2. f(t) = (et +1, e2t +1, et2 +1) 3. f(t) = (cos2t,sin2t,t) 4. f(t) = (sin2t,2sin2 t,2cost) For Exercises 5-6, find the velocity v(t) and acceleration a(t) of an object with the given position vector r(t). 5. r(t) = (t,t−sint,1−cost) 6. r(t) = (3cost,2sint,1) B 7. Let f(t) = cost 1+ a2t2 , sint 1+ a2t2 , −at 1+ a2t2 , with a = 0. (a) Show that f(t) = 1 for all t. (b) Show directly that f′ (t)···f(t) = 0 for all t. 8. If f′ (t) = 0 for all t in some interval (a,b), show that f(t) is a constant vector in (a,b). 15See pp. 27-30 in FARIN. 1.9 Arc Length 59 1.9 Arc Length Let r(t) = (x(t), y(t), z(t)) be the position vector of an object moving in R3 . Since v(t) is the speed of the object at time t, it seems natural to define the distance s traveled by the object from time t = a to t = b as the definite integral s = b a v(t) dt = b a x′(t)2 + y′(t)2 + z′(t)2 dt, (1.40) which is analogous to the case from single-variable calculus for parametric functions in R2 . This is indeed how we will define the distance traveled and, in general, the arc length of a curve in R3 . Definition 1.13. Let f(t) = (x(t), y(t), z(t)) be a curve in R3 whose domain includes the inter- val [a,b]. Suppose that in the interval (a,b) the first derivative of each component function x(t), y(t) and z(t) exists and is continuous, and that no section of the curve is repeated. Then the arc length L of the curve from t = a to t = b is L = b a f′ (t) dt = b a x′(t)2 + y′(t)2 + z′(t)2 dt (1.41) A real-valued function whose first derivative is continuous is called continuously differ- entiable (or a C 1 function), and a function whose derivatives of all orders are continuous is called smooth (or a C ∞ function). All the functions we will consider will be smooth. A smooth curve f(t) is one whose derivative f′ (t) is never the zero vector and whose component functions are all smooth. Note that we did not prove that the formula in the above definition actually gives the length of a section of a curve. A rigorous proof requires dealing with some subtleties, nor- mally glossed over in calculus texts, which are beyond the scope of this book.16 Example 1.41. Find the length L of the helix f(t) = (cost,sint,t) from t = 0 to t = 2π. Solution: By formula (1.41), we have L = 2π 0 (−sint)2 +(cost)2 +12 dt = 2π 0 sin2 t+cos2 t+1dt = 2π 0 2dt = 2(2π−0) = 2 2π Similar to the case in R2 , if there are values of t in the interval [a,b] where the derivative of a component function is not continuous then it is often possible to partition [a,b] into subintervals where all the component functions are continuously differentiable (except at the endpoints, which can be ignored). The sum of the arc lengths over the subintervals will be the arc length over [a,b]. 16In particular, Duhamel's principle is needed. See the proof in TAYLOR and MANN, § 14.2 and § 18.2. 60 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE Notice that the curve traced out by the function f(t) = (cost,sint,t) from Example 1.41 is also traced out by the function g(t) = (cos2t,sin2t,2t). For example, over the interval [0,π], g(t) traces out the same section of the curve as f(t) does over the interval [0,2π]. Intuitively, this says that g(t) traces the curve twice as fast as f(t). This makes sense since, viewing the functions as position vectors and their derivatives as velocity vectors, the speeds of f(t) and g(t) are f′ (t) = 2 and g′ (t) = 2 2, respectively. We say that g(t) and f(t) are different parametrizations of the same curve. Definition 1.14. Let C be a smooth curve in R3 represented by a function f(t) defined on an interval [a,b], and let α : [c,d] → [a,b] be a smooth one-to-one mapping of an interval [c,d] onto [a,b]. Then the function g : [c,d] → R3 defined by g(s) = f(α(s)) is a parametrization of C with parameter s. If α is strictly increasing on [c,d] then we say that g(s) is equivalent to f(t). s t f(t) [c,d] [a,b] R3α f g(s) = f(α(s)) = f(t) Note that the differentiability of g(s) follows from a version of the Chain Rule for vector- valued functions (the proof is left as an exercise): Theorem 1.21. Chain Rule: If f(t) is a differentiable vector-valued function of t, and t = α(s) is a differentiable scalar function of s, then f(s) = f(α(s)) is a differentiable vector-valued function of s, and df ds = df dt dt ds (1.42) for any s where the composite function f(α(s)) is defined. Example 1.42. The following are all equivalent parametrizations of the same curve: f(t) = (cost,sint,t) for t in [0,2π] g(s) = (cos2s,sin2s,2s) for s in [0,π] h(s) = (cos2πs,sin2πs,2πs) for s in [0,1] To see that g(s) is equivalent to f(t), define α : [0,π] → [0,2π] by α(s) = 2s. Then α is smooth, one-to-one, maps [0,π] onto [0,2π], and is strictly increasing (since α′ (s) = 2 > 0 for all s). Likewise, defining α : [0,1] → [0,2π] by α(s) = 2πs shows that h(s) is equivalent to f(t). 1.9 Arc Length 61 A curve can have many parametrizations, with different speeds, so which one is the best to use? In some situations the arc length parametrization can be useful. The idea behind this is to replace the parameter t, for any given smooth parametrization f(t) defined on [a,b], by the parameter s given by s = s(t) = t a f′ (u) du. (1.43) In terms of motion along a curve, s is the distance traveled along the curve after time t has elapsed. So the new parameter will be distance instead of time. There is a natural correspondence between s and t: from a starting point on the curve, the distance traveled along the curve (in one direction) is uniquely determined by the amount of time elapsed, and vice versa. Since s is the arc length of the curve over the interval [a,t] for each t in [a,b], then it is a function of t. By the Fundamental Theorem of Calculus, its derivative is s′ (t) = ds dt = d dt t a f′ (u) du = f′ (t) for all t in [a,b]. Since f(t) is smooth, then f′ (t) > 0 for all t in [a,b]. Thus s′ (t) > 0 and hence s(t) is strictly increasing on the interval [a,b]. Recall that this means that s is a one-to-one mapping of the interval [a,b] onto the interval [s(a),s(b)]. But we see that s(a) = a a f′ (u) du = 0 and s(b) = b a f′ (u) du = L = arc length from t = a to t = b s t [0,L] [a,b] α(s) s(t) Figure 1.9.1 t = α(s) So the function s : [a,b] → [0,L] is a one-to-one, differentiable mapping onto the interval [0,L]. From single-variable calculus, we know that this means that there exists an inverse function α : [0,L] → [a,b] that is differentiable and the inverse of s : [a,b] → [0,L]. That is, for each t in [a,b] there is a unique s in [0,L] such that s = s(t) and t = α(s). And we know that the derivative of α is α′ (s) = 1 s′(α(s)) = 1 f′(α(s)) So define the arc length parametrization f : [0,L] → R3 by f(s) = f(α(s)) for all s in [0,L]. Then f(s) is smooth, by the Chain Rule. In fact, f(s) has unit speed: f′ (s) = f′ (α(s))α′ (s) by the Chain Rule, so = f′ (α(s)) 1 f′(α(s)) , so f′ (s) = 1 for all s in [0,L]. So the arc length parametrization traverses the curve at a "normal" rate. 62 CHAPTER 1. VECTORS IN EUCLIDEAN SPACE In practice, parametrizing a curve f(t) by arc length requires you to evaluate the integral s = t a f′ (u) du in some closed form (as a function of t) so that you could then solve for t in terms of s. If that can be done, you would then substitute the expression for t in terms of s (which we called α(s)) into the formula for f(t) to get f(s). Example 1.43. Parametrize the helix f(t) = (cost,sint,t), for t in [0,2π], by arc length. Solution: By Example 1.41 and formula (1.43), we have s = t 0 f′ (u) du = t 0 2du = 2t for all t in [0,2π]. So we can solve for t in terms of s: t = α(s) = s 2 . ∴ f(s) = cos s 2 ,sin s 2 , s 2 for all s in [0,2 2π]. Note that f′ (s) = 1. Arc length plays an important role when discussing curvature and moving frame fields, in the field of mathematics known as differential geometry.17 The methods involve using an arc length parametrization, which often leads to an integral that is either difficult or impossible to evaluate in a simple closed form. The simple integral in Example 1.43 is the exception, not the norm. In general, arc length parametrizations are more useful for theoretical purposes than for practical computations.18 Curvature and moving frame fields can be defined without using arc length, which makes their computation much easier, and these definitions can be shown to be equivalent to those using arc length. We will leave this to the exercises. The arc length for curves given in other coordinate systems can also be calculated: Theorem 1.22. Suppose that r = r(t), θ = θ(t) and z = z(t) are the cylindrical coordinates of a curve f(t), for t in [a,b]. Then the arc length L of the curve over [a,b] is L = b a r′(t)2 + r(t)2θ′(t)2 + z′(t)2 dt (1.44) Proof: The Cartesian coordinates (x(t), y(t), z(t)) of a point on the curve are given by x(t) = r(t)cosθ(t), y(t) = r(t)sinθ(t), z(t) = z(t) so differentiating the above expressions for x(t) and y(t) with respect to t gives x′ (t) = r′ (t)cosθ(t)− r(t)θ′ (t)sinθ(t), y′ (t) = r′ (t)sinθ(t)+ r(t)θ′ (t)cosθ(t) 17See O'NEILL for an introduction to elementary differential geometry. 18For example, the usual parametrizations of Bézier curves, which we discussed in Section 1.8, are polynomial functions in R3. This makes their computation relatively simple, which, in CAD, is desirable. But their arc length parametrizations are not only not polynomials, they are in fact usually impossible to calculate at all. 2 Functions of Several Variables 2.1 Functions of Two or Three Variables In Section 1.8 we discussed vector-valued functions of a single real variable. We will now examine real-valued functions of a point (or vector) in R2 or R3 . For the most part these functions will be defined on sets of points in R2 , but there will be times when we will use points in R3 , and there will also be times when it will be convenient to think of the points as vectors (or terminal points of vectors). A real-valued function f defined on a subset D of R2 is a rule that assigns to each point (x, y) in D a real number f (x, y). The largest possible set D in R2 on which f is defined is called the domain of f , and the range of f is the set of all real numbers f (x, y) as (x, y) varies over the domain D. A similar definition holds for functions f (x, y, z) defined on points (x, y, z) in R3 . Example 2.1. The domain of the function f (x, y) = xy is all of R2 , and the range of f is all of R. Example 2.2. The domain of the function f (x, y) = 1 x− y is all of R2 except the points (x, y) for which x = y. That is, the domain is the set D = {(x, y) : x = y}. The range of f is all real numbers except 0. Example 2.3. The domain of the function f (x, y) = 1− x2 − y2 is the set D = {(x, y) : x2 + y2 ≤ 1}, since the quantity inside the square root is nonnegative if and only if 1−(x2 + y2 ) ≥ 0. We see that D consists of all points on and inside the unit circle in R2 (D is sometimes called the closed unit disk). The range of f is the interval [0,1] in R. 65 66 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES Example 2.4. The domain of the function f (x, y, z) = ex+y−z is all of R3 , and the range of f is all positive real numbers. A function f (x, y) defined in R2 is often written as z = f (x, y), as was mentioned in Section 1.1, so that the graph of f (x, y) is the set {(x, y, z) : z = f (x, y)} in R3 . So we see that this graph is a surface in R3 , since it satisfies an equation of the form F(x, y, z) = 0 (namely, F(x, y, z) = f (x, y)− z). The traces of this surface in the planes z = c, where c varies over R, are called the level curves of the function. Equivalently, the level curves are the solution sets of the equations f (x, y) = c, for c in R. Level curves are often projected onto the xy-plane to give an idea of the various "elevation" levels of the surface (as is done in topography). Example 2.5. The graph of the function f (x, y) = sin x2 + y2 x2 + y2 is shown below. Note that the level curves (shown both on the surface and projected onto the xy-plane) are groups of concentric circles. -10 -5 0 5 10 -10 -5 0 5 10 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 z x y z Figure 2.1.1 The function f (x, y) = sin x2+y2 x2+y2 You may be wondering what happens to the function in Example 2.5 at the point (x, y) = (0,0), since both the numerator and denominator are 0 at that point. The function is not defined at (0,0), but the limit of the function exists (and equals 1) as (x, y) approaches (0,0). We will now state explicitly what is meant by the limit of a function of two variables. 2.1 Functions of Two or Three Variables 67 Definition 2.1. Let (a,b) be a point in R2 , and let f (x, y) be a real-valued function defined on some set containing (a,b) (but not necessarily defined at (a,b) itself). Then we say that the limit of f (x, y) equals L as (x, y) approaches (a,b), written as lim (x,y)→(a,b) f (x, y) = L , (2.1) if given any ǫ > 0, there exists a δ > 0 such that \| f (x, y)− L\| < ǫ whenever 0 < (x− a)2 +(y− b)2 < δ. A similar definition can be made for functions of three variables. The idea behind the above definition is that the values of f (x, y) can get arbitrarily close to L (i.e. within ǫ of L) if we pick (x, y) sufficiently close to (a,b) (i.e. inside a circle centered at (a,b) with some sufficiently small radius δ). If you recall the "epsilon-delta" proofs of limits of real-valued functions of a single variable, you may remember how awkward they can be, and how they can usually only be done easily for simple functions. In general, the multivariable cases are at least equally awkward to go through, so we will not bother with such proofs. Instead, we will simply state that when the function f (x, y) is given by a single formula and is defined at the point (a,b) (e.g. is not some indeterminate form like 0/0) then you can just substitute (x, y) = (a,b) into the formula for f (x, y) to find the limit. Example 2.6. lim (x,y)→(1,2) xy x2 + y2 = (1)(2) 12 +22 = 2 5 since f (x, y) = xy x2+y2 is properly defined at the point (1,2). The major difference between limits in one variable and limits in two or more variables has to do with how a point is approached. In the single-variable case, the statement "x → a" means that x gets closer to the value a from two possible directions along the real number line (see Figure 2.1.2(a)). In two dimensions, however, (x, y) can approach a point (a,b) along an infinite number of paths (see Figure 2.1.2(b)). 0 xa xx (a) x → a in R x y 0 (a,b) (b) (x, y) → (a,b) in R2 Figure 2.1.2 "Approaching" a point in different dimensions 2.1 Functions of Two or Three Variables 69 Example 2.8. Show that lim (x,y)→(0,0) y4 x2 + y2 = 0. Since substituting (x, y) = (0,0) into the function gives the indeterminate form 0/0, we need an alternate method for evaluating this limit. We will use Theorem 2.1(e). First, notice that y4 = y2 4 and so 0 ≤ y4 ≤ x2 + y2 4 for all (x, y). But x2 + y2 4 = (x2 + y2 )2 . Thus, for all (x, y) = (0,0) we have y4 x2 + y2 ≤ (x2 + y2 )2 x2 + y2 = x2 + y2 → 0 as (x, y) → (0,0). Therefore lim (x,y)→(0,0) y4 x2 + y2 = 0. Continuity can be defined similarly as in the single-variable case. Definition 2.2. A real-valued function f (x, y) with domain D in R2 is continuous at the point (a,b) in D if lim (x,y)→(a,b) f (x, y) = f (a,b). We say that f (x, y) is a continuous function if it is continuous at every point in its domain D. Unless indicated otherwise, you can assume that all the functions we deal with are con- tinuous. In fact, we can modify the function from Example 2.8 so that it is continuous on all of R2 . Example 2.9. Define a function f (x, y) on all of R2 as follows: f (x, y) =    0 if (x, y) = (0,0) y4 x2 + y2 if (x, y) = (0,0) Then f (x, y) is well-defined for all (x, y) in R2 (i.e. there are no indeterminate forms for any (x, y)), and we see that lim (x,y)→(a,b) f (x, y) = b4 a2 + b2 = f (a,b) for (a,b) = (0,0). So since lim (x,y)→(0,0) f (x, y) = 0 = f (0,0) by Example 2.8, then f (x, y) is continuous on all of R2 . 2.2 Partial Derivatives 71 2.2 Partial Derivatives Now that we have an idea of what functions of several variables are, and what a limit of such a function is, we can start to develop an idea of a derivative of a function of two or more variables. We will start with the notion of a partial derivative. Definition 2.3. Let f (x, y) be a real-valued function with domain D in R2 , and let (a,b) be a point in D. Then the partial derivative of f at (a,b) with respect to x, denoted by ∂f ∂x (a,b), is defined as ∂f ∂x (a,b) = lim h→0 f (a+ h,b)− f (a,b) h (2.2) and the partial derivative of f at (a,b) with respect to y, denoted by ∂f ∂y (a,b), is defined as ∂f ∂y (a,b) = lim h→0 f (a,b + h)− f (a,b) h . (2.3) Note: The symbol ∂ is pronounced "del".1 Recall that the derivative of a function f (x) can be interpreted as the rate of change of that function in the (positive) x direction. From the definitions above, we can see that the partial derivative of a function f (x, y) with respect to x or y is the rate of change of f (x, y) in the (positive) x or y direction, respectively. What this means is that the partial derivative of a function f (x, y) with respect to x can be calculated by treating the y variable as a constant, and then simply differentiating f (x, y) as if it were a function of x alone, using the usual rules from single-variable calculus. Likewise, the partial derivative of f (x, y) with respect to y is obtained by treating the x variable as a constant and then differentiating f (x, y) as if it were a function of y alone. Example 2.10. Find ∂f ∂x (x, y) and ∂f ∂y (x, y) for the function f (x, y) = x2 y+ y3 . Solution: Treating y as a constant and differentiating f (x, y) with respect to x gives ∂f ∂x (x, y) = 2xy and treating x as a constant and differentiating f (x, y) with respect to y gives ∂f ∂y (x, y) = x2 +3y2 . 1It is not a Greek letter. The symbol was first used by the mathematicians A. Clairaut and L. Euler around 1740, to distinguish it from the letter d used for the "usual" derivative. 2.3 Tangent Plane to a Surface 75 2.3 Tangent Plane to a Surface In the previous section we mentioned that the partial derivatives ∂f ∂x and ∂f ∂y can be thought of as the rate of change of a function z = f (x, y) in the positive x and y directions, respectively. Recall that the derivative dy dx of a function y = f (x) has a geometric meaning, namely as the slope of the tangent line to the graph of f at the point (x, f (x)) in R2 . There is a similar geometric meaning to the partial derivatives ∂f ∂x and ∂f ∂y of a function z = f (x, y): given a point (a,b) in the domain D of f (x, y), the trace of the surface described by z = f (x, y) in the plane y = b is a curve in R3 through the point (a,b, f (a,b)), and the slope of the tangent line Lx to that curve at that point is ∂f ∂x (a,b). Similarly, ∂f ∂y (a,b) is the slope of the tangent line Ly to the trace of the surface z = f (x, y) in the plane x = a (see Figure 2.3.1). y z x 0 (a,b) D Lx b (a,b, f (a,b)) slope = ∂f ∂x (a,b) z = f (x, y) (a) Tangent line Lx in the plane y = b y z x 0 (a,b) D Ly a (a,b, f (a,b)) slope = ∂f ∂y (a,b) z = f (x, y) (b) Tangent line Ly in the plane x = a Figure 2.3.1 Partial derivatives as slopes Since the derivative dy dx of a function y = f (x) is used to find the tangent line to the graph of f (which is a curve in R2 ), you might expect that partial derivatives can be used to define a tangent plane to the graph of a surface z = f (x, y). This indeed turns out to be the case. First, we need a definition of a tangent plane. The intuitive idea is that a tangent plane "just touches" a surface at a point. The formal definition mimics the intuitive notion of a tangent line to a curve. Definition 2.4. Let z = f (x, y) be the equation of a surface S in R3 , and let P = (a,b, c) be a point on S. Let T be a plane which contains the point P, and let Q = (x, y, z) represent a generic point on the surface S. If the (acute) angle between the vector −−→ PQ and the plane T approaches zero as the point Q approaches P along the surface S, then we call T the tangent plane to S at P. Note that since two lines in R3 determine a plane, then the two tangent lines to the surface z = f (x, y) in the x and y directions described in Figure 2.3.1 are contained in the tangent plane at that point, if the tangent plane exists at that point. The existence of those two 76 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES tangent lines does not by itself guarantee the existence of the tangent plane. It is possible that if we take the trace of the surface in the plane x− y = 0 (which makes a 45◦ angle with the positive x-axis), the resulting curve in that plane may have a tangent line which is not in the plane determined by the other two tangent lines, or it may not have a tangent line at all at that point. Luckily, it turns out4 that if ∂f ∂x and ∂f ∂y exist in a region around a point (a,b) and are continuous at (a,b) then the tangent plane to the surface z = f (x, y) will exist at the point (a,b, f (a,b)). In this text, those conditions will always hold. y z x 0 (a,b, f (a,b)) z = f (x, y) T Lx Ly Figure 2.3.2 Tangent plane Suppose that we want an equation of the tangent plane T to the surface z = f (x, y) at a point (a,b, f (a,b)). Let Lx and Ly be the tangent lines to the traces of the surface in the planes y = b and x = a, respectively (as in Figure 2.3.2), and suppose that the conditions for T to exist do hold. Then the equation for T is A(x− a)+B(y− b)+C(z − f (a,b)) = 0 (2.4) where n = (A,B,C) is a normal vector to the plane T. Since T contains the lines Lx and Ly, then all we need are vectors vx and vy that are parallel to Lx and Ly, respectively, and then let n = vx ×××vy. x z 0 vx = (1,0, ∂f ∂x (a,b)) ∂f ∂x (a,b) 1 Figure 2.3.3 Since the slope of Lx is ∂f ∂x (a,b), then the vector vx = (1,0, ∂f ∂x (a,b)) is parallel to Lx (since vx lies in the xz-plane and lies in a line with slope ∂f ∂x (a,b) 1 = ∂f ∂x (a,b). See Figure 2.3.3). Similarly, the vector vy = (0,1, ∂f ∂y (a,b)) is parallel to Ly. Hence, the vector n = vx ×××vy = i j k 1 0 ∂f ∂x (a,b) 0 1 ∂f ∂y (a,b) = − ∂f ∂x (a,b)i− ∂f ∂y (a,b)j+k is normal to the plane T. Thus the equation of T is − ∂f ∂x (a,b)(x− a)− ∂f ∂y (a,b)(y− b)+ z − f (a,b) = 0 . (2.5) Multiplying both sides by −1, we have the following result: The equation of the tangent plane to the surface z = f (x, y) at the point (a,b, f (a,b)) is ∂f ∂x (a,b)(x− a)+ ∂f ∂y (a,b)(y− b)− z + f (a,b) = 0 (2.6) 4See TAYLOR and MANN, § 6.4. 78 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES 2.4 Directional Derivatives and the Gradient For a function z = f (x, y), we learned that the partial derivatives ∂f ∂x and ∂f ∂y represent the (instantaneous) rate of change of f in the positive x and y directions, respectively. What about other directions? It turns out that we can find the rate of change in any direction using a more general type of derivative called a directional derivative. Definition 2.5. Let f (x, y) be a real-valued function with domain D in R2 , and let (a,b) be a point in D. Let v be a unit vector in R2 . Then the directional derivative of f at (a,b) in the direction of v, denoted by Dv f (a,b), is defined as Dv f (a,b) = lim h→0 f ((a,b)+ hv)− f (a,b) h (2.8) Notice in the definition that we seem to be treating the point (a,b) as a vector, since we are adding the vector hv to it. But this is just the usual idea of identifying vectors with their terminal points, which the reader should be used to by now. If we were to write the vector v as v = (v1,v2), then Dv f (a,b) = lim h→0 f (a+ hv1,b + hv2)− f (a,b) h . (2.9) From this we can immediately recognize that the partial derivatives ∂f ∂x and ∂f ∂y are special cases of the directional derivative with v = i = (1,0) and v = j = (0,1), respectively. That is, ∂f ∂x = Di f and ∂f ∂y = Dj f . Since there are many vectors with the same direction, we use a unit vector in the definition, as that represents a "standard" vector for a given direction. If f (x, y) has continuous partial derivatives ∂f ∂x and ∂f ∂y (which will always be the case in this text), then there is a simple formula for the directional derivative: Theorem 2.2. Let f (x, y) be a real-valued function with domain D in R2 such that the partial derivatives ∂f ∂x and ∂f ∂y exist and are continuous in D. Let (a,b) be a point in D, and let v = (v1,v2) be a unit vector in R2 . Then Dv f (a,b) = v1 ∂f ∂x (a,b)+ v2 ∂f ∂y (a,b) . (2.10) Proof: Note that if v = i = (1,0) then the above formula reduces to Dv f (a,b) = ∂f ∂x (a,b), which we know is true since Di f = ∂f ∂x , as we noted earlier. Similarly, for v = j = (0,1) the formula reduces to Dv f (a,b) = ∂f ∂y (a,b), which is true since Dj f = ∂f ∂y . So since i = (1,0) and j = (0,1) are the only unit vectors in R2 with a zero component, then we need only show the formula holds for unit vectors v = (v1,v2) with v1 = 0 and v2 = 0. So fix such a vector v and fix a number h = 0. 2.4 Directional Derivatives and the Gradient 81 The value of f (x, y) is constant along a level curve, so since v is a tangent vector to this curve, then the rate of change of f in the direction of v is 0, i.e. Dv f = 0. But we know that Dv f = v···∇f = v ∇f cosθ, where θ is the angle between v and ∇f . So since v = 1 then Dv f = ∇f cosθ. So since ∇f = 0 then Dv f = 0 ⇒ cosθ = 0 ⇒ θ = 90◦ . In other words, ∇f ⊥ v, which means that ∇f is normal to the level curve. In general, for any unit vector v in R2 , we still have Dv f = ∇f cosθ, where θ is the angle between v and ∇f . At a fixed point (x, y) the length ∇f is fixed, and the value of Dv f then varies as θ varies. The largest value that Dv f can take is when cosθ = 1 (θ = 0◦ ), while the smallest value occurs when cosθ = −1 (θ = 180◦ ). In other words, the value of the function f increases the fastest in the direction of ∇f (since θ = 0◦ in that case), and the value of f decreases the fastest in the direction of −∇f (since θ = 180◦ in that case). We have thus proved the following theorem: Theorem 2.4. Let f (x, y) be a continuously differentiable real-valued function, with ∇f = 0. Then: (a) The gradient ∇f is normal to any level curve f (x, y) = c. (b) The value of f (x, y) increases the fastest in the direction of ∇f . (c) The value of f (x, y) decreases the fastest in the direction of −∇f . Example 2.16. In which direction does the function f (x, y) = xy2 + x3 y increase the fastest from the point (1,2)? In which direction does it decrease the fastest? Solution: Since ∇f = (y2 + 3x2 y,2xy + x3 ), then ∇f (1,2) = (10,5) = 0. A unit vector in that direction is v = ∇f ∇f = 2 5 , 1 5 . Thus, f increases the fastest in the direction of 2 5 , 1 5 and decreases the fastest in the direction of −2 5 , −1 5 . Though we proved Theorem 2.4 for functions of two variables, a similar argument can be used to show that it also applies to functions of three or more variables. Likewise, the directional derivative in the three-dimensional case can also be defined by the formula Dv f = v···∇f . Example 2.17. The temperature T of a solid is given by the function T(x, y, z) = e−x + e−2y + e4z , where x, y, z are space coordinates relative to the center of the solid. In which direction from the point (1,1,1) will the temperature decrease the fastest? Solution: Since ∇f = (−e−x ,−2e−2y ,4e4z ), then the temperature will decrease the fastest in the direction of −∇f (1,1,1) = (e−1 ,2e−2 ,−4e4 ). 2.5 Maxima and Minima 83 2.5 Maxima and Minima The gradient can be used to find extreme points of real-valued functions of several variables, that is, points where the function has a local maximum or local minimum. We will consider only functions of two variables; functions of three or more variables require methods using linear algebra. Definition 2.7. Let f (x, y) be a real-valued function, and let (a,b) be a point in the domain of f . We say that f has a local maximum at (a,b) if f (x, y) ≤ f (a,b) for all (x, y) inside some disk of positive radius centered at (a,b), i.e. there is some sufficiently small r > 0 such that f (x, y) ≤ f (a,b) for all (x, y) for which (x− a)2 +(y− b)2 < r2 . Likewise, we say that f has a local minimum at (a,b) if f (x, y) ≥ f (a,b) for all (x, y) inside some disk of positive radius centered at (a,b). If f (x, y) ≤ f (a,b) for all (x, y) in the domain of f , then f has a global maximum at (a,b). If f (x, y) ≥ f (a,b) for all (x, y) in the domain of f , then f has a global minimum at (a,b). Suppose that (a,b) is a local maximum point for f (x, y), and that the first-order partial derivatives of f exist at (a,b). We know that f (a,b) is the largest value of f (x, y) as (x, y) goes in all directions from the point (a,b), in some sufficiently small disk centered at (a,b). In particular, f (a,b) is the largest value of f in the x direction (around the point (a,b)), that is, the single-variable function g(x) = f (x,b) has a local maximum at x = a. So we know that g′ (a) = 0. Since g′ (x) = ∂f ∂x (x,b), then ∂f ∂x (a,b) = 0. Similarly, f (a,b) is the largest value of f near (a,b) in the y direction and so ∂f ∂y (a,b) = 0. We thus have the following theorem: Theorem 2.5. Let f (x, y) be a real-valued function such that both ∂f ∂x (a,b) and ∂f ∂y (a,b) exist. Then a necessary condition for f (x, y) to have a local maximum or minimum at (a,b) is that ∇f (a,b) = 0. Note: Theorem 2.5 can be extended to apply to functions of three or more variables. A point (a,b) where ∇f (a,b) = 0 is called a critical point for the function f (x, y). So given a function f (x, y), to find the critical points of f you have to solve the equations ∂f ∂x (x, y) = 0 and ∂f ∂y (x, y) = 0 simultaneously for (x, y). Similar to the single-variable case, the necessary condition that ∇f (a,b) = 0 is not always sufficient to guarantee that a critical point is a local maximum or minimum. Example 2.18. The function f (x, y) = xy has a critical point at (0,0): ∂f ∂x = y = 0 ⇒ y = 0, and ∂f ∂y = x = 0 ⇒ x = 0, so (0,0) is the only critical point. But clearly f does not have a local maximum or minimum at (0,0) since any disk around (0,0) contains points (x, y) where the values of x and y have the same sign (so that f (x, y) = xy > 0 = f (0,0)) and different signs (so that f (x, y) = xy < 0 = f (0,0)). In fact, along the path y = x in R2 , f (x, y) = x2 , which has a 84 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES local minimum at (0,0), while along the path y = −x we have f (x, y) = −x2 , which has a local maximum at (0,0). So (0,0) is an example of a saddle point, i.e. it is a local maximum in one direction and a local minimum in another direction. The graph of f (x, y) is shown in Figure 2.5.1, which is a hyperbolic paraboloid. -10 -5 0 5 10 -10 -5 0 5 10 -100 -50 0 50 100 z x y z Figure 2.5.1 f (x, y) = xy, saddle point at (0,0) The following theorem gives sufficient conditions for a critical point to be a local maximum or minimum of a smooth function (i.e. a function whose partial derivatives of all orders exist and are continuous), which we will not prove here.6 Theorem 2.6. Let f (x, y) be a smooth real-valued function, with a critical point at (a,b) (i.e. ∇f (a,b) = 0). Define D = ∂2 f ∂x2 (a,b) ∂2 f ∂y2 (a,b)− ∂2 f ∂y∂x (a,b) 2 Then (a) if D > 0 and ∂2 f ∂x2 (a,b) > 0, then f has a local minimum at (a,b) (b) if D > 0 and ∂2 f ∂x2 (a,b) < 0, then f has a local maximum at (a,b) (c) if D < 0, then f has neither a local minimum nor a local maximum at (a,b) (d) if D = 0, then the test fails. 6See TAYLOR and MANN, § 7.6. 2.6 Unconstrained Optimization: Numerical Methods 89 2.6 Unconstrained Optimization: Numerical Methods The types of problems that we solved in the previous section were examples of unconstrained optimization problems. That is, we tried to find local (and perhaps even global) maximum and minimum points of real-valued functions f (x, y), where the points (x, y) could be any points in the domain of f . The method we used required us to find the critical points of f , which meant having to solve the equation ∇f = 0, which in general is a system of two equa- tions in two unknowns (x and y). While this was relatively simple for the examples we did, in general this will not be the case. If the equations involve polynomials in x and y of degree three or higher, or complicated expressions involving trigonometric, exponential, or loga- rithmic functions, then solving even one such equation, let alone two, could be impossible by elementary means.7 For example, if one of the equations that had to be solved was x3 +9x−2 = 0 , you may have a hard time getting the exact solutions. Trial and error would not help much, especially since the only real solution8 turns out to be 3 28+1− 3 28−1. In a situation such as this, the only choice may be to find a solution using some numerical method which gives a sequence of numbers which converge to the actual solution. For example, Newton's method for solving equations f (x) = 0, which you probably learned in single-variable calcu- lus. In this section we will describe another method of Newton for finding critical points of real-valued functions of two variables. Let f (x, y) be a smooth real-valued function, and define D(x, y) = ∂2 f ∂x2 (x, y) ∂2 f ∂y2 (x, y)− ∂2 f ∂y∂x (x, y) 2 . Newton's algorithm: Pick an initial point (x0, y0). For n = 0,1,2,3,..., define: xn+1 = xn − ∂2 f ∂y2 (xn, yn) ∂2 f ∂x∂y (xn, yn) ∂f ∂y (xn, yn) ∂f ∂x (xn, yn) D(xn, yn) , yn+1 = yn − ∂2 f ∂x2 (xn, yn) ∂2 f ∂x∂y (xn, yn) ∂f ∂x (xn, yn) ∂f ∂y (xn, yn) D(xn, yn) (2.14) Then the sequence of points (xn, yn)∞ n=1 converges to a critical point. If there are several critical points, then you will have to try different initial points to find them. 7This is also a problem for the equivalent method (the Second Derivative Test) in single-variable calculus, though one that is not usually emphasized. 8There are also two nonreal, complex number solutions. Cubic polynomial equations in one variable can be solved using Cardan's formulas, which are not quite as simple as the familiar quadratic formula. See USPENSKY for more details. There are formulas for solving polynomial equations of degree 4, but it can be proved that there is no general formula for solving equations for polynomials of degree five or higher. 90 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES Example 2.23. Find all local maxima and minima of f (x, y) = x3 − xy− x+ xy3 − y4 . Solution: First calculate the necessary partial derivatives: ∂f ∂x = 3x2 − y−1+ y3 , ∂f ∂y = −x+3xy2 −4y3 ∂2 f ∂x2 = 6x , ∂2 f ∂y2 = 6xy−12y2 , ∂2 f ∂y∂x = −1+3y2 Notice that solving ∇f = 0 would involve solving two third-degree polynomial equations in x and y, which in this case can not be done easily. We need to pick an initial point (x0, y0) for our algorithm. Looking at the graph of z = f (x, y) over a large region may help (see Figure 2.6.1 below), though it may be hard to tell where the critical points are. -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 -350000 -300000 -250000 -200000 -150000 -100000 -50000 0 50000 z x y z Figure 2.6.1 f (x, y) = x3 − xy− x+ xy3 − y4 for −20 ≤ x ≤ 20 and −20 ≤ y ≤ 20 Notice in the formulas (2.14) that we divide by D, so we should pick an initial point where D is not zero. And we can see that D(0,0) = (0)(0)−(−1)2 = −1 = 0, so take (0,0) as our initial point. Since it may take a large number of iterations of Newton's algorithm to be sure that we are close enough to the actual critical point, and since the computations are quite tedious, we will let a computer do the computing. For this, we will write a simple program, using the Java programming language, which will take a given initial point as a parameter and then perform 100 iterations of Newton's algorithm. In each iteration the new point will be printed, so that we can see if there is convergence. The full code is shown in Listing 2.1. 92 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES To use this program, you should first save the code in Listing 2.1 in a plain text file called newton.java. You will need the Java Development Kit9 to compile the code. In the directory where newton.java is saved, run this command at a command prompt to compile the code: javac newton.java Then run the program with the initial point (0,0) with this command: java newton 0 0 Below is the output of the program using (0,0) as the initial point, truncated to show the first 10 lines and the last 5 lines: java newton 0 0 Initial point: (0.0,0.0) n = 1: (0.0,-1.0) n = 2: (1.0,-0.5) n = 3: (0.6065857885615251,-0.44194107452339687) n = 4: (0.484506572966545,-0.405341511995805) n = 5: (0.47123972682634485,-0.3966334583092305) n = 6: (0.47113558510349535,-0.39636450001936047) n = 7: (0.4711356343449705,-0.3963643379632247) n = 8: (0.4711356343449874,-0.39636433796318005) n = 9: (0.4711356343449874,-0.39636433796318005) n = 10: (0.4711356343449874,-0.39636433796318005) ... n = 96: (0.4711356343449874,-0.39636433796318005) n = 97: (0.4711356343449874,-0.39636433796318005) n = 98: (0.4711356343449874,-0.39636433796318005) n = 99: (0.4711356343449874,-0.39636433796318005) n = 100: (0.4711356343449874,-0.39636433796318005) As you can see, we appear to have converged fairly quickly (after only 8 iterations) to what appears to be an actual critical point (up to Java's level of precision), namely the point (0.4711356343449874,−0.39636433796318005). It is easy to confirm that ∇f = 0 at this point, either by evaluating ∂f ∂x and ∂f ∂y at the point ourselves or by modifying our program to also print the values of the partial derivatives at the point. It turns out that both partial derivatives are indeed close enough to zero to be considered zero: ∂f ∂x (0.4711356343449874,−0.39636433796318005) = 4.85722573273506×10−17 ∂f ∂y (0.4711356343449874,−0.39636433796318005) = −8.326672684688674×10−17 We also have D(0.4711356343449874,−0.39636433796318005) = −8.776075636032301 < 0, so by Theorem 2.6 we know that (0.4711356343449874,−0.39636433796318005) is a saddle point. 9Available for free at 94 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES -1 -0.8 -0.6 -0.4 -0.2 0 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 z x y z (−0.67,0.42,0.57) Figure 2.6.2 f (x, y) = x3 − xy− x+ xy3 − y4 for −1 ≤ x ≤ 0 and 0 ≤ y ≤ 1 We can summarize our findings for the function f (x, y) = x3 − xy− x+ xy3 − y4 : (0.4711356343449874,−0.39636433796318005) : saddle point (−0.6703832459238667,0.42501465652420045) : local maximum (−7.540962756992551,−5.595509445899435) : saddle point The derivation of Newton's algorithm, and the proof that it converges (given a "reason- able" choice for the initial point) requires techniques beyond the scope of this text. See RALSTON and RABINOWITZ for more detail and for discussion of other numerical methods. Our description of Newton's algorithm is the special two-variable case of a more general algorithm that can be applied to functions of n ≥ 2 variables. In the case of functions which have a global maximum or minimum, Newton's algorithm can be used to find those points. In general, global maxima and minima tend to be more interesting than local versions, at least in practical applications. A maximization problem can always be turned into a minimization problem (why?), so a large number of methods have been developed to find the global minimum of functions of any number of variables. This field of study is called nonlinear programming. Many of these methods are based on the steepest descent technique, which is based on an idea that we discussed in Section 2.4. Recall that the negative gradient −∇f gives the direction of the fastest rate of decrease of a function f . The crux of the steepest descent idea, then, is that starting from some initial point, you move a certain amount in the direction of −∇f at that point. Wherever that takes you 2.6 Unconstrained Optimization: Numerical Methods 95 becomes your new point, and you then just keep repeating that procedure until eventually (hopefully) you reach the point where f has its smallest value. There is a "pure" steepest descent method, and a multitude of variations on it that improve the rate of convergence, ease of calculation, etc. In fact, Newton's algorithm can be interpreted as a modified steepest descent method. For more discussion of this, and of nonlinear programming in general, see BAZARAA, SHERALI and SHETTY. Exercises C 1. Recall Example 2.21 from the previous section, where we showed that the point (2,1) was a global minimum for the function f (x, y) = (x −2)4 +(x −2y)2 . Notice that our computer program can be modified fairly easily to use this function (just change the return values in the fx, fy, fxx, fyy and fxy function definitions to use the appropriate partial derivative). Either modify that program or write one of your own in a programming language of your choice to show that Newton's algorithm does lead to the point (2,1). First use the initial point (0,3), then use the initial point (3,2), and compare the results. Make sure that your program attempts to do 100 iterations of the algorithm. Did anything strange happen when your program ran? If so, how do you explain it? (Hint: Something strange should happen.) 2. There is a version of Newton's algorithm for solving a system of two equations f1(x, y) = 0 and f2(x, y) = 0 , where f1(x, y) and f2(x, y) are smooth real-valued functions: Pick an initial point (x0, y0). For n = 0,1,2,3,..., define: xn+1 = xn − f1(xn, yn) f2(xn, yn) ∂f1 ∂y (xn, yn) ∂f2 ∂y (xn, yn) D(xn, yn) , yn+1 = yn + f1(xn, yn) f2(xn, yn) ∂f1 ∂x (xn, yn) ∂f2 ∂x (xn, yn) D(xn, yn) , where D(xn, yn) = ∂f1 ∂x (xn, yn) ∂f2 ∂y (xn, yn)− ∂f1 ∂y (xn, yn) ∂f2 ∂x (xn, yn) . Then the sequence of points (xn, yn)∞ n=1 converges to a solution. Write a computer program that uses this algorithm to find approximate solutions to the system of equations f1(x, y) = sin(xy)− x− y = 0 and f2(x, y) = e2x −2x+3y = 0 . Show that you get two different solutions when using (0,0) and (1,1) for the initial point (x0, y0). 96 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES 2.7 Constrained Optimization: Lagrange Multipliers In Sections 2.5 and 2.6 we were concerned with finding maxima and minima of functions without any constraints on the variables (other than being in the domain of the function). What would we do if there were constraints on the variables? The following example illus- trates a simple case of this type of problem. Example 2.24. For a rectangle whose perimeter is 20 m, find the dimensions that will max- imize the area. Solution: The area A of a rectangle with width x and height y is A = xy. The perimeter P of the rectangle is then given by the formula P = 2x+2y. Since we are given that the perimeter P = 20, this problem can be stated as: Maximize : f (x, y) = xy given : 2x+2y = 20 The reader is probably familiar with a simple method, using single-variable calculus, for solving this problem. Since we must have 2x + 2y = 20, then we can solve for, say, y in terms of x using that equation. This gives y = 10− x, which we then substitute into f to get f (x, y) = xy = x(10− x) = 10x − x2 . This is now a function of x alone, so we now just have to maximize the function f (x) = 10x− x2 on the interval [0,10]. Since f ′ (x) = 10−2x = 0 ⇒ x = 5 and f ′′ (5) = −2 < 0, then the Second Derivative Test tells us that x = 5 is a local maximum for f , and hence x = 5 must be the global maximum on the interval [0,10] (since f = 0 at the endpoints of the interval). So since y = 10− x = 5, then the maximum area occurs for a rectangle whose width and height both are 5 m. Notice in the above example that the ease of the solution depended on being able to solve for one variable in terms of the other in the equation 2x+2y = 20. But what if that were not possible (which is often the case)? In this section we will use a general method, called the Lagrange multiplier method10 , for solving constrained optimization problems: Maximize (or minimize) : f (x, y) (or f (x, y, z)) given : g(x, y) = c (or g(x, y, z) = c) for some constant c The equation g(x, y) = c is called the constraint equation, and we say that x and y are con- strained by g(x, y) = c. Points (x, y) which are maxima or minima of f (x, y) with the con- dition that they satisfy the constraint equation g(x, y) = c are called constrained maximum or constrained minimum points, respectively. Similar definitions hold for functions of three variables. The Lagrange multiplier method for solving such problems can now be stated: 10Named after the French mathematician Joseph Louis Lagrange (1736-1813). 2.7 Constrained Optimization: Lagrange Multipliers 97 Theorem 2.7. Let f (x, y) and g(x, y) be smooth functions, and suppose that c is a scalar constant such that ∇g(x, y) = 0 for all (x, y) that satisfy the equation g(x, y) = c. Then to solve the constrained optimization problem Maximize (or minimize) : f (x, y) given : g(x, y) = c , find the points (x, y) that solve the equation ∇f (x, y) = λ∇g(x, y) for some constant λ (the number λ is called the Lagrange multiplier). If there is a constrained maximum or mini- mum, then it must be such a point. A rigorous proof of the above theorem requires use of the Implicit Function Theorem, which is beyond the scope of this text.11 Note that the theorem only gives a necessary con- dition for a point to be a constrained maximum or minimum. Whether a point (x, y) that satisfies ∇f (x, y) = λ∇g(x, y) for some λ actually is a constrained maximum or minimum can sometimes be determined by the nature of the problem itself. For instance, in Example 2.24 it was clear that there had to be a global maximum. So how can you tell when a point that satisfies the condition in Theorem 2.7 really is a constrained maximum or minimum? The answer is that it depends on the constraint func- tion g(x, y), together with any implicit constraints. It can be shown12 that if the constraint equation g(x, y) = c (plus any hidden constraints) describes a bounded set B in R2 , then the constrained maximum or minimum of f (x, y) will occur either at a point (x, y) satisfying ∇f (x, y) = λ∇g(x, y) or at a "boundary" point of the set B. In Example 2.24 the constraint equation 2x+2y = 20 describes a line in R2 , which by itself is not bounded. However, there are "hidden" constraints, due to the nature of the problem, namely 0 ≤ x, y ≤ 10, which cause that line to be restricted to a line segment in R2 (including the endpoints of that line segment), which is bounded. Example 2.25. For a rectangle whose perimeter is 20 m, use the Lagrange multiplier method to find the dimensions that will maximize the area. Solution: As we saw in Example 2.24, with x and y representing the width and height, respectively, of the rectangle, this problem can be stated as: Maximize : f (x, y) = xy given : g(x, y) = 2x+2y = 20 Then solving the equation ∇f (x, y) = λ∇g(x, y) for some λ means solving the equations 11See TAYLOR and MANN, § 6.8 for more detail. 12Again, see TAYLOR and MANN. 98 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES ∂f ∂x = λ ∂g ∂x and ∂f ∂y = λ ∂g ∂y , namely: y = 2λ , x = 2λ The general idea is to solve for λ in both equations, then set those expressions equal (since they both equal λ) to solve for x and y. Doing this we get y 2 = λ = x 2 ⇒ x = y , so now substitute either of the expressions for x or y into the constraint equation to solve for x and y: 20 = g(x, y) = 2x+2y = 2x+2x = 4x ⇒ x = 5 ⇒ y = 5 There must be a maximum area, since the minimum area is 0 and f (5,5) = 25 > 0, so the point (5,5) that we found (called a constrained critical point) must be the constrained maxi- mum. ∴ The maximum area occurs for a rectangle whose width and height both are 5 m. Example 2.26. Find the points on the circle x2 + y2 = 80 which are closest to and farthest from the point (1,2). Solution: The distance d from any point (x, y) to the point (1,2) is d = (x−1)2 +(y−2)2 , and minimizing the distance is equivalent to minimizing the square of the distance. Thus the problem can be stated as: Maximize (and minimize) : f (x, y) = (x−1)2 +(y−2)2 given : g(x, y) = x2 + y2 = 80 Solving ∇f (x, y) = λ∇g(x, y) means solving the following equations: 2(x−1) = 2λx , 2(y−2) = 2λy Note that x = 0 since otherwise we would get −2 = 0 in the first equation. Similarly, y = 0. So we can solve both equations for λ as follows: x−1 x = λ = y−2 y ⇒ xy− y = xy−2x ⇒ y = 2x 2.7 Constrained Optimization: Lagrange Multipliers 99 x y 0 (4,8) (1,2) (−4,−8) x2 + y2 = 80 Figure 2.7.1 Substituting this into g(x, y) = x2 + y2 = 80 yields 5x2 = 80, so x = ±4. So the two constrained critical points are (4,8) and (−4,−8). Since f (4,8) = 45 and f (−4,−8) = 125, and since there must be points on the circle closest to and farthest from (1,2), then it must be the case that (4,8) is the point on the circle clos- est to (1,2) and (−4,−8) is the farthest from (1,2) (see Figure 2.7.1). Notice that since the constraint equation x2 +y2 = 80 describes a circle, which is a bounded set in R2 , then we were guaranteed that the constrained critical points we found were indeed the constrained maximum and minimum. The Lagrange multiplier method can be extended to functions of three variables. Example 2.27. Maximize (and minimize) : f (x, y, z) = x+ z given : g(x, y, z) = x2 + y2 + z2 = 1 Solution: Solve the equation ∇f (x, y, z) = λ∇g(x, y, z): 1 = 2λx 0 = 2λy 1 = 2λz The first equation implies λ = 0 (otherwise we would have 1 = 0), so we can divide by λ in the second equation to get y = 0 and we can divide by λ in the first and third equations to get x = 1 2λ = z. Substituting these expressions into the constraint equation g(x, y, z) = x2 + y2 + z2 = 1 yields the constrained critical points 1 2 ,0, 1 2 and −1 2 ,0, −1 2 . Since f 1 2 ,0, 1 2 > f −1 2 ,0, −1 2 , and since the constraint equation x2 + y2 + z2 = 1 describes a sphere (which is bounded) in R3 , then 1 2 ,0, 1 2 is the constrained maximum point and −1 2 ,0, −1 2 is the constrained minimum point. So far we have not attached any significance to the value of the Lagrange multiplier λ. We needed λ only to find the constrained critical points, but made no use of its value. It turns out that λ gives an approximation of the change in the value of the function f (x, y) that we wish to maximize or minimize, when the constant c in the constraint equation g(x, y) = c is changed by 1. 100 CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES For example, in Example 2.25 we showed that the constrained optimization problem Maximize : f (x, y) = xy given : g(x, y) = 2x+2y = 20 had the solution (x, y) = (5,5), and that λ = x/2 = y/2. Thus, λ = 2.5. In a similar fashion we could show that the constrained optimization problem Maximize : f (x, y) = xy given : g(x, y) = 2x+2y = 21 has the solution (x, y) = (5.25,5.25). So we see that the value of f (x, y) at the constrained maximum increased from f (5,5) = 25 to f (5.25,5.25) = 27.5625, i.e. it increased by 2.5625 when we increased the value of c in the constraint equation g(x, y) = c from c = 20 to c = 21. Notice that λ = 2.5 is close to 2.5625, that is, λ ≈ ∆f = f (new max. pt)− f (old max. pt) . Finally, note that solving the equation ∇f (x, y) = λ∇g(x, y) means having to solve a system of two (possibly nonlinear) equations in three unknowns, which as we have seen before, may not be possible to do. And the 3-variable case can get even more complicated. All of this somewhat restricts the usefulness of Lagrange's method to relatively simple functions. Luckily there are many numerical methods for solving constrained optimization problems, though we will not discuss them here.13 Exercises A 1. Find the constrained maxima and minima of f (x, y) = 2x+ y given that x2 + y2 = 4. 2. Find the constrained maxima and minima of f (x, y) = xy given that x2 +3y2 = 6. 3. Find the points on the circle x2 + y2 = 100 which are closest to and farthest from the point (2,3). B 4. Find the constrained maxima and minima of f (x, y, z) = x+ y2 +2z given that 4x2 +9y2 − 36z2 = 36. 5. Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid x2 a2 + y2 b2 + z2 c2 = 1 . 13See BAZARAA, SHERALI and SHETTY. 3 Multiple Integrals 3.1 Double Integrals In single-variable calculus, differentiation and integration are thought of as inverse opera- tions. For instance, to integrate a function f (x) it is necessary to find the antiderivative of f , that is, another function F(x) whose derivative is f (x). Is there a similar way of defining in- tegration of real-valued functions of two or more variables? The answer is yes, as we will see shortly. Recall also that the definite integral of a nonnegative function f (x) ≥ 0 represented the area "under" the curve y = f (x). As we will now see, the double integral of a nonnegative real-valued function f (x, y) ≥ 0 represents the volume "under" the surface z = f (x, y). Let f (x, y) be a continuous function such that f (x, y) ≥ 0 for all (x, y) on the rectangle R = {(x, y) : a ≤ x ≤ b, c ≤ y ≤ d} in R2 . We will often write this as R = [a,b]×[c,d]. For any number x∗ in the interval [a,b], slice the surface z = f (x, y) with the plane x = x∗ parallel to the yz-plane. Then the trace of the surface in that plane is the curve f (x∗, y), where x∗ is fixed and only y varies. The area A under that curve (i.e. the area of the region between the curve and the xy-plane) as y varies over the interval [c,d] then depends only on the value of x∗. So using the variable x instead of x∗, let A(x) be that area (see Figure 3.1.1). y z x 0 A(x) R a x b c d z = f (x, y) Figure 3.1.1 The area A(x) varies with x Then A(x) = d c f (x, y)dy since we are treating x as fixed, and only y varies. This makes sense since for a fixed x the function f (x, y) is a continuous function of y over the interval [c,d], so we know that the area under the curve is the definite integral. The area A(x) is a function of x, so by the "slice" or cross-section method from single-variable calculus we know that the volume V of the solid under the surface z = f (x, y) but above the xy-plane over the 101 102 CHAPTER 3. MULTIPLE INTEGRALS rectangle R is the integral over [a,b] of that cross-sectional area A(x): V = b a A(x)dx = b a d c f (x, y)dy dx (3.1) We will always refer to this volume as "the volume under the surface". The above expression uses what are called iterated integrals. First the function f (x, y) is integrated as a func- tion of y, treating the variable x as a constant (this is called integrating with respect to y). That is what occurs in the "inner" integral between the square brackets in equation (3.1). This is the first iterated integral. Once that integration is performed, the result is then an expression involving only x, which can then be integrated with respect to x. That is what occurs in the "outer" integral above (the second iterated integral). The final result is then a number (the volume). This process of going through two iterations of integrals is called double integration, and the last expression in equation (3.1) is called a double integral. Notice that integrating f (x, y) with respect to y is the inverse operation of taking the partial derivative of f (x, y) with respect to y. Also, we could just as easily have taken the area of cross-sections under the surface which were parallel to the xz-plane, which would then depend only on the variable y, so that the volume V would be V = d c b a f (x, y)dx dy . (3.2) It turns out that in general1 the order of the iterated integrals does not matter. Also, we will usually discard the brackets and simply write V = d c b a f (x, y)dxdy , (3.3) where it is understood that the fact that dx is written before dy means that the function f (x, y) is first integrated with respect to x using the "inner" limits of integration a and b, and then the resulting function is integrated with respect to y using the "outer" limits of integration c and d. This order of integration can be changed if it is more convenient. Example 3.1. Find the volume V under the plane z = 8x+6y over the rectangle R = [0,1]× [0,2]. 1due to Fubini's Theorem. See Ch. 18 in TAYLOR and MANN. 3.2 Double Integrals Over a General Region 105 3.2 Double Integrals Over a General Region In the previous section we got an idea of what a double integral over a rectangle represents. We can now define the double integral of a real-valued function f (x, y) over more general regions in R2 . Suppose that we have a region R in the xy-plane that is bounded on the left by the vertical line x = a, bounded on the right by the vertical line x = b (where a < b), bounded below by a curve y = g1(x), and bounded above by a curve y = g2(x), as in Figure 3.2.1(a). We will assume that g1(x) and g2(x) do not intersect on the open interval (a,b) (they could intersect at the endpoints x = a and x = b, though). a b x y 0 y = g2(x) y = g1(x) R (a) Vertical slice: b a g2(x) g1(x) f (x, y)dydx x y 0 x = h1(y) x = h2(y) R c d (b) Horizontal slice: d c h2(y) h1(y) f (x, y)dxdy Figure 3.2.1 Double integral over a nonrectangular region R Then using the slice method from the previous section, the double integral of a real-valued function f (x, y) over the region R, denoted by R f (x, y)dA, is given by R f (x, y)dA = b a g2(x) g1(x) f (x, y)dy dx (3.4) This means that we take vertical slices in the region R between the curves y = g1(x) and y = g2(x). The symbol dA is sometimes called an area element or infinitesimal, with the A signifying area. Note that f (x, y) is first integrated with respect to y, with functions of x as the limits of integration. This makes sense since the result of the first iterated integral will have to be a function of x alone, which then allows us to take the second iterated integral with respect to x. Similarly, if we have a region R in the xy-plane that is bounded on the left by a curve x = h1(y), bounded on the right by a curve x = h2(y), bounded below by the horizontal line 108 CHAPTER 3. MULTIPLE INTEGRALS ∆xi ∆yj is the base area of a parallelepiped, as shown in Figure 3.2.5(b). Then the total vol- ume under the surface is approximately the sum of the volumes of all such parallelepipeds, namely j i f (xi∗, yj∗)∆xi ∆yj , (3.6) where the summation occurs over the indices of the subrectangles inside R. If we take smaller and smaller subrectangles, so that the length of the largest diagonal of the subrect- angles goes to 0, then the subrectangles begin to fill more and more of the region R, and so the above sum approaches the actual volume under the surface z = f (x, y) over the region R. We then define R f (x, y)dA as the limit of that double summation (the limit is taken over all subdivisions of the rectangle [a,b]×[c,d] as the largest diagonal of the subrectangles goes to 0). a bxi xi+1 x y 0 d c yj yj+1 (xi∗, yj∗) (a) Subrectangles inside the region R y z x 0 R xi xi+1 yj yj+1 z = f (x, y) ∆yj ∆xi (xi∗, yj∗) f (xi∗, yj∗) (b) Parallelepiped over a subrectangle, with volume f (xi∗, yj∗)∆xi ∆yj Figure 3.2.5 Double integral over a general region R A similar definition can be made for a function f (x, y) that is not necessarily always non- negative: just replace each mention of volume by the negative volume in the description above when f (x, y) < 0. In the case of a region of the type shown in Figure 3.2.1, using the def- inition of the Riemann integral from single-variable calculus, our definition of R f (x, y)dA reduces to a sequence of two iterated integrals. Finally, the region R does not have to be bounded. We can evaluate improper double integrals (i.e. over an unbounded region, or over a region which contains points where the function f (x, y) is not defined) as a sequence of iterated improper single-variable integrals. 110 CHAPTER 3. MULTIPLE INTEGRALS 3.3 Triple Integrals Our definition of a double integral of a real-valued function f (x, y) over a region R in R2 can be extended to define a triple integral of a real-valued function f (x, y, z) over a solid S in R3 . We simply proceed as before: the solid S can be enclosed in some rectangular parallelepiped, which is then divided into subparallelepipeds. In each subparallelepiped inside S, with sides of lengths ∆x, ∆y and ∆z, pick a point (x∗, y∗, z∗). Then define the triple integral of f (x, y, z) over S, denoted by S f (x, y, z)dV, by S f (x, y, z)dV = lim f (x∗, y∗, z∗)∆x∆y∆z , (3.7) where the limit is over all divisions of the rectangular parallelepiped enclosing S into sub- parallelepipeds whose largest diagonal is going to 0, and the triple summation is over all the subparallelepipeds inside S. It can be shown that this limit does not depend on the choice of the rectangular parallelepiped enclosing S. The symbol dV is often called the volume element. Physically, what does the triple integral represent? We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out that the triple integral simply generalizes this idea: it can be thought of as representing the hypervolume under a three-dimensional hypersurface w = f (x, y, z) whose graph lies in R4 . In general, the word "volume" is often used as a general term to signify the same concept for any n- dimensional object (e.g. length in R1 , area in R2 ). It may be hard to get a grasp on the concept of the "volume" of a four-dimensional object, but at least we now know how to calculate that volume! In the case where S is a rectangular parallelepiped [x1,x2] × [y1, y2] × [z1, z2], that is, S = {(x, y, z) : x1 ≤ x ≤ x2, y1 ≤ y ≤ y2, z1 ≤ z ≤ z2}, the triple integral is a sequence of three iterated integrals, namely S f (x, y, z)dV = z2 z1 y2 y1 x2 x1 f (x, y, z)dxdydz , (3.8) where the order of integration does not matter. This is the simplest case. A more complicated case is where S is a solid which is bounded below by a surface z = g1(x, y), bounded above by a surface z = g2(x, y), y is bounded between two curves h1(x) and h2(x), and x varies between a and b. Then S f (x, y, z)dV = b a h2(x) h1(x) g2(x,y) g1(x,y) f (x, y, z)dz dydx . (3.9) Notice in this case that the first iterated integral will result in a function of x and y (since its limits of integration are functions of x and y), which then leaves you with a double integral of 3.4 Numerical Approximation of Multiple Integrals 113 3.4 Numerical Approximation of Multiple Integrals As you have seen, calculating multiple integrals is tricky even for simple functions and regions. For complicated functions, it may not be possible to evaluate one of the iterated in- tegrals in a simple closed form. Luckily there are numerical methods for approximating the value of a multiple integral. The method we will discuss is called the Monte Carlo method. The idea behind it is based on the concept of the average value of a function, which you learned in single-variable calculus. Recall that for a continuous function f (x), the average value ¯f of f over an interval [a,b] is defined as ¯f = 1 b − a b a f (x)dx . (3.11) The quantity b − a is the length of the interval [a,b], which can be thought of as the "volume" of the interval. Applying the same reasoning to functions of two or three variables, we define the average value of f (x, y) over a region R to be ¯f = 1 A(R) R f (x, y)dA , (3.12) where A(R) is the area of the region R, and we define the average value of f (x, y, z) over a solid S to be ¯f = 1 V(S) S f (x, y, z)dV , (3.13) where V(S) is the volume of the solid S. Thus, for example, we have R f (x, y)dA = A(R) ¯f . (3.14) The average value of f (x, y) over R can be thought of as representing the sum of all the values of f divided by the number of points in R. Unfortunately there are an infinite number (in fact, uncountably many) points in any region, i.e. they can not be listed in a discrete sequence. But what if we took a very large number N of random points in the region R (which can be generated by a computer) and then took the average of the values of f for those points, and used that average as the value of ¯f ? This is exactly what the Monte Carlo method does. So in formula (3.14) the approximation we get is R f (x, y)dA ≈ A(R) ¯f ± A(R) f 2 −( ¯f )2 N , (3.15) where ¯f = N i=1 f (xi, yi) N and f 2 = N i=1(f (xi, yi))2 N , (3.16) 114 CHAPTER 3. MULTIPLE INTEGRALS with the sums taken over the N random points (x1, y1), ..., (xN , yN ). The ± "error term" in formula (3.15) does not really provide hard bounds on the approximation. It represents a single standard deviation from the expected value of the integral. That is, it provides a likely bound on the error. Due to its use of random points, the Monte Carlo method is an example of a probabilistic method (as opposed to deterministic methods such as Newton's method, which use a specific formula for generating points). For example, we can use formula (3.15) to approximate the volume V under the plane z = 8x + 6y over the rectangle R = [0,1] × [0,2]. In Example 3.1 in Section 3.1, we showed that the actual volume is 20. Below is a code listing (montecarlo.java) for a Java program that calculates the volume, using a number of points N that is passed on the command line as a parameter. //Program to approximate the double integral of f(x,y)=8x+6y //over the rectangle [0,1]x[0,2]. public class montecarlo " +/- " + vol()Math.sqrt((mf2 - Math.pow(mf,2))/N)); //Print the result } //The volume of the rectangle [0,1]x[0,2] public static double vol() { return 12; } } Listing 3.1 Program listing for montecarlo.java The results of running this program with various numbers of random points (e.g. java montecarlo 100) are shown below: 3.4 Numerical Approximation of Multiple Integrals 115 N = 10: 19.36543087722646 +/- 2.7346060413546147 N = 100: 21.334419561385353 +/- 0.7547037194998519 N = 1000: 19.807662237526227 +/- 0.26701709691370235 N = 10000: 20.080975812043256 +/- 0.08378816229769506 N = 100000: 20.009403854556716 +/- 0.026346782289498317 N = 1000000: 20.000866994982314 +/- 0.008321168748642816 As you can see, the approximation is fairly good. As N → ∞, it can be shown that the Monte Carlo approximation converges to the actual volume (on the order of O( N), in com- putational complexity terminology). In the above example the region R was a rectangle. To use the Monte Carlo method for a nonrectangular (bounded) region R, only a slight modification is needed. Pick a rectangle ˜R that encloses R, and generate random points in that rectangle as before. Then use those points in the calculation of ¯f only if they are inside R. There is no need to calculate the area of R for formula (3.15) in this case, since the exclusion of points not inside R allows you to use the area of the rectangle ˜R instead, similar to before. For instance, in Example 3.4 we showed that the volume under the surface z = 8x + 6y over the nonrectangular region R = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2x2 } is 6.4. Since the rectangle ˜R = [0,1]× [0,2] contains R, we can use the same program as before, with the only change being a check to see if y < 2x2 for a random point (x, y) in [0,1] × [0,2]. Listing 3.2 below contains the code (montecarlo2.java): //Program to approximate the double integral of f(x,y)=8x+6y over the //region bounded by x=0, x=1, y=0, and y=2x^2 public class montecarlo2if (y < 2Math.pow(x,2)) { //The point is in the region 3.5 Change of Variables in Multiple Integrals 117 3.5 Change of Variables in Multiple Integrals Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The an- swer is yes, though it is a bit more complicated than the substitution method which you learned in single-variable calculus. Recall that if you are given, for example, the definite integral 2 1 x3 x2 −1dx , then you would make the substitution u = x2 −1 ⇒ x2 = u +1 du = 2xdx which changes the limits of integration x = 1 ⇒ u = 0 x = 2 ⇒ u = 3 so that we get 2 1 x3 x2 −1dx = 2 1 1 2 x2 ·2x x2 −1dx = 3 0 1 2 (u +1) u du = 1 2 3 0 u3/2 + u1/2 du , which can be easily integrated to give = 14 3 5 . Let us take a different look at what happened when we did that substitution, which will give some motivation for how substitution works in multiple integrals. First, we let u = x2 − 1. On the interval of integration [1,2], the function x → x2 −1 is strictly increasing (and maps [1,2] onto [0,3]) and hence has an inverse function (defined on the interval [0,3]). That is, on [0,3] we can define x as a function of u, namely x = g(u) = u +1 . Then substituting that expression for x into the function f (x) = x3 x2 −1 gives f (x) = f (g(u)) = (u +1)3/2 u , 118 CHAPTER 3. MULTIPLE INTEGRALS and we see that dx du = g′ (u) ⇒ dx = g′ (u)du dx = 1 2 (u +1)−1/2 du , so since g(0) = 1 ⇒ 0 = g−1 (1) g(3) = 2 ⇒ 3 = g−1 (2) then performing the substitution as we did earlier gives 2 1 f (x)dx = 2 1 x3 x2 −1dx = 3 0 1 2 (u +1) u du , which can be written as = 3 0 (u +1)3/2 u · 1 2 (u +1)−1/2 du , which means 2 1 f (x)dx = g−1 (2) g−1(1) f (g(u)) g′ (u)du . In general, if x = g(u) is a one-to-one, differentiable function from an interval [c,d] (which you can think of as being on the "u-axis") onto an interval [a,b] (on the x-axis), which means that g′ (u) = 0 on the interval (c,d), so that a = g(c) and b = g(d), then c = g−1 (a) and d = g−1 (b), and b a f (x)dx = g−1 (b) g−1(a) f (g(u)) g′ (u)du . (3.17) This is called the change of variable formula for integrals of single-variable functions, and it is what you were implicitly using when doing integration by substitution. This formula turns out to be a special case of a more general formula which can be used to evaluate multiple integrals. We will state the formulas for double and triple integrals involving real-valued functions of two and three variables, respectively. We will assume that all the functions involved are continuously differentiable and that the regions and solids involved all have "reasonable" boundaries. The proof of the following theorem is beyond the scope of the text.2 2See TAYLOR and MANN, § 15.32 and § 15.62 for all the details. 3.5 Change of Variables in Multiple Integrals 119 Theorem 3.1. Change of Variables Formula for Multiple Integrals Let x = x(u,v) and y = y(u,v) define a one-to-one mapping of a region R′ in the uv-plane onto a region R in the xy-plane such that the determinant J(u,v) = ∂x ∂u ∂x ∂v ∂y ∂u ∂y ∂v (3.18) is never 0 in R′ . Then R f (x, y)dA(x, y) = R′ f (x(u,v), y(u,v))\|J(u,v)\|dA(u,v) . (3.19) We use the notation dA(x, y) and dA(u,v) to denote the area element in the (x, y) and (u,v) coordinates, respectively. Similarly, if x = x(u,v,w), y = y(u,v,w) and z = z(u,v,w) define a one-to-one mapping of a solid S′ in uvw-space onto a solid S in xyz-space such that the determinant J(u,v,w) = ∂x ∂u ∂x ∂v ∂x ∂w ∂y ∂u ∂y ∂v ∂y ∂w ∂z ∂u ∂z ∂v ∂z ∂w (3.20) is never 0 in S′ , then S f (x, y, z)dV(x, y, z) = S′ f (x(u,v,w), y(u,v,w), z(u,v,w))\|J(u,v,w)\|dV(u,v,w) . (3.21) The determinant J(u,v) in formula (3.18) is called the Jacobian of x and y with respect to u and v, and is sometimes written as J(u,v) = ∂(x, y) ∂(u,v) . (3.22) Similarly, the Jacobian J(u,v,w) of three variables is sometimes written as J(u,v,w) = ∂(x, y, z) ∂(u,v,w) . (3.23) Notice that formula (3.19) is saying that dA(x, y) = \|J(u,v)\|dA(u,v), which you can think of as a two-variable version of the relation dx = g′ (u)du in the single-variable case. The following example shows how the change of variables formula is used. 124 CHAPTER 3. MULTIPLE INTEGRALS 3.6 Application: Center of Mass a b x y 0 y = f (x) R (¯x, ¯y) Figure 3.6.1 Center of mass of R Recall from single-variable calculus that for a region R = {(x, y) : a ≤ x ≤ b,0 ≤ y ≤ f (x)} in R2 that represents a thin, flat plate (see Figure 3.6.1), where f (x) is a con- tinuous function on [a,b], the center of mass of R has coordinates (¯x, ¯y) given by ¯x = My M and ¯y = Mx M , where Mx = b a (f (x))2 2 dx , My = b a xf (x)dx , M = b a f (x)dx , (3.27) assuming that R has uniform density, i.e the mass of R is uniformly distributed over the region. In this case the area M of the region is considered the mass of R (the density is constant, and taken as 1 for simplicity). In the general case where the density of a region (or lamina) R is a continuous function δ = δ(x, y) of the coordinates (x, y) of points inside R (where R can be any region in R2 ) the coordinates (¯x, ¯y) of the center of mass of R are given by ¯x = My M and ¯y = Mx M , (3.28) where My = R xδ(x, y)dA , Mx = R yδ(x, y)dA , M = R δ(x, y)dA , (3.29) The quantities Mx and My are called the moments (or first moments) of the region R about the x-axis and y-axis, respectively. The quantity M is the mass of the region R. To see this, think of taking a small rectangle inside R with dimensions ∆x and ∆y close to 0. The mass of that rectangle is approximately δ(x∗, y∗)∆x∆y, for some point (x∗, y∗) in that rectangle. Then the mass of R is the limit of the sums of the masses of all such rectangles inside R as the diagonals of the rectangles approach 0, which is the double integral R δ(x, y)dA. Note that the formulas in (3.27) represent a special case when δ(x, y) = 1 throughout R in the formulas in (3.29). Example 3.13. Find the center of mass of the region R = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2x2 }, if the density function at (x, y) is δ(x, y) = x+ y. 128 CHAPTER 3. MULTIPLE INTEGRALS 3.7 Application: Probability and Expected Value In this section we will briefly discuss some applications of multiple integrals in the field of probability theory. In particular we will see ways in which multiple integrals can be used to calculate probabilities and expected values. Probability Suppose that you have a standard six-sided (fair) die, and you let a variable X represent the value rolled. Then the probability of rolling a 3, written as P(X = 3), is 1 6 , since there are six sides on the die and each one is equally likely to be rolled, and hence in particular the 3 has a one out of six chance of being rolled. Likewise the probability of rolling at most a 3, written as P(X ≤ 3), is 3 6 = 1 2 , since of the six numbers on the die, there are three equally likely numbers (1, 2, and 3) that are less than or equal to 3. Note that P(X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3). We call X a discrete random variable on the sample space (or probability space) Ω consisting of all possible outcomes. In our case, Ω = {1,2,3,4,5,6}. An event A is a subset of the sample space. For example, in the case of the die, the event X ≤ 3 is the set {1,2,3}. Now let X be a variable representing a random real number in the interval (0,1). Note that the set of all real numbers between 0 and 1 is not a discrete (or countable) set of values, i.e. it can not be put into a one-to-one correspondence with the set of positive integers.3 In this case, for any real number x in (0,1), it makes no sense to consider P(X = x) since it must be 0 (why?). Instead, we consider the probability P(X ≤ x), which is given by P(X ≤ x) = x. The reasoning is this: the interval (0,1) has length 1, and for x in (0,1) the interval (0,x) has length x. So since X represents a random number in (0,1), and hence is uniformly distributed over (0,1), then P(X ≤ x) = length of (0,x) length of (0,1) = x 1 = x . We call X a continuous random variable on the sample space Ω = (0,1). An event A is a subset of the sample space. For example, in our case the event X ≤ x is the set (0,x). In the case of a discrete random variable, we saw how the probability of an event was the sum of the probabilities of the individual outcomes comprising that event (e.g. P(X ≤ 3) = P(X = 1)+ P(X = 2)+ P(X = 3) in the die example). For a continuous random variable, the probability of an event will instead be the integral of a function, which we will now describe. Let X be a continuous real-valued random variable on a sample space Ω in R. For sim- 3For a proof see p. 9-10 in KAMKE, E., Theory of Sets, New York: Dover, 1950. 134 CHAPTER 3. MULTIPLE INTEGRALS and similarly (see Exercise 3) it can be shown that EY = 1 0 y 0 n(n−1)y(y− x)n−2 dxdy = n n+1 . So, for example, if you were to repeatedly take samples of n = 3 random real numbers from (0,1), and each time store the minimum and maximum values in the sample, then the aver- age of the minimums would approach 1 4 and the average of the maximums would approach 3 4 as the number of samples grows. It would be relatively simple (see Exercise 4) to write a computer program to test this. Exercises B 1. Evaluate the integral ∞ −∞ e−x2 dx using anything you have learned so far. 2. For σ > 0 and µ > 0, evaluate ∞ −∞ 1 σ 2π e−(x−µ)2 /2σ2 dx. 3. Show that EY = n n+1 in Example 3.18 C 4. Write a computer program (in the language of your choice) that verifies the results in Example 3.18 for the case n = 3 by taking large numbers of samples. 5. Repeat Exercise 4 for the case when n = 4. 6. For continuous random variables X, Y with joint p.d.f. f (x, y), define the second moments E(X2 ) and E(Y 2 ) by E(X2 ) = ∞ −∞ ∞ −∞ x2 f (x, y)dxdy and E(Y 2 ) = ∞ −∞ ∞ −∞ y2 f (x, y)dxdy , and the variances Var(X) and Var(Y ) by Var(X) = E(X2 )−(EX)2 and Var(Y ) = E(Y 2 )−(EY )2 . Find Var(X) and Var(Y ) for X and Y as in Example 3.18. 7. Continuing Exercise 6, the correlation ρ between X and Y is defined as ρ = E(XY )−(EX)(EY ) Var(X)Var(Y ) , where E(XY ) = ∞ −∞ ∞ −∞ xy f (x, y)dxdy. Find ρ for X and Y as in Example 3.18. (Note: The quantity E(XY )−(EX)(EY ) is called the covariance of X and Y .) 8. In Example 3.17 would the answer change if the interval (0,100) is used instead of (0,1)? Explain. 4 Line and Surface Integrals 4.1 Line Integrals In single-variable calculus you learned how to integrate a real-valued function f (x) over an interval [a,b] in R1 . This integral (usually called a Riemann integral) can be thought of as an integral over a path in R1 , since an interval (or collection of intervals) is really the only kind of "path" in R1 . You may also recall that if f (x) represented the force applied along the x-axis to an object at position x in [a,b], then the work W done in moving that object from position x = a to x = b was defined as the integral: W = b a f (x)dx In this section, we will see how to define the integral of a function (either real-valued or vector-valued) of two variables over a general path (i.e. a curve) in R2 . This definition will be motivated by the physical notion of work. We will begin with real-valued functions of two variables. In physics, the intuitive idea of work is that Work = Force × Distance . Suppose that we want to find the total amount W of work done in moving an object along a curve C in R2 with a smooth parametrization x = x(t), y = y(t), a ≤ t ≤ b, with a force f (x, y) which varies with the position (x, y) of the object and is applied in the direction of motion along C (see Figure 4.1.1 below). x y 0 C t = a t = b ∆si ≈ ∆xi 2 +∆yi 2 t = ti t = ti+1 ∆yi ∆xi Figure 4.1.1 Curve C : x = x(t), y = y(t) for t in [a,b] We will assume for now that the function f (x, y) is continuous and real-valued, so we only consider the magnitude of the force. Partition the interval [a,b] as follows: a = t0 < t1 < t2 < ··· < tn−1 < tn = b , for some integer n ≥ 2 135 136 CHAPTER 4. LINE AND SURFACE INTEGRALS As we can see from Figure 4.1.1, over a typical subinterval [ti,ti+1] the distance ∆si traveled along the curve is approximately ∆xi 2 +∆yi 2 , by the Pythagorean Theorem. Thus, if the subinterval is small enough then the work done in moving the object along that piece of the curve is approximately Force × Distance ≈ f (xi∗, yi∗) ∆xi 2 +∆yi 2 , (4.1) where (xi∗, yi∗) = (x(ti∗), y(ti∗)) for some ti∗ in [ti,ti+1], and so W ≈ n−1 i=0 f (xi∗, yi∗) ∆xi 2 +∆yi 2 (4.2) is approximately the total amount of work done over the entire curve. But since ∆xi 2 +∆yi 2 = ∆xi ∆ti 2 + ∆yi ∆ti 2 ∆ti , where ∆ti = ti+1 − ti, then W ≈ n−1 i=0 f (xi∗, yi∗) ∆xi ∆ti 2 + ∆yi ∆ti 2 ∆ti . (4.3) Taking the limit of that sum as the length of the largest subinterval goes to 0, the sum over all subintervals becomes the integral from t = a to t = b, ∆xi ∆ti and ∆yi ∆ti become x′ (t) and y′ (t), respectively, and f (xi∗, yi∗) becomes f (x(t), y(t)), so that W = b a f (x(t), y(t)) x′(t)2 + y′(t)2 dt . (4.4) The integral on the right side of the above equation gives us our idea of how to define, for any real-valued function f (x, y), the integral of f (x, y) along the curve C, called a line integral: Definition 4.1. For a real-valued function f (x, y) and a curve C in R2 , parametrized by x = x(t), y = y(t), a ≤ t ≤ b, the line integral of f (x, y) along C with respect to arc length s is C f (x, y)ds = b a f (x(t), y(t)) x′(t)2 + y′(t)2 dt . (4.5) The symbol ds is the differential of the arc length function s = s(t) = t a x′(u)2 + y′(u)2 du , (4.6) 4.1 Line Integrals 137 which you may recognize from Section 1.9 as the length of the curve C over the interval [a,t], for all t in [a,b]. That is, ds = s′ (t)dt = x′(t)2 + y′(t)2 dt , (4.7) by the Fundamental Theorem of Calculus. For a general real-valued function f (x, y), what does the line integral C f (x, y)ds rep- resent? The preceding discussion of ds gives us a clue. You can think of differentials as infinitesimal lengths. So if you think of f (x, y) as the height of a picket fence along C, then f (x, y)ds can be thought of as approximately the area of a section of that fence over some infinitesimally small section of the curve, and thus the line integral C f (x, y)ds is the total area of that picket fence (see Figure 4.1.2). x y 0 C ds f (x, y) Figure 4.1.2 Area of shaded rectangle = height×width ≈ f (x, y)ds Example 4.1. Use a line integral to show that the lateral surface area A of a right circular cylinder of radius r and height h is 2πrh. y z x 0 r h = f (x, y) C : x2 + y2 = r2 Figure 4.1.3 Solution: We will use the right circular cylinder with base circle C given by x2 + y2 = r2 and with height h in the positive z direction (see Figure 4.1.3). Parametrize C as follows: x = x(t) = rcost , y = y(t) = rsint , 0 ≤ t ≤ 2π Let f (x, y) = h for all (x, y). Then A = C f (x, y)ds = b a f (x(t), y(t)) x′(t)2 + y′(t)2 dt = 2π 0 h (−rsint)2 +(rcost)2 dt = h 2π 0 r sin2 t+cos2 t dt = rh 2π 0 1dt = 2πrh 138 CHAPTER 4. LINE AND SURFACE INTEGRALS Note in Example 4.1 that if we had traversed the circle C twice, i.e. let t vary from 0 to 4π, then we would have gotten an area of 4πrh, i.e. twice the desired area, even though the curve itself is still the same (namely, a circle of radius r). Also, notice that we traversed the circle in the counter-clockwise direction. If we had gone in the clockwise direction, using the parametrization x = x(t) = rcos(2π− t) , y = y(t) = rsin(2π− t) , 0 ≤ t ≤ 2π , (4.8) then it is easy to verify (see Exercise 12) that the value of the line integral is unchanged. In general, it can be shown (see Exercise 15) that reversing the direction in which a curve C is traversed leaves C f (x, y)ds unchanged, for any f (x, y). If a curve C has a parametriza- tion x = x(t), y = y(t), a ≤ t ≤ b, then denote by −C the same curve as C but traversed in the opposite direction. Then −C is parametrized by x = x(a+ b − t) , y = y(a+ b − t) , a ≤ t ≤ b , (4.9) and we have C f (x, y)ds = −C f (x, y)ds . (4.10) Notice that our definition of the line integral was with respect to the arc length parameter s. We can also define C f (x, y)dx = b a f (x(t), y(t))x′ (t)dt (4.11) as the line integral of f (x, y) along C with respect to x, and C f (x, y)dy = b a f (x(t), y(t)) y′ (t)dt (4.12) as the line integral of f (x, y) along C with respect to y. In the derivation of the formula for a line integral, we used the idea of work as force multiplied by distance. However, we know that force is actually a vector. So it would be helpful to develop a vector form for a line integral. For this, suppose that we have a function f(x, y) defined on R2 by f(x, y) = P(x, y)i + Q(x, y)j for some continuous real-valued functions P(x, y) and Q(x, y) on R2 . Such a function f is called a vector field on R2 . It is defined at points in R2 , and its values are vectors in R2 . For a curve C with a smooth parametrization x = x(t), y = y(t), a ≤ t ≤ b, let r(t) = x(t)i + y(t)j 4.1 Line Integrals 139 be the position vector for a point (x(t), y(t)) on C. Then r′ (t) = x′ (t)i+ y′ (t)j and so C P(x, y)dx + C Q(x, y)dy = b a P(x(t), y(t))x′ (t)dt + b a Q(x(t), y(t)) y′ (t)dt = b a (P(x(t), y(t))x′ (t)+Q(x(t), y(t)) y′ (t))dt = b a f(x(t), y(t))···r′ (t)dt by definition of f(x, y). Notice that the function f(x(t), y(t))···r′ (t) is a real-valued function on [a,b], so the last integral on the right looks somewhat similar to our earlier definition of a line integral. This leads us to the following definition: Definition 4.2. For a vector field f(x, y) = P(x, y)i + Q(x, y)j and a curve C with a smooth parametrization x = x(t), y = y(t), a ≤ t ≤ b, the line integral of f along C is C f··· dr = C P(x, y)dx + C Q(x, y)dy (4.13) = b a f(x(t), y(t))···r′ (t)dt , (4.14) where r(t) = x(t)i+ y(t)j is the position vector for points on C. We use the notation dr = r′ (t)dt = dxi+ dyj to denote the differential of the vector-valued function r. The line integral in Definition 4.2 is often called a line integral of a vector field to distinguish it from the line integral in Definition 4.1 which is called a line integral of a scalar field. For convenience we will often write C P(x, y)dx + C Q(x, y)dy = C P(x, y)dx+Q(x, y)dy , where it is understood that the line integral along C is being applied to both P and Q. The quantity P(x, y)dx+Q(x, y)dy is known as a differential form. For a real-valued function F(x, y), the differential of F is dF = ∂F ∂x dx+ ∂F ∂y dy. A differential form P(x, y)dx+Q(x, y)dy is called exact if it equals dF for some function F(x, y). Recall that if the points on a curve C have position vector r(t) = x(t)i+ y(t)j, then r′ (t) is a tangent vector to C at the point (x(t), y(t)) in the direction of increasing t (which we call the direction of C). Since C is a smooth curve, then r′ (t) = 0 on [a,b] and hence T(t) = r′ (t) r′(t) is the unit tangent vector to C at (x(t), y(t)). Putting Definitions 4.1 and 4.2 together we get the following theorem: 4.1 Line Integrals 141 So in both cases, if the vector field f(x, y) = (x2 + y2 )i+2xyj represents the force moving an object from (0,0) to (1,2) along the given curve C, then the work done is 13 3 . This may lead you to think that work (and more generally, the line integral of a vector field) is independent of the path taken. However, as we will see in the next section, this is not always the case. Although we defined line integrals over a single smooth curve, if C is a piecewise smooth curve, that is C = C1 ∪C2 ∪...∪Cn is the union of smooth curves C1,...,Cn, then we can define C f··· dr = C1 f··· dr1 + C2 f··· dr2 +...+ Cn f··· drn where each ri is the position vector of the curve Ci. Example 4.3. Evaluate C(x2 + y2 )dx+2xydy, where C is the polygonal path from (0,0) to (0,2) to (1,2). x y 0 (1,2)2 1 C1 C2 Figure 4.1.5 Solution: Write C = C1 ∪ C2, where C1 is the curve given by x = 0, y = t, 0 ≤ t ≤ 2 and C2 is the curve given by x = t, y = 2, 0 ≤ t ≤ 1 (see Figure 4.1.5). Then C (x2 + y2 )dx+2xydy = C1 (x2 + y2 )dx+2xydy + C2 (x2 + y2 )dx+2xydy = 2 0 (02 + t2 )(0)+2(0)t(1) dt + 1 0 (t2 +4)(1)+2t(2)(0) dt = 2 0 0dt+ 1 0 (t2 +4)dt = t3 3 +4t 1 0 = 1 3 +4 = 13 3 Line integral notation varies quite a bit. For example, in physics it is common to see the notation b a f ··· dl, where it is understood that the limits of integration a and b are for the underlying parameter t of the curve, and the letter l signifies length. Also, the formulation C f ··· Tds from Theorem 4.1 is often preferred in physics since it emphasizes the idea of integrating the tangential component f···T of f in the direction of T (i.e. in the direction of C), which is a useful physical interpretation of line integrals. 144 CHAPTER 4. LINE AND SURFACE INTEGRALS The above formula can be interpreted in terms of the work done by a force f(x, y) (treated as a vector) moving an object along a curve C: the total work performed moving the object along C from its initial point to its terminal point, and then back to the initial point moving backwards along the same path, is zero. This is because when force is considered as a vector, direction is accounted for. The preceding discussion shows the importance of always taking the direction of the curve into account when using line integrals of vector fields. For this reason, the curves in line integrals are sometimes referred to as directed curves or oriented curves. Recall that our definition of a line integral required that we have a parametrization x = x(t), y = y(t), a ≤ t ≤ b for the curve C. But as we know, any curve has infinitely many parametrizations. So could we get a different value for a line integral using some other parametrization of C, say, x = ˜x(u), y = ˜y(u), c ≤ u ≤ d ? If so, this would mean that our definition is not well-defined. Luckily, it turns out that the value of a line integral of a vector field is unchanged as long as the direction of the curve C is preserved by whatever parametrization is chosen: Theorem 4.2. Let f(x, y) = P(x, y)i+Q(x, y)j be a vector field, and let C be a smooth curve parametrized by x = x(t), y = y(t), a ≤ t ≤ b. Suppose that t = α(u) for c ≤ u ≤ d, such that a = α(c), b = α(d), and α′ (u) > 0 on the open interval (c,d) (i.e. α(u) is strictly increasing on [c,d]). Then C f··· dr has the same value for the parametrizations x = x(t), y = y(t), a ≤ t ≤ b and x = ˜x(u) = x(α(u)), y = ˜y(u) = y(α(u)), c ≤ u ≤ d. Proof: Since α(u) is strictly increasing and maps [c,d] onto [a,b], then we know that t = α(u) has an inverse function u = α−1 (t) defined on [a,b] such that c = α−1 (a), d = α−1 (b), and du dt = 1 α′(u) . Also, dt = α′ (u)du, and by the Chain Rule ˜x′ (u) = d ˜x du = d du (x(α(u))) = dx dt dt du = x′ (t)α′ (u) ⇒ x′ (t) = ˜x′ (u) α′(u) so making the susbstitution t = α(u) gives b a P(x(t), y(t))x′ (t)dt = α−1 (b) α−1(a) P(x(α(u)), y(α(u))) ˜x′ (u) α′(u) (α′ (u)du) = d c P(˜x(u), ˜y(u)) ˜x′ (u)du , which shows that C P(x, y)dx has the same value for both parametrizations. A similar argument shows that C Q(x, y)dy has the same value for both parametrizations, and hence C f··· dr has the same value. QED Notice that the condition α′ (u) > 0 in Theorem 4.2 means that the two parametrizations move along C in the same direction. That was not the case with the "reverse" parametriza- tion for −C: for u = a+ b − t we have t = α(u) = a+ b − u ⇒ α′ (u) = −1 < 0. 146 CHAPTER 4. LINE AND SURFACE INTEGRALS So far, the examples we have seen of line integrals (e.g. Example 4.2) have had the same value for different curves joining the initial point to the terminal point. That is, the line integral has been independent of the path joining the two points. As we mentioned before, this is not always the case. The following theorem gives a necessary and sufficient condition for this path independence: Theorem 4.3. In a region R, the line integral C f··· dr is independent of the path between any two points in R if and only if C f··· dr = 0 for every closed curve C which is contained in R. Proof: Suppose that C f··· dr = 0 for every closed curve C which is contained in R. Let P1 and P2 be two distinct points in R. Let C1 be a curve in R going from P1 to P2, and let C2 be another curve in R going from P1 to P2, as in Figure 4.2.2. C1 C2 P1 P2 Figure 4.2.2 Then C = C1 ∪−C2 is a closed curve in R (from P1 to P1), and so C f··· dr = 0. Thus, 0 = C f··· dr = C1 f··· dr + −C2 f··· dr = C1 f··· dr − C2 f··· dr , and so C1 f··· dr = C2 f··· dr. This proves path independence. Conversely, suppose that the line integral C f···dr is independent of the path between any two points in R. Let C be a closed curve contained in R. Let P1 and P2 be two distinct points on C. Let C1 be a part of the curve C that goes from P1 to P2, and let C2 be the remaining part of C that goes from P1 to P2, again as in Figure 4.2.2. Then by path independence we have C1 f··· dr = C2 f··· dr C1 f··· dr − C2 f··· dr = 0 C1 f··· dr + −C2 f··· dr = 0 , so C f··· dr = 0 since C = C1 ∪−C2 . QED 4.2 Properties of Line Integrals 147 Clearly, the above theorem does not give a practical way to determine path independence, since it is impossible to check the line integrals around all possible closed curves in a region. What it mostly does is give an idea of the way in which line integrals behave, and how seem- ingly unrelated line integrals can be related (in this case, a specific line integral between two points and all line integrals around closed curves). For a more practical method for determining path independence, we first need a version of the Chain Rule for multivariable functions: Theorem 4.4. (Chain Rule) If z = f (x, y) is a continuously differentiable function of x and y, and both x = x(t) and y = y(t) are differentiable functions of t, then z is a differentiable function of t, and dz dt = ∂z ∂x dx dt + ∂z ∂y dy dt (4.19) at all points where the derivatives on the right are defined. The proof is virtually identical to the proof of Theorem 2.2 from Section 2.4 (which uses the Mean Value Theorem), so we omit it.1 We will now use this Chain Rule to prove the following sufficient condition for path independence of line integrals: Theorem 4.5. Let f(x, y) = P(x, y)i+Q(x, y)j be a vector field in some region R, with P and Q continuously differentiable functions on R. Let C be a smooth curve in R parametrized by x = x(t), y = y(t), a ≤ t ≤ b. Suppose that there is a real-valued function F(x, y) such that ∇F = f on R. Then C f··· dr = F(B) − F(A) , (4.20) where A = (x(a), y(a)) and B = (x(b), y(b)) are the endpoints of C. Thus, the line integral is independent of the path between its endpoints, since it depends only on the values of F at those endpoints. Proof: By definition of C f··· dr, we have C f··· dr = b a P(x(t), y(t))x′ (t)+Q(x(t), y(t)) y′ (t) dt = b a ∂F ∂x dx dt + ∂F ∂y dy dt dt (since ∇F = f ⇒ ∂F ∂x = P and ∂F ∂y = Q) = b a F ′ (x(t), y(t))dt (by the Chain Rule in Theorem 4.4) = F(x(t), y(t)) b a = F(B) − F(A) by the Fundamental Theorem of Calculus. QED 1See TAYLOR and MANN, § 6.5. 148 CHAPTER 4. LINE AND SURFACE INTEGRALS Theorem 4.5 can be thought of as the line integral version of the Fundamental Theorem of Calculus. A real-valued function F(x, y) such that ∇F(x, y) = f(x, y) is called a potential for f. A conservative vector field is one which has a potential. Example 4.5. Recall from Examples 4.2 and 4.3 in Section 4.1 that the line integral C(x2 + y2 )dx + 2xydy was found to have the value 13 3 for three different curves C going from the point (0,0) to the point (1,2). Use Theorem 4.5 to show that this line integral is indeed path independent. Solution: We need to find a real-valued function F(x, y) such that ∂F ∂x = x2 + y2 and ∂F ∂y = 2xy . Suppose that ∂F ∂x = x2 + y2 , Then we must have F(x, y) = 1 3 x3 + xy2 + g(y) for some function g(y). So ∂F ∂y = 2xy+ g′ (y) satisfies the condition ∂F ∂y = 2xy if g′ (y) = 0, i.e. g(y) = K, where K is a constant. Since any choice for K will do (why?), we pick K = 0. Thus, a potential F(x, y) for f(x, y) = (x2 + y2 )i+2xyj exists, namely F(x, y) = 1 3 x3 + xy2 . Hence the line integral C(x2 + y2 )dx+2xydy is path independent. Note that we can also verify that the value of the line integral of f along any curve C going from (0,0) to (1,2) will always be 13 3 , since by Theorem 4.5 C f··· dr = F(1,2) − F(0,0) = 1 3 (1)3 +(1)(2)2 −(0+0) = 1 3 +4 = 13 3 . A consequence of Theorem 4.5 in the special case where C is a closed curve, so that the endpoints A and B are the same point, is the following important corollary: Corollary 4.6. If a vector field f has a potential in a region R, then C f···dr = 0 for any closed curve C in R (i.e. C ∇F ··· dr = 0 for any real-valued function F(x, y)). Example 4.6. Evaluate C xdx+ ydy for C : x = 2cost, y = 3sint, 0 ≤ t ≤ 2π. Solution: The vector field f(x, y) = xi+ yj has a potential F(x, y): ∂F ∂x = x ⇒ F(x, y) = 1 2 x2 + g(y) ,so ∂F ∂y = y ⇒ g′ (y) = y ⇒ g(y) = 1 2 y2 + K 150 CHAPTER 4. LINE AND SURFACE INTEGRALS 4.3 Green's Theorem We will now see a way of evaluating the line integral of a smooth vector field around a simple closed curve. A vector field f(x, y) = P(x, y)i + Q(x, y)j is smooth if its component functions P(x, y) and Q(x, y) are smooth. We will use Green's Theorem (sometimes called Green's Theorem in the plane) to relate the line integral around a closed curve with a double integral over the region inside the curve: Theorem 4.7. (Green's Theorem) Let R be a region in R2 whose boundary is a simple closed curve C which is piecewise smooth. Let f(x, y) = P(x, y)i+Q(x, y)j be a smooth vector field defined on both R and C. Then C f··· dr = R ∂Q ∂x − ∂P ∂y dA , (4.21) where C is traversed so that R is always on the left side of C. Proof: We will prove the theorem in the case for a simple region R, that is, where the boundary curve C can be written as C = C1 ∪C2 in two distinct ways: C1 = the curve y = y1(x) from the point X1 to the point X2 (4.22) C2 = the curve y = y2(x) from the point X2 to the point X1, (4.23) where X1 and X2 are the points on C farthest to the left and right, respectively; and C1 = the curve x = x1(y) from the point Y2 to the point Y1 (4.24) C2 = the curve x = x2(y) from the point Y1 to the point Y2, (4.25) where Y1 and Y2 are the lowest and highest points, respectively, on C. See Figure 4.3.1. a b x y y = y2(x) y = y1(x) x = x2(y) x = x1(y) Y2 Y1 X2 X1 R C d c Figure 4.3.1 Integrate P(x, y) around C using the representation C = C1 ∪C2 given by (4.23) and (4.24). 4.3 Green's Theorem 153 x y 0 C1 C2 1 1 1/2 1/2 R Figure 4.3.3 The annulus R If we modify the region R to be the annulus R = {(x, y) : 1/4 ≤ x2 + y2 ≤ 1} (see Figure 4.3.3), and take the "boundary" C of R to be C = C1 ∪ C2, where C1 is the unit circle x2 + y2 = 1 traversed counterclockwise and C2 is the circle x2 + y2 = 1/4 traversed clockwise, then it can be shown (see Exercise 8) that C f··· dr = 0 . We would still have R ∂Q ∂x − ∂P ∂y dA = 0, so for this R we would have C f··· dr = R ∂Q ∂x − ∂P ∂y dA , which shows that Green's Theorem holds for the annular region R. It turns out that Green's Theorem can be extended to multiply connected regions, that is, regions like the annulus in Example 4.8, which have one or more regions cut out from the interior, as opposed to discrete points being cut out. For such regions, the "outer" boundary and the "inner" boundaries are traversed so that R is always on the left side. C1 C2 R1 R2 (a) Region R with one hole C1 C2C3 R1 R2 (b) Region R with two holes Figure 4.3.4 Multiply connected regions The intuitive idea for why Green's Theorem holds for multiply connected regions is shown in Figure 4.3.4 above. The idea is to cut "slits" between the boundaries of a multiply con- nected region R so that R is divided into subregions which do not have any "holes". For example, in Figure 4.3.4(a) the region R is the union of the regions R1 and R2, which are divided by the slits indicated by the dashed lines. Those slits are part of the boundary of both R1 and R2, and we traverse then in the manner indicated by the arrows. Notice that along each slit the boundary of R1 is traversed in the opposite direction as that of R2, which 154 CHAPTER 4. LINE AND SURFACE INTEGRALS means that the line integrals of f along those slits cancel each other out. Since R1 and R2 do not have holes in them, then Green's Theorem holds in each subregion, so that bdy of R1 f··· dr = R1 ∂Q ∂x − ∂P ∂y dA and bdy of R2 f··· dr = R2 ∂Q ∂x − ∂P ∂y dA . But since the line integrals along the slits cancel out, we have C1∪C2 f··· dr = bdy of R1 f··· dr + bdy of R2 f··· dr , and so C1∪C2 f··· dr = R1 ∂Q ∂x − ∂P ∂y dA + R2 ∂Q ∂x − ∂P ∂y dA = R ∂Q ∂x − ∂P ∂y dA , which shows that Green's Theorem holds in the region R. A similar argument shows that the theorem holds in the region with two holes shown in Figure 4.3.4(b). We know from Corollary 4.6 that when a smooth vector field f(x, y) = P(x, y)i+Q(x, y)j on a region R (whose boundary is a piecewise smooth, simple closed curve C) has a potential in R, then C f···dr = 0. And if the potential F(x, y) is smooth in R, then ∂F ∂x = P and ∂F ∂y = Q, and so we know that ∂2 F ∂y∂x = ∂2 F ∂x∂y ⇒ ∂P ∂y = ∂Q ∂x in R. Conversely, if ∂P ∂y = ∂Q ∂x in R then C f··· dr = R ∂Q ∂x − ∂P ∂y dA = R 0dA = 0 . For a simply connected region R (i.e. a region with no holes), the following can be shown: The following statements are equivalent for a simply connected region R in R2 : (a) f(x, y) = P(x, y)i+Q(x, y)j has a smooth potential F(x, y) in R (b) C f··· dr is independent of the path for any curve C in R (c) C f··· dr = 0 for every simple closed curve C in R (d) ∂P ∂y = ∂Q ∂x in R (in this case, the differential form P dx+Q dy is exact) 156 CHAPTER 4. LINE AND SURFACE INTEGRALS 4.4 Surface Integrals and the Divergence Theorem In Section 4.1 we learned how to integrate along a curve. We will now learn how to perform integration over a surface in R3 , such as a sphere or a paraboloid. Recall from Section 1.8 how we identified points (x, y, z) on a curve C in R3 , parametrized by x = x(t), y = y(t), z = z(t), a ≤ t ≤ b, with the terminal points of the position vector r(t) = x(t)i+ y(t)j+ z(t)k for t in [a,b]. The idea behind a parametrization of a curve is that it "transforms" a subset of R1 (nor- mally an interval [a,b]) into a curve in R2 or R3 (see Figure 4.4.1). a t b R1 y z x 0 (x(a), y(a), z(a)) (x(t), y(t), z(t)) (x(b), y(b), z(b))r(t) Cx = x(t) y = y(t) z = z(t) Figure 4.4.1 Parametrization of a curve C in R3 Similar to how we used a parametrization of a curve to define the line integral along the curve, we will use a parametrization of a surface to define a surface integral. We will use two variables, u and v, to parametrize a surface Σ in R3 : x = x(u,v), y = y(u,v), z = z(u,v), for (u,v) in some region R in R2 (see Figure 4.4.2). u v R R2 (u,v) y z x 0 Σ r(u,v) x = x(u,v) y = y(u,v) z = z(u,v) Figure 4.4.2 Parametrization of a surface Σ in R3 In this case, the position vector of a point on the surface Σ is given by the vector-valued function r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k for (u,v) in R. 4.4 Surface Integrals and the Divergence Theorem 157 Since r(u,v) is a function of two variables, define the partial derivatives ∂r ∂u and ∂r ∂v for (u,v) in R by ∂r ∂u (u,v) = ∂x ∂u (u,v)i + ∂y ∂u (u,v)j + ∂z ∂u (u,v)k , and ∂r ∂v (u,v) = ∂x ∂v (u,v)i + ∂y ∂v (u,v)j + ∂z ∂v (u,v)k . The parametrization of Σ can be thought of as "transforming" a region in R2 (in the uv- plane) into a 2-dimensional surface in R3 . This parametrization of the surface is sometimes called a patch, based on the idea of "patching" the region R onto Σ in the grid-like manner shown in Figure 4.4.2. In fact, those gridlines in R lead us to how we will define a surface integral over Σ. Along the vertical gridlines in R, the variable u is constant. So those lines get mapped to curves on Σ, and the variable u is constant along the position vector r(u,v). Thus, the tangent vector to those curves at a point (u,v) is ∂r ∂v . Similarly, the horizontal gridlines in R get mapped to curves on Σ whose tangent vectors are ∂r ∂u . Now take a point (u,v) in R as, say, the lower left corner of one of the rectangular grid sections in R, as shown in Figure 4.4.2. Suppose that this rectangle has a small width and height of ∆u and ∆v, respectively. The corner points of that rectangle are (u,v), (u + ∆u,v), (u +∆u,v+∆v) and (u,v+∆v). So the area of that rectangle is A = ∆u∆v. Then that rectangle gets mapped by the parametrization onto some section of the surface Σ which, for ∆u and ∆v small enough, will have a surface area (call it dσ) that is very close to the area of the parallelogram which has adjacent sides r(u+∆u,v)−r(u,v) (corresponding to the line segment from (u,v) to (u + ∆u,v) in R) and r(u,v + ∆v) − r(u,v) (corresponding to the line segment from (u,v) to (u,v +∆v) in R). But by combining our usual notion of a partial derivative (see Definition 2.3 in Section 2.2) with that of the derivative of a vector-valued function (see Definition 1.12 in Section 1.8) applied to a function of two variables, we have ∂r ∂u ≈ r(u +∆u,v)−r(u,v) ∆u , and ∂r ∂v ≈ r(u,v+∆v)−r(u,v) ∆v , and so the surface area element dσ is approximately (r(u +∆u,v)−r(u,v))×××(r(u,v+∆v)−r(u,v)) ≈ (∆u ∂r ∂u )×××(∆v ∂r ∂v ) = ∂r ∂u ××× ∂r ∂v ∆u∆v by Theorem 1.13 in Section 1.4. Thus, the total surface area S of Σ is approximately the sum of all the quantities ∂r ∂u ××× ∂r ∂v ∆u∆v, summed over the rectangles in R. Taking the limit of that sum as the diagonal of the largest rectangle goes to 0 gives S = R ∂r ∂u ××× ∂r ∂v du dv . (4.26) 158 CHAPTER 4. LINE AND SURFACE INTEGRALS We will write the double integral on the right using the special notation Σ dσ = R ∂r ∂u ××× ∂r ∂v du dv . (4.27) This is a special case of a surface integral over the surface Σ, where the surface area element dσ can be thought of as 1dσ. Replacing 1 by a general real-valued function f (x, y, z) defined in R3 , we have the following: Definition 4.3. Let Σ be a surface in R3 parametrized by x = x(u,v), y = y(u,v), z = z(u,v), for (u,v) in some region R in R2 . Let r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k be the position vector for any point on Σ, and let f (x, y, z) be a real-valued function defined on some subset of R3 that contains Σ. The surface integral of f (x, y, z) over Σ is Σ f (x, y, z)dσ = R f (x(u,v), y(u,v), z(u,v)) ∂r ∂u ××× ∂r ∂v du dv . (4.28) In particular, the surface area S of Σ is S = Σ 1dσ . (4.29) Example 4.9. A torus T is a surface obtained by revolving a circle of radius a in the yz-plane around the z-axis, where the circle's center is at a distance b from the z-axis (0 < a < b), as in Figure 4.4.3. Find the surface area of T. y z 0 a (y− b)2 + z2 = a2 u b (a) Circle in the yz-plane x y z v a (x,y,z) (b) Torus T Figure 4.4.3 Solution: For any point on the circle, the line segment from the center of the circle to that point makes an angle u with the y-axis in the positive y direction (see Figure 4.4.3(a)). And as the circle revolves around the z-axis, the line segment from the origin to the center of that 162 CHAPTER 4. LINE AND SURFACE INTEGRALS Theorem 4.8. (Divergence Theorem) Let Σ be a closed surface in R3 which bounds a solid S, and let f(x, y, z) = f1(x, y, z)i+ f2(x, y, z)j+ f3(x, y, z)k be a vector field defined on some subset of R3 that contains Σ. Then Σ f··· dσ = S div f dV , (4.31) where div f = ∂f1 ∂x + ∂f2 ∂y + ∂f3 ∂z (4.32) is called the divergence of f. The proof of the Divergence Theorem is very similar to the proof of Green's Theorem, i.e. it is first proved for the simple case when the solid S is bounded above by one surface, bounded below by another surface, and bounded laterally by one or more surfaces. The proof can then be extended to more general solids.3 Example 4.11. Evaluate Σ f ··· dσ, where f(x, y, z) = xi + yj + zk and Σ is the unit sphere x2 + y2 + z2 = 1. Solution: We see that div f = 1+1+1 = 3, so Σ f··· dσ = S div f dV = S 3 dV = 3 S 1 dV = 3vol(S) = 3· 4π(1)3 3 = 4π . In physical applications, the surface integral Σ f··· dσ is often referred to as the flux of f through the surface Σ. For example, if f represents the velocity field of a fluid, then the flux is the net quantity of fluid to flow through the surface Σ per unit time. A positive flux means there is a net flow out of the surface (i.e. in the direction of the outward unit normal vector n), while a negative flux indicates a net flow inward (in the direction of −n). The term divergence comes from interpreting div f as a measure of how much a vector field "diverges" from a point. This is best seen by using another definition of div f which is equivalent4 to the definition given by formula (4.32). Namely, for a point (x, y, z) in R3 , div f(x, y, z) = lim V→0 1 V Σ f··· dσ , (4.33) 3See TAYLOR and MANN, § 15.6 for the details. 4See SCHEY, p. 36-39, for an intuitive discussion of this. 4.4 Surface Integrals and the Divergence Theorem 163 where V is the volume enclosed by a closed surface Σ around the point (x, y, z). In the limit, V → 0 means that we take smaller and smaller closed surfaces around (x, y, z), which means that the volumes they enclose are going to zero. It can be shown that this limit is independent of the shapes of those surfaces. Notice that the limit being taken is of the ratio of the flux through a surface to the volume enclosed by that surface, which gives a rough measure of the flow "leaving" a point, as we mentioned. Vector fields which have zero divergence are often called solenoidal fields. The following theorem is a simple consequence of formula (4.33). Theorem 4.9. If the flux of a vector field f is zero through every closed surface containing a given point, then div f = 0 at that point. Proof: By formula (4.33), at the given point (x, y, z) we have div f(x, y, z) = lim V→0 1 V Σ f··· dσ for closed surfaces Σ containing (x, y, z), so = lim V→0 1 V (0) by our assumption that the flux through each Σ is zero, so = lim V→0 0 = 0 . QED Lastly, we note that sometimes the notation Σ f (x, y, z)dσ and Σ f··· dσ is used to denote surface integrals of scalar and vector fields, respectively, over closed sur- faces. Especially in physics texts, it is common to see simply Σ instead of Σ . Exercises A For Exercises 1-4, use the Divergence Theorem to evaluate the surface integral Σ f ··· dσ of the given vector field f(x, y, z) over the surface Σ. 1. f(x, y, z) = xi+2yj+3zk, Σ : x2 + y2 + z2 = 9 2. f(x, y, z) = xi+ yj+ zk, Σ : boundary of the solid cube S = {(x, y, z) : 0 ≤ x, y, z ≤ 1} 3. f(x, y, z) = x3 i+ y3 j+ z3 k, Σ : x2 + y2 + z2 = 1 4. f(x, y, z) = 2i+3j+5k, Σ : x2 + y2 + z2 = 1 164 CHAPTER 4. LINE AND SURFACE INTEGRALS B 5. Show that the flux of any constant vector field through any closed surface is zero. 6. Evaluate the surface integral from Exercise 2 without using the Divergence Theorem, i.e. using only Definition 4.3, as in Example 4.10. Note that there will be a different outward unit normal vector to each of the six faces of the cube. 7. Evaluate the surface integral Σ f··· dσ, where f(x, y, z) = x2 i+ xyj+ zk and Σ is the part of the plane 6x + 3y + 2z = 6 with x ≥ 0, y ≥ 0, and z ≥ 0, with the outward unit normal n pointing in the positive z direction. 8. Use a surface integral to show that the surface area of a sphere of radius r is 4πr2 . (Hint: Use spherical coordinates to parametrize the sphere.) 9. Use a surface integral to show that the surface area of a right circular cone of radius R and height h is πR h2 + R2. (Hint: Use the parametrization x = rcosθ, y = rsinθ, z = h R r, for 0 ≤ r ≤ R and 0 ≤ θ ≤ 2π.) 10. The ellipsoid x2 a2 + y2 b2 + z2 c2 = 1 can be parametrized using ellipsoidal coordinates x = asinφ cosθ , y = bsinφ sinθ , z = ccosφ , for 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Show that the surface area S of the ellipsoid is S = π 0 2π 0 sinφ a2b2 cos2 φ+ c2(a2 sin2 θ + b2 cos2 θ)sin2 φ dθ dφ . (Note: The above double integral can not be evaluated by elementary means. For specific values of a, b and c it can be evaluated using numerical methods. An alternative is to express the surface area in terms of elliptic integrals.5 ) C 11. Use Definition 4.3 to prove that the surface area S over a region R in R2 of a surface z = f (x, y) is given by the formula S = R 1+ ∂f ∂x 2 + ∂f ∂y 2 dA . (Hint: Think of the parametrization of the surface.) 5BOWMAN, F., Introduction to Elliptic Functions, with Applications, New York: Dover, 1961, § III.7. 4.5 Stokes' Theorem 165 4.5 Stokes' Theorem So far the only types of line integrals which we have discussed are those along curves in R2 . But the definitions and properties which were covered in Sections 4.1 and 4.2 can easily be extended to include functions of three variables, so that we can now discuss line integrals along curves in R3 . Definition 4.5. For a real-valued function f (x, y, z) and a curve C in R3 , parametrized by x = x(t), y = y(t), z = z(t), a ≤ t ≤ b, the line integral of f (x, y, z) along C with respect to arc length s is C f (x, y, z)ds = b a f (x(t), y(t), z(t)) x′(t)2 + y′(t)2 + z′(t)2 dt . (4.34) The line integral of f (x, y, z) along C with respect to x is C f (x, y, z)dx = b a f (x(t), y(t), z(t))x′ (t)dt . (4.35) The line integral of f (x, y, z) along C with respect to y is C f (x, y, z)dy = b a f (x(t), y(t), z(t)) y′ (t)dt . (4.36) The line integral of f (x, y, z) along C with respect to z is C f (x, y, z)dz = b a f (x(t), y(t), z(t)) z′ (t)dt . (4.37) Similar to the two-variable case, if f (x, y, z) ≥ 0 then the line integral C f (x, y, z)ds can be thought of as the total area of the "picket fence" of height f (x, y, z) at each point along the curve C in R3 . Vector fields in R3 are defined in a similar fashion to those in R2 , which allows us to define the line integral of a vector field along a curve in R3 . Definition 4.6. For a vector field f(x, y, z) = P(x, y, z)i+Q(x, y, z)j+R(x, y, z)k and a curve C in R3 with a smooth parametrization x = x(t), y = y(t), z = z(t), a ≤ t ≤ b, the line integral of f along C is C f··· dr = C P(x, y, z)dx + C Q(x, y, z)dy + C R(x, y, z)dz (4.38) = b a f(x(t), y(t), z(t))···r′ (t)dt , (4.39) where r(t) = x(t)i+ y(t)j+ z(t)k is the position vector for points on C. 168 CHAPTER 4. LINE AND SURFACE INTEGRALS So by Theorem 4.12 we know that C f··· dr = F(B) − F(A) , where A = (x(0), y(0), z(0)) and B = (x(8π), y(8π), z(8π)), so = F(8πsin8π,8πcos8π,8π) − F(0sin0,0cos0,0) = F(0,8π,8π) − F(0,0,0) = 0+ (8π)2 2 +(8π)2 −(0+0+0) = 96π2 . We will now discuss a generalization of Green's Theorem in R2 to orientable surfaces in R3 , called Stokes' Theorem. A surface Σ in R3 is orientable if there is a continuous vector field N in R3 such that N is nonzero and normal to Σ (i.e. perpendicular to the tangent plane) at each point of Σ. We say that such an N is a normal vector field. y z x 0 N −N Figure 4.5.2 For example, the unit sphere x2 +y2 +z2 = 1 is orientable, since the continuous vector field N(x, y, z) = xi+ yj+zk is nonzero and normal to the sphere at each point. In fact, −N(x, y, z) is another normal vector field (see Figure 4.5.2). We see in this case that N(x, y, z) is what we have called an outward normal vector, and −N(x, y, z) is an inward normal vector. These "outward" and "inward" normal vec- tor fields on the sphere correspond to an "outer" and "inner" side, respectively, of the sphere. That is, we say that the sphere is a two- sided surface. Roughly, "two-sided" means "orientable". Other ex- amples of two-sided, and hence orientable, surfaces are cylinders, paraboloids, ellipsoids, and planes. You may be wondering what kind of surface would not have two sides. An example is the Möbius strip, which is constructed by taking a thin rectangle and connecting its ends at the opposite corners, resulting in a "twisted" strip (see Figure 4.5.3). A B A B −→ (a) Connect A to A and B to B along the ends A → A → (b) Not orientable Figure 4.5.3 Möbius strip If you imagine walking along a line down the center of the Möbius strip, as in Figure 4.5.3(b), then you arrive back at the same place from which you started but upside down! That is, your orientation changed even though your motion was continuous along that center 4.5 Stokes' Theorem 169 line. Informally, thinking of your vertical direction as a normal vector field along the strip, there is a discontinuity at your starting point (and, in fact, at every point) since your vertical direction takes two different values there. The Möbius strip has only one side, and hence is nonorientable.7 For an orientable surface Σ which has a boundary curve C, pick a unit normal vector n such that if you walked along C with your head pointing in the direction of n, then the surface would be on your left. We say in this situation that n is a positive unit normal vector and that C is traversed n-positively. We can now state Stokes' Theorem: Theorem 4.14. (Stokes' Theorem) Let Σ be an orientable surface in R3 whose boundary is a simple closed curve C, and let f(x, y, z) = P(x, y, z)i +Q(x, y, z)j + R(x, y, z)k be a smooth vector field defined on some subset of R3 that contains Σ. Then C f··· dr = Σ (curl f)···ndσ , (4.45) where curl f = ∂R ∂y − ∂Q ∂z i + ∂P ∂z − ∂R ∂x j + ∂Q ∂x − ∂P ∂y k , (4.46) n is a positive unit normal vector over Σ, and C is traversed n-positively. Proof: As the general case is beyond the scope of this text, we will prove the theorem only for the special case where Σ is the graph of z = z(x, y) for some smooth real-valued function z(x, y), with (x, y) varying over a region D in R2 . y z x 0 n (x, y)D CD C Σ : z = z(x, y) Figure 4.5.4 Projecting Σ onto the xy-plane, we see that the closed curve C (the boundary curve of Σ) projects onto a closed curve CD which is the boundary curve of D (see Fig- ure 4.5.4). Assuming that C has a smooth parametriza- tion, its projection CD in the xy-plane also has a smooth parametrization, say CD : x = x(t) , y = y(t) , a ≤ t ≤ b , and so C can be parametrized (in R3 ) as C : x = x(t) , y = y(t) , z = z(x(t), y(t)) , a ≤ t ≤ b , since the curve C is part of the surface z = z(x, y). Now, by the Chain Rule (Theorem 4.4 in Section 4.2), for z = z(x(t), y(t)) as a function of t, we know that z′ (t) = ∂z ∂x x′ (t) + ∂z ∂y y′ (t) , 7For further discussion of orientability, see O'NEILL, § IV.7. 174 CHAPTER 4. LINE AND SURFACE INTEGRALS In physical applications, for a simple closed curve C the line integral C f···dr is often called the circulation of f around C. For example, if E represents the electrostatic field due to a point charge, then it turns out8 that curl E = 0, which means that the circulation C E···dr = 0 by Stokes' Theorem. Vector fields which have zero curl are often called irrotational fields. In fact, the term curl was created by the 19th century Scottish physicist James Clerk Maxwell in his study of electromagnetism, where it is used extensively. In physics, the curl is interpreted as a measure of circulation density. This is best seen by using another definition of curl f which is equivalent9 to the definition given by formula (4.46). Namely, for a point (x, y, z) in R3 , n···(curl f)(x, y, z) = lim S→0 1 S C f··· dr , (4.50) where S is the surface area of a surface Σ containing the point (x, y, z) and with a simple closed boundary curve C and positive unit normal vector n at (x, y, z). In the limit, think of the curve C shrinking to the point (x, y, z), which causes Σ, the surface it bounds, to have smaller and smaller surface area. That ratio of circulation to surface area in the limit is what makes the curl a rough measure of circulation density (i.e. circulation per unit area). x y 0 f Figure 4.5.6 Curl and rotation An idea of how the curl of a vector field is related to rotation is shown in Figure 4.5.6. Suppose we have a vector field f(x, y, z) which is always parallel to the xy-plane at each point (x, y, z) and that the vectors grow larger the further the point (x, y, z) is from the y- axis. For example, f(x, y, z) = (1+ x2 )j. Think of the vector field as representing the flow of water, and imagine dropping two wheels with paddles into that water flow, as in Fig- ure 4.5.6. Since the flow is stronger (i.e. the magnitude of f is larger) as you move away from the y-axis, then such a wheel would ro- tate counterclockwise if it were dropped to the right of the y-axis, and it would rotate clockwise if it were dropped to the left of the y-axis. In both cases the curl would be nonzero (curl f(x, y, z) = 2xk in our example) and would obey the right-hand rule, that is, curl f(x, y, z) points in the direction of your thumb as you cup your right hand in the direction of the rota- tion of the wheel. So the curl points outward (in the positive z-direction) if x > 0 and points inward (in the negative z-direction) if x < 0. Notice that if all the vectors had the same di- rection and the same magnitude, then the wheels would not rotate and hence there would be no curl (which is why such fields are called irrotational, meaning no rotation). 8See Ch. 2 in REITZ, MILFORD and CHRISTY. 9See SCHEY, p. 78-81, for the derivation. 4.6 Gradient, Divergence, Curl and Laplacian 177 4.6 Gradient, Divergence, Curl and Laplacian In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. We will then show how to write these quantities in cylindrical and spherical coordinates. For a real-valued function f (x, y, z) on R3 , the gradient ∇f (x, y, z) is a vector-valued func- tion on R3 , that is, its value at a point (x, y, z) is the vector ∇f (x, y, z) = ∂f ∂x , ∂f ∂y , ∂f ∂z = ∂f ∂x i + ∂f ∂y j + ∂f ∂z k in R3 , where each of the partial derivatives is evaluated at the point (x, y, z). So in this way, you can think of the symbol ∇ as being "applied" to a real-valued function f to produce a vector ∇f . It turns out that the divergence and curl can also be expressed in terms of the symbol ∇. This is done by thinking of ∇ as a vector in R3 , namely ∇ = ∂ ∂x i + ∂ ∂y j + ∂ ∂z k . (4.51) Here, the symbols ∂ ∂x , ∂ ∂y and ∂ ∂z are to be thought of as "partial derivative operators" that will get "applied" to a real-valued function, say f (x, y, z), to produce the partial derivatives ∂f ∂x , ∂f ∂y and ∂f ∂z . For instance, ∂ ∂x "applied" to f (x, y, z) produces ∂f ∂x . Is ∇ really a vector? Strictly speaking, no, since ∂ ∂x , ∂ ∂y and ∂ ∂z are not actual numbers. But it helps to think of ∇ as a vector, especially with the divergence and curl, as we will soon see. The process of "applying" ∂ ∂x , ∂ ∂y , ∂ ∂z to a real-valued function f (x, y, z) is normally thought of as multiplying the quantities: ∂ ∂x (f ) = ∂f ∂x , ∂ ∂y (f ) = ∂f ∂y , ∂ ∂z (f ) = ∂f ∂z For this reason, ∇ is often referred to as the "del operator", since it "operates" on functions. For example, it is often convenient to write the divergence div f as ∇···f, since for a vector field f(x, y, z) = f1(x, y, z)i+ f2(x, y, z)j+ f3(x, y, z)k, the dot product of f with ∇ (thought of as a vector) makes sense: ∇···f = ∂ ∂x i + ∂ ∂y j + ∂ ∂z k ···(f1(x, y, z)i + f2(x, y, z)j + f3(x, y, z)k) = ∂ ∂x (f1) + ∂ ∂y (f2) + ∂ ∂z (f3) = ∂f1 ∂x + ∂f2 ∂y + ∂f3 ∂z = div f 180 CHAPTER 4. LINE AND SURFACE INTEGRALS Corollary 4.18. The flux of the curl of a smooth vector field f(x, y, z) through any closed surface is zero. Proof: Let Σ be a closed surface which bounds a solid S. The flux of ∇×××f through Σ is Σ (∇×××f)··· dσ = S ∇···(∇×××f) dV (by the Divergence Theorem) = S 0 dV (by Theorem 4.17) = 0 . QED There is another method for proving Theorem 4.15 which can be useful, and is often used in physics. Namely, if the surface integral Σ f (x, y, z)dσ = 0 for all surfaces Σ in some solid region (usually all of R3 ), then we must have f (x, y, z) = 0 throughout that region. The proof is not trivial, and physicists do not usually bother to prove it. But the result is true, and can also be applied to double and triple integrals. For instance, to prove Theorem 4.15, assume that f (x, y, z) is a smooth real-valued func- tion on R3 . Let C be a simple closed curve in R3 and let Σ be any capping surface for C (i.e. Σ is orientable and its boundary is C). Since ∇f is a vector field, then Σ (∇×××(∇f ))···ndσ = C ∇f ··· dr by Stokes' Theorem, so = 0 by Corollary 4.13. Since the choice of Σ was arbitrary, then we must have (∇×××(∇f ))···n = 0 throughout R3 , where n is any unit vector. Using i, j and k in place of n, we see that we must have ∇×××(∇f ) = 0 in R3 , which completes the proof. Example 4.18. A system of electric charges has a charge density ρ(x, y, z) and produces an electrostatic field E(x, y, z) at points (x, y, z) in space. Gauss' Law states that Σ E··· dσ = 4π S ρ dV for any closed surface Σ which encloses the charges, with S being the solid region enclosed by Σ. Show that ∇···E = 4πρ. This is one of Maxwell's Equations.10 10In Gaussian (or CGS) units. 4.6 Gradient, Divergence, Curl and Laplacian 183 Goal: Show that the gradient of a real-valued function F(ρ,θ,φ) in spherical coordinates is: ∇F = ∂F ∂ρ eρ + 1 ρ sinφ ∂F ∂θ eθ + 1 ρ ∂F ∂φ eφ Idea: In the Cartesian gradient formula ∇F(x, y, z) = ∂F ∂x i+ ∂F ∂y j+ ∂F ∂z k, put the Cartesian ba- sis vectors i, j, k in terms of the spherical coordinate basis vectors eρ, eθ, eφ and functions of ρ, θ and φ. Then put the partial derivatives ∂F ∂x , ∂F ∂y , ∂F ∂z in terms of ∂F ∂ρ , ∂F ∂θ , ∂F ∂φ and functions of ρ, θ and φ. Step 1: Get formulas for eρ, eθ, eφ in terms of i, j, k. We can see from Figure 4.6.2 that the unit vector eρ in the ρ direction at a general point (ρ,θ,φ) is eρ = r r , where r = xi + yj + zk is the position vector of the point in Cartesian coordinates. Thus, eρ = r r = xi+ yj+ zk x2 + y2 + z2 , so using x = ρ sinφcosθ, y = ρ sinφsinθ, z = ρ cosφ, and ρ = x2 + y2 + z2, we get: eρ = sinφ cosθi + sinφ sinθj + cosφk Now, since the angle θ is measured in the xy-plane, then the unit vector eθ in the θ direction must be parallel to the xy-plane. That is, eθ is of the form ai+ bj+0k. To figure out what a and b are, note that since eθ ⊥ eρ, then in particular eθ ⊥ eρ when eρ is in the xy-plane. That occurs when the angle φ is π/2. Putting φ = π/2 into the formula for eρ gives eρ = cosθi+sinθj+0k, and we see that a vector perpendicular to that is −sinθi+cosθj+0k. Since this vector is also a unit vector and points in the (positive) θ direction, it must be eθ: eθ = −sinθi + cosθj + 0k Lastly, since eφ = eθ ×××eρ, we get: eφ = cosφ cosθi + cosφ sinθj − sinφk Step 2: Use the three formulas from Step 1 to solve for i, j, k in terms of eρ, eθ, eφ. This comes down to solving a system of three equations in three unknowns. There are many ways of doing this, but we will do it by combining the formulas for eρ and eφ to eliminate k, which will give us an equation involving just i and j. This, with the formula for eθ, will then leave us with a system of two equations in two unknowns (i and j), which we will use to solve first for j then for i. Lastly, we will solve for k. First, note that sinφeρ + cosφeφ = cosθi + sinθj Appendix B We will prove the right-hand rule for the cross product of two vectors in R3 . For any vectors v and w in R3 , define a new vector, n(v,w), as follows: 1. If v and w are nonzero and not parallel, and θ is the angle between them, then n(v,w) is the vector in R3 such that: (a) the magnitude of n(v,w) is v w sinθ, (b) n(v,w) is perpendicular to the plane containing v and w, and (c) v, w, n(v,w) form a right-handed system. 2. If v and w are nonzero and parallel, then n(v,w) = 0. 3. If either v or w is 0, then n(v,w) = 0. The goal is to show that n(v,w) = v×××w for all v, w in R3 , which would prove the right-hand rule for the cross product (by part 1(c) of our definition). To do this, we will perform the following steps: Step 1: Show that n(v,w) = v×××w if v and w are any two of the basis vectors i, j, k. This was already shown in Example 1.11 in Section 1.4. Step 2: Show that n(av,bw) = ab(v×××w) for any scalars a, b if v and w are any two of the basis vectors i, j, k. If either a = 0 or b = 0 then n(av,bw) = 0 = ab(v×××w), so the result holds. So assume that a = 0 and b = 0. Let v and w be any two of the basis vectors i, j, k. For example, we will show that the result holds for v = i and w = k (the other possibilities follow in a similar fashion). For av = ai and bw = bk, the angle θ between av and bw is 90◦ . Hence the magnitude of n(av,bw), by definition, is ai bk sin90◦ = \|ab\|. Also, by definition, n(av,bw) is per- pendicular to the plane containing ai and bk, namely, the xz-plane. Thus, n(av,bw) must be a scalar multiple of j. Since its magnitude is \|ab\|, then n(av,bw) must be either \|ab\|j or −\|ab\|j. There are four possibilities for the combinations of signs for a and b. We will consider the case when a > 0 and b > 0 (the other three possibilities are handled similarly). 192 193 In this case, n(av,bw) must be either abj or −abj. Now, since i, j, k form a right-handed system, then i, k, j form a left-handed system, and so i, k, −j form a right-handed system. Thus, ai, bk, −abj form a right-handed system (since a > 0, b > 0, and ab > 0). So since, by definition, ai, bk, n(ai,bk) form a right-handed system, and since n(ai,bk) has to be either abj or −abj, this means that we must have n(ai,bk) = −abj. But we know that ai ××× bk = ab(i ××× k) = ab(−j) = −abj. Therefore, n(ai,bk) = ab(i ××× k), which is what we needed to show. ∴ n(av,bw) = ab(v×××w) Step 3: Show that n(u,v+w) = n(u,v)+n(u,w) for any vectors u, v, w. If u = 0 then the result holds trivially since n(u,v+w), n(u,v) and n(u,w) are all the zero vector. If v = 0, then the result follows easily since n(u,v + w) = n(u,0 + w) = n(u,w) = 0+n(u,w) = n(u,0) = n(u,w) = n(u,v)+n(u,w). A similar argument shows that the result holds if w = 0. So now assume that u, v and w are all nonzero vectors. We will describe a geometric construction of n(u,v), which is shown in the figure below. Let P be a plane perpendicular to u. Multiply the vector v by the positive scalar u , then project the vector u v straight down onto the plane P. You can think of this projection vector (denoted by pro jP u v) as the shadow of the vector u v on the plane P, with the light source directly overhead the terminal point of u v. If θ is the angle between u and v, then we see that pro jP u v has magnitude u v sinθ, which is the magnitude of n(u,v). So rotating pro jP u v by 90◦ in a counter-clockwise direction in the plane P gives a vector whose magnitude is the same as that of n(u,v) and which is perpendicular to pro jP u v (and hence perpendicular to v). Since this vector is in P then it is also perpendicular to u. And we can see that u, v and this vector form a right-handed system. Hence this vector must be n(u,v). Note that this holds even if u ∥ v, since in that case θ = 0◦ and so sinθ = 0 which means that n(u,v) has magnitude 0, which is what we would expect. u v pro jP u v u v n(u,v) θ θ P Now apply this same geometric construction to get n(u,w) and n(u,v+w). Since u (v+ w) is the sum of the vectors u v and u w, then the projection vector pro jP u (v+w) is the sum of the projection vectors pro jP u v and pro jP u w (to see this, using the shadow 194 Appendix B: Proof of the Right-Hand Rule for the Cross Product analogy again and the parallelogram rule for vector addition, think of how projecting a parallelogram onto a plane gives you a parallelogram in that plane). So then rotating all three projection vectors by 90◦ in a counter-clockwise direction in the plane P preserves that sum (see the figure below), which means that n(u,v+w) = n(u,v)+n(u,w). u v w v+w u (v+w) pro jP u v pro jP u w pro jP u (v+w) u v u w n(u,v) n(u,w) n(u,v+w) θ θ P Step 4: Show that n(w,v) = −n(v,w) for any vectors v, w. If v and w are nonzero and parallel, or if either is 0, then n(w,v) = 0 = −n(v,w), so the result holds. So assume that v and w are nonzero and not parallel. Then n(w,v) has magnitude w v sinθ, which is the same as the magnitude of n(v,w), and hence is the same as the magnitude of −n(v,w). By definition, n(v,w) is perpendicular to the plane containing w and v, and hence so is −n(v,w). Also, v, w, n(v,w) form a right-handed system, and so w, v, n(v,w) form a left-handed system, and hence w, v, −n(v,w) form a right-handed system. Thus, we have shown that −n(v,w) is a vector with the same magnitude as n(w,v) and is perpendicular to the plane containing w and v, and that w, v, −n(v,w) form a right-handed system. So by definition this means that −n(v,w) must be n(w,v). Step 5: Show that n(v,w) = v×××w for all vectors v, w. Write v = v1 i+ v2 j+ v3 k and w = w1 i+ w2 j+ w3 k. Then by Steps 3 and 4, we have Appendix C 3D Graphing with Gnuplot Gnuplot is a free, open-source software package for producing a variety of graphs. Versions are available for many operating systems. Below is a very brief tutorial on how to use Gnuplot to graph functions of several variables. INSTALLATION 1. Go to and follow the links to download the lat- est version for your operating system. For Windows, you should get the Zip file with a name such as gp420win32.zip, which is version 4.2.0. All the examples we will discuss require at least version 4.2.0. 2. Install the downloaded file. For example, in Windows you would unzip the Zip file you downloaded in Step 1 into some folder (use the "Use folder names" option if extracting with WinZip). RUNNING GNUPLOT 1. In Windows, run wgnuplot.exe from the folder (or bin folder) where you installed Gnu- plot. In Linux, just type gnuplot in a terminal window. 2. You should now get a Gnuplot terminal with a gnuplot> command prompt. In Windows this will appear in a new window, while in Linux it will appear in the terminal window where the gnuplot command was run. For Windows, if the font is unreadable you can change it by right-clicking on the text part of the Gnuplot window and selecting the "Choose Font.." option. For example, the font "Courier", style "Regular", size "12" is usually a good choice (that choice can be saved for future sessions by right-clicking in the Gnuplot window again and selecting the option to update wgnuplot.ini). 3. At the gnuplot> command prompt you can now run graphing commands, which we will now describe. GRAPHING FUNCTIONS The usual way to create 3D graphs in Gnuplot is with the splot command: splot <range> <comma-separated list of functions> 196 198 Appendix C: 3D Graphing with Gnuplot Note that we had to type 2x*2 to multiply 2 times x2 . For clarity, parentheses can be used to make sure the operations are being performed in the correct order: splot [-1:1][-2:2] 2(x2) + y2 In the above example, to also plot the function z = ex+y on the same graph, put a comma after the first function then append the new function: splot [-1:1][-2:2] 2(x2) + y2, exp(x+y) By default, the x-axis and y-axis are not shown in the graph. To display the axes, use this command before the splot command: set zeroaxis Also, by default the x- and y-axes are switched from their usual position. To show the axes with the orientation which we have used throughout the text, use this command: set view 60,120,1,1 Also, to label the axes, use these commands: set xlabel "x" set ylabel "y" set zlabel "z" To show the level curves of the surface z = f (x, y) on both the surface and projected onto the xy-plane, use this command: set contour both The default mesh size for the grid on the surface is 10 units. To get more of a colored/shaded surface, increase the mesh size (to, say, 25) like this: set isosamples 25 Putting all this together, we get the following graph with these commands: set zeroaxis set view 60,120,1,1 set xlabel "x" set ylabel "y" set zlabel "z" set contour both set isosamples 25 splot [-1:1][-2:2] 2(x2) + y2, exp(x+y) 199 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 5 10 15 20 25 z 2∗ x∗∗2+ y∗∗2 6 5 4 3 2 1 exp(x+ y) 20 15 10 5 x y z The numbers listed below the functions in the key in the upper right corner of the graph are the "levels" of the level curves of the corresponding surface. That is, they are the num- bers c such that f (x, y) = c. Because of the large number of level curves, the key was put outside the graph with the set key outside command. If you do not want the function key displayed, it can be turned off with this command: unset key PARAMETRIC FUNCTIONS Gnuplot has the ability to graph surfaces given in various parametric forms. For example, for a surface parametrized in cylindrical coordinates x = rcosθ , y = rsinθ , z = z you would do the following: set mapping cylindrical set parametric splot [a:b][c:d] vcos(u),vsin(u),f(u,v) where the variable u represents θ, with a ≤ u ≤ b, the variable v represents r, with c ≤ v ≤ d, and z = f (u,v) is some function of u and v. Example C.2. The graph of the helicoid z = θ in Example 1.34 from Section 1.7 (p. 49) was created using the following commands: 200 Appendix C: 3D Graphing with Gnuplot set mapping cylindrical set parametric set view 60,120,1,1 set xyplane 0 set xlabel "x" set ylabel "y" set zlabel "z" unset key set isosamples 15 splot [0:4pi][0:2] vcos(u),v*sin(u),u The command set xyplane 0 moves the z-axis so that z = 0 aligns with the xy-plane (which is not the default in Gnuplot). Looking at the graph, you will see that r varies from 0 to 2, and θ varies from 0 to 4π. PRINTING AND SAVING In Windows, to print a graph from Gnuplot right-click on the titlebar of the graph's window, select "Options" and then the "Print.." option. If that does not work on your version of Gnuplot, then go to the File menu on the main Gnuplot menubar, select "Output Device ...", and enter pdf in the Terminal type? textfield, hit OK. That will allow you to print the graph as a PDF file. To save a graph, say, as a PNG file, go to the File menu on the main Gnuplot menubar, select "Output Device ...", and enter png in the Terminal type? textfield, hit OK. Then, in the File menu again, select the "Output ..." option and enter a filename (say, graph.png) in the Output filename? textfield, hit OK. Now run your splot command again and you should see a file called graph.png in the current directory (usually the directory where wgnuplot.exe is located, though you can change that setting using the "Change Directory ..." option in the File menu). In Linux, to save the graph as a file called graph.png, you would issue the following com- mands: set terminal png set output 'graph.png' and then run your splot command. There are many terminal types (which determine the output format). Run the command set terminal to see all the possible types. In Linux, the postscript terminal type is popular, since the print quality is high and there are many PostScript viewers available. To quit Gnuplot, type quit at the gnuplot> command prompt. 202 GNU Free Documentation License could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. 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Pearson Debuts Interactive NovaNET Geometry Pearson has launched a new online geometry course for its NovaNET 15.0 service targeted toward students in grades 6 through 12 and adult education. Person's NovaNET is an online, standards-based courseware system designed for middle- and high-school students. Aligned to the 2007 Prentice Hall Geometry textbook, the new NovaNET Geometry course includes 77 multimedia lessons and includes instructional strategies for each. Additional features include: Interactive practices; Feedback and remediation; Ongoing, formative and summative assessments for each lesson; and Support for special needs students, including struggling readers. According to Pearson, the previous geometry course remains available, but the new version is designed for split-semester geometry schedules divided into Geometry A and B
fifth edition continues to improve on the features that have made it the market leader. The text offers a flexible organization, enabling instructors to adapt the book to their particular courses. The book is both complete and careful, and it continues to maintain its emphasis on algorithms and applications. Excellent exercise sets allow students to perfect skills as they practice. This new edition continues to feature numerous computer science applications-making this the ideal text for preparing students for advanced study. I will once again be teaching discrete mathematics this summer, so I am searching through the mathematical publishing pathways looking for a suitable textbook. Therefore, that is the context within which I examined this book. It certainly is the largest discrete book that I have encountered; including the appendices and problem solutions, there are over one thousand pages. Grimaldi has tried to include every topic that falls under the discrete mathematics tent. Therefore, this is a book that could be used for a two semester sequence in discrete mathematics. When examining discrete books for possible adoption I start with the simple premise that logic, set theory and functions and relations must be covered very early. In my ideal world, they are the first three chapters. Set theory and relations are so fundamental a part of other areas that I am surprised when authors don't cover them first. The first chapter in this book covers basic counting principles. While this doesn't break too much from my ideal sequence, I see no overpowering reason why fundamental counting should be before set theory. Given that the rules of counting for sums and products can easily be related to sets, there is a strong justification for putting set theory first.Read more › This is a bad book if you are not already familiar with the basic concepts of the material. The author was more interested in showing worked examples than explaining concepts, and the more difficult problems in the exercise sections do not have solutions in the back of the book, so even 'self-learning' is extremely hard. Unless you have a very good teacher, you will not benefit from the way the material is presented inside this book. 'Solutions' and 'examples' are presented 'as is' without explanations. One of my friends into math did mention it's not a bad reference guide for proofs, but he was as unimpressed with this book as a learning tool as I was. The level of rigor is very high, but the simple explanations to go with it are not present. I advise finding a good source on the subject instead of this unfriendly text, which has a target audience of math professionals. I bought this book as a supplement to a summer course in Discrete Math, and since this was my first ever exposure to mathematical proof and dialog, I first thought this book mostly alien, with occaisional sections of brevity; it did help me fill in some gaps left behind in Rosen's book, especially on some basic proofs dealing with integers and with combinatorial reasoning--something this book is REALLY good at... I'm in my first course of Combinatorics with a teacher that assumes we know alot more calculus than we do. We use Tucker's Applied combinatorics 5th, and I was cruising along just fine until we hit Generating Functions. Brick wall. Rosen's book didn't cover it (well; there's a great page of known identities, but not an intro-level version), neither did Epp, so I dusted this tome off my shelf and cracked it open... section 9.1 presents Generating functions on such an easy to use language and analytic explanation that I went from getting every problem wrong in Tucker's book to getting them all right; all due to the clarity of exposition. I've also found that as my 'mathematical maturity' has grown in the last year, so has the comprehensibility of this text. It may be too deep for a beginner--I would agree that it would be too much for all but your brightest minus an excellent teacher--but this book teaches 'real math' and does so very well. In conclusion, if you have the available student loan $$ and want a very good supplementary book that you really can take with you to higher classes, put this at the top of your list. I also own Epp and Rosen's discrete math texts, and have to say that for me ultimately I needed all three as a beginner; plus a few extra books from the library for special topics. But what I learned stayed with me and all three have their positives and negatives, but if I were to choose only one to stay on my shelf, THIS would be the one. It is common to feel you need someone to explain what you are reading while studying from a book and even more if the subject is mathematics. That is what surprises readers while starting to explore this interesting book. At the beginning it is hard to believe how simple it becomes to understand the different topics. That is a consequence of the easy way readers assimilate what is learnt by analyzing general and particular examples. That is the way in which the book presents the different units: the usual incomprehensible explanations are replaced by a lot of short examples which are easily understandable. Students not only feel they understand what they read but also enjoy and are attracted by a subject that is nice when comprehended. Even if it seems to be too long, its more than eight-hundred pages do not reflect the period of time which takes to learn each unit. They are considerably short and are also divided in sections that reduce the difficulty of continuous reading, especially after having stopped for a wile, leaving aside the need to go over the last pages. I consider this is a recommendable book for those students who are studying all the mathematic points which are analyzed in the volume. I believe it is the best complement for daily classes or a good option to study on your own.
16.69 FREE None(1 Copy): Fair Water damaged Satisfaction 100% guaranteed. Bookmans AZVery Good 020540844359.05 FREE New: New 0205408443 WE HAVE NUMEROUS COPIES-PAPERBACK-light shelf wear to cover, edges, and corners, insignificant tear on top of back cover. Walker Bookstore AZ, USA $72.95 FREE Used Very Good(2 Copies): Very good More Books FL, USA $100.40 FREE Used Very Good(2 Copies): Very good Great customer service. You will be happy! booklab NY, USA $100.40 FREE New: New More Books FL, USA $155.07 FREE New: New Great customer service. You will be happy! booklab NY, USA $155.07 FREE About the Book The resource math teachers have been waiting for is finally here! Volume Two of the Van de Walle Professional Mathematics Series provides practical guidance along with proven strategies for practicing teachers of grades 3 through 5. In addition to many of the popular topics and features from John Van de Walle's market-leading textbook, "Elementary and Middle School Mathematics," this volume offers brand-new material specifically written for these grades. The expanded grade-specific coverage and unique page design allow readers to quickly and easily locate information to implement in the classroom. Nearly 200 grade-appropriate activities are included. The student-centered, problem-based approach will help students develop real understanding and confidence in mathematics, making this series indispensable for teachers! Big Ideas provide clear and succinct explanations of the most critical concepts in 3-5 mathematics. Problem-based activities in Chapters 2-12 provide numerous engaging tasks to help students develop understanding. Assessment Notes illustrate how assessment can be an integral part of instruction and suggest practical assessment strategies. Expanded Lessons elaborate on one activity in each chapter, providing examples for creating step-by-step lesson plans for classroom implementation. A Companion Website ( provides access to more than 50 reproducible blackline masters to utilize in the classroom. The are provided in the appendix for teachers' reference. About the Authors John Van de Walle is Professor Emeritus at Virginia Commonwealth University. He is a co-author of "Scott Foresman-Addison Wesley Mathematics," a K-to-6 textbook series, and the author of "Elementary and Middle School Mathematics: Teaching Developmentally," the best-selling text and resource book on which this professional series is based. LouAnn Lovin is a former classroom teacher and is currently an assistant professor in mathematics education at James Madison University, where she teaches mathematics methods and mathematics content courses for Pre-K-8 prospective teachers and is involved in the mathematical professional development of teachers in grades 4-8. Collect all three volumes in the Van de Walle Professional Mathematics Series! Each volume provides in-depth coverage at specific grade levels. Learn more about the series at
in the series of highly respected Swokowski/Cole mathematics texts retains the elements that have made it so popular with instructors and students alike: its exposition is clear, the time-tested exercise sets feature a variety of applications, its uncluttered layout is appealing, and the difficulty level of problems is appropriate and consistent. The goal of this text is to prepare students for further courses in mathematics.This book is set apart from the competition in a number of ways: it is mathematically sound, it focuses on preparing students for further courses in mathematics, and it has excellent problem sets. This edition has been improved in many respects. All of the chapters include numerous technology inserts with specific keystrokes for the TI-83 Plus and the TI-86, ideal for students who are working with a calculator for the first time. The new design of the text makes the technology inserts easily identifiable, so if a professor prefers to skip these sections it is simple to do so.
MATH 205A: First Half of Elementary Algebra This course is the first half of the Elementary Algebra course. It will cover signed numbers, evaluation of expressions, ratios and proportions, solving linear equations, and applications. Graphing of lines, the slope of a line, graphing linear equations, solving systems of equations, basic rules of exponents, and operations on polynomials will be covered. Sect# Type Room Instructor Units Days Time Start-End Footnotes 0825 LEC PB5 LOCKHART L 2.5 MW 0230P - 0435P Class meets 09/04/07 - 10/25/07 PB3 LOCKHART L 2.5 TuTh MATH 205B: Second Half of Elementary Algebra Prerequisite: Math 205A with a grade of 'C' or better. Advisory: Concurrent enrollment in Guidance 563B is advised. Transferable: GAV-GE: B4 This course contains the material covered in the second half of the Elementary Algebra Course. It will cover factoring, polynomials, solving quadratic equations by factoring, rational expressions and equations, complex fractions, radicals and radical equations, solving quadratic equations by completing the square and the quadratic formula. Application problems are integrated throughout the topics.
Algebra 1 Description Students learn how algebra relates to the physical world with an outstanding textbook presenting mathematics as a study of absolutes. Concepts are developed and mastered through an abundance of worked examples and student exercises. Designed to be used in grades 8 or 9 and is 374 pages.
Abstract: This article compares two algorithms that compute values of the sine and cosine functions using analytic geometry and the unit circle definition of sine and cosine. One algorithm uses chords on the circle and the other uses tangents. > This material is only available to signed-in subscribers. Additional history about the chordic and CORDIC algorithms (PDF) The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
Presented by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course follows up on a previous course, Algebra 1A, which "develops algebraic fluency by providing students... Developed by Tina Fujita, James Hawker, and John Whitlock of Hillsborough Community College, these five curriculum guides integrate mathematical and biological concepts. These guides can be used in mathematics courses... The introduction to this site remarks, "If you need help in college algebra, you have come to the right place." Their statement is accurate, as the staff members at the West Texas A&M University's Virtual Math Lab have... Presented by HippoCampus, a project of the Monterey Institute for Technology and Education, this free online course "is a study of the basic skills and concepts of elementary algebra, including language and operations...
Buy New Textbook Used Textbook We're Sorry Sold Out eTextbook We're Sorry Not Available More New and Used from Private Sellers Starting at $149John Squires and Karen Wyrick have drawn upon their successes in the classroom and the lab as inspiration for MyMathLab for Developmental Math: Prealgebra, Introductory Algebra & Intermediate Algebra . This new MyMathLab® eCourse offers students a guided learning path through content that has been organized into small, manageable mini-modules. This course structure includes pre-made tutorials and assessments for every topic in the course, giving instructors an eCourse that can be easily set up and customized for a variety of learning environments. This package consists of the MyMathLab access kit only, and does not include any supplementary material. Author Biography John Squires has been teaching math for over 20 years. He was the architect of the nationally acclaimed "Do the Math" program at Cleveland State Community College and is now head of the math department at Chattanooga State Community College, where he is implementing course redesign throughout the department. John is the 2010 Cross Scholar for the League for Innovation and the author of the 13th Cross Paper which focuses on course redesign. As a redesign scholar for The National Center for Academic Transformation (NCAT), John speaks frequently on course redesign and has worked with both colleges and high schools on using technology to improve student learning. Karen Wyrick is the current chair of the math department at Cleveland State Community College and has been teaching math there for over 18 years. She is an outstanding instructor, as students have selected her as the college's best instructor more than once, and she was recently awarded a 2011 AMATYC Teaching Excellence Award. Karen played an integral role in Cleveland State's Bellwether Award-winning "Do the Math" redesign project, and she speaks frequently on course redesign at colleges throughout the nation and also serves as a redesign scholar for The National Center for Academic Transformation (NCAT). Table of Contents Mini-Module 1: Whole Numbers 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing Whole Numbers 1.8 More with Multiplying and Dividing 1.9 Exponents 1.10 Order of Operations and Whole Numbers 1.11 More Problem Solving Mini-Module 2: Integers 2.1 Understanding Integers 2.2 Adding Integers 2.3 Subtracting Integers 2.4 Multiplying and Dividing Integers 2.5 Exponents and Integers 2.6 Order of Operation and Integers Mini-Module 3: Introduction to Algebra 3.1 Variables and Expressions 3.2 Like Terms 3.3 Distributing 3.4 Simplifying Expressions 3.5 Translating Words into Symbols Mini-Module 4: Equations 4.1 Equations and Solutions 4.2 Solving Equations by Adding or Subtracting 4.3 Solving Equations by Multiplying or Dividing 4.4 Solving Equations - Two Steps 4.5 Solving Equations - Multiple Steps 4.6 Translating Words into Equations 4.7 Applications of Equations Mini-Module 5: Factors and Fractions 5.1 Factors 5.2 Prime Factorization 5.3 Understanding Fractions 5.4 Simplifying Fractions - GCF and Factors Method 5.5 Simplifying Fractions - Prime Factors Method 5.6 Multiplying Fractions 5.7 Dividing Fractions Mini-Module 6: LCM and Fractions 6.1 Finding the LCM - List Method 6.2 Finding the LCM - GCF Method 6.3 Finding the LCM - Prime Factor Method 6.4 Writing Fractions with an LCD 6.5 Adding and Subtracting Like Fractions 6.6 Adding and Subtracting Unlike Fractions Mini-Module 7: Mixed Numbers 7.1 Changing a Mixed Number to an Improper Fraction 7.2 Changing an Improper Fraction to a Mixed Number 7.3 Multiplying Mixed Numbers 7.4 Dividing Mixed Numbers 7.5 Adding Mixed Numbers 7.6 Subtracting Mixed Numbers 7.7 Adding and Subtracting Mixed Numbers—Improper Fractions Mini-Module 8: Operations with Decimals 8.1 Decimal Notation 8.2 Comparing Decimals 8.3 Rounding Decimals 8.4 Adding and Subtracting Decimals 8.5 Multiplying Decimals 8.6 Dividing Decimals Mini-Module 9: More with Fractions and Decimals 9.1 Order of Operations and Fractions 9.2 Order of Operations and Decimals 9.3 Converting Fractions to Decimals 9.4 Converting Decimals to Fractions 9.5 Solving Equations Involving Fractions 9.6 Solving Equations Involving Decimals Mini-Module 10: Ratios, Rates, and Percents 10.1 Ratios 10.2 Rates 10.3 Proportions 10.4 Percent Notation 10.5 Percent and Decimal Conversions 10.6 Percent and Fraction Conversions 10.7 The Percent Equation 10.8 The Percent Proportion 10.9 Percent Applications Mini-Module 11: Introduction to Geometry 11.1 Lines and Angles 11.2 Figures 11.3 Perimeter - Definitions and Units 11.4 Finding Perimeter 11.5 Area - Definitions and Units 11.6 Finding Area 11.7 Understanding Circles 11.8 Finding Circumference 11.9 Finding Area—Circles Mini-Module 12: More Geometry 12.1 Volume - Definitions and Units 12.2 Finding Volume 12.3 Square Roots 12.4 The Pythagorean Theorem 12.5 Similar Figures 12.6 Finding Missing Lengths 12.7 Congruent Triangles 12.8 Applications of Equations and Geometric Figures Mini-Module 13: Statistics 13.1 Bar Graphs 13.2 Line Graphs 13.3 Circle Graphs 13.4 Mean 13.5 Median 13.6 Mode 13.7 Introduction to Probability Mini-Module 14: Real Numbers 14.1 Introduction to Real Numbers 14.2 Inequalities and Absolute Value 14.3 Adding Real Numbers 14.4 Subtracting Real Numbers 14.5 Multiplying Real Numbers 14.6 Dividing Real Numbers 14.7 Properties of Real Numbers 14.8 Exponents and the Order of Operations Mini-Module 15: Algebraic Expressions and Solving Linear Equations 15.1 Evaluating Algebraic Expressions 15.2 Simplifying Expressions 15.3 Translating Words into Symbols and Equations 15.4 Linear Equations and Solutions 15.5 Using the Addition and Multiplication Properties 15.6 Using the Addition and Multiplication Properties Together Mini-Module 16: Solving More Linear Equations and Inequalities 16.1 Solving Equations with Variables on Both Sides 16.2 Solving Equations with Parentheses 16.3 Solving Equations with Fractions 16.4 Solving a Variety of Equations 16.5 Solving Equations and Formulas for a Variable 16.6 Solving and Graphing Linear Inequalities in One Variable 16.7 Applications of Linear Equations and Inequalities Mini-Module 17: Introduction to Graphing Linear Equations 17.1 The Rectangular Coordinate System 17.2 Graphing Linear Equations by Plotting Points 17.3 Graphing Linear Equations Using Intercepts 17.4 Graphing Linear Equations of the Form x=a, y=b, and y=mx 17.5 Applications of Graphing Linear Equations Mini-Module 18: Slope, Equations of Lines, and Linear Inequalities in Two Variables
High School Algebra: Tutorials, Study Guides and Web Resources for Students, Teachers and Parents Algebra dates back to ancient times when Babylonians solved quadratic equations using almost the same methods that high schoolers are taught today. Our High School Algebra Web Guideexplains math concepts for parents; offers homework help, extra practice and tutorials to the math-phobic student; and gives teachers tools to help students truly understand algebra. Ask your child's teacher what the class is studying and supplement those subjects at home. The "Teaching High School Algebra" section of this Web guide will be of particular interest to many parents and can provide inspiration for ways to get your child interested in algebra. Don't worry if you feel a little rusty working with a subject you haven't touched in years; many sites with algebra resources have "pre-algebra" sections that can provide a refresher course. Having your children teach concepts to you is another great way to get a refresher and can help your children clarify concepts in their own minds. The "High School Algebra Help" section of this guide is full of interesting sites to help explain and interest students in algebra. You may want to browse that section for sites to show your child or sites that you and your child can use together. Algebrahelp.com offers a basic online tutorial that can be helpful for parents with little or no algebra background. Also check out the "Algebra Study Tips" section for guidelines on helping your students work well at home. Homeschool Math collects links to mostly free algebra tutorials and lessons, in addition to other helpful tools such as the "Online Equation Editor," which parents can use to write their own equations at home. Math.com has a parent's section designed to support parents while helping their kids with homework. It includes clear and direct explanations of algebraic principles; a section for formulas, dictionaries, biographies and math tables; and a guide to tutoring options.
Spring 2013 MTH 121 - Trigonometry (4.0 units) Section 4444 Class Begins 22/01/2013 Course Description This course will explore the mathematical uses and implications of triangles with its focus on the six trigonometric functions, the inverse trigonometric functions, and their graphs. Students will learn to solve triangles, apply trigonometry to physical phenomena, and work with the trigonometric functions in an algebraic setting. Topics will also include De Moivre's Theorem and applications with vectors. A graphing calculator will be required for the course. Estimated Time per Week: Students can expect to spend approximately 12 hours per week reading, working on22), read the orientation, and get the Course ID. Then go to register using your Access Code and the Course ID, and begin working on the first week's assignment and discussion. Assignments & Tests: Approximately daily homework; approximately weekly quizzes; a cumulative Final Exam. See syllabus or Assignments in MyLab for more detail. Additional Comments: After an initial login to eTudes to read the orientation and get the access code, the entire course will be conducted online through the MyLab system. Students are required to have Internet access, an active email account, send emails and private messages, and work independently on a schedule
I School Vision u0026 Mission We are fully committed to offering an all-round education enhanced with the gospel spirit and the virtues of humility, respect, kindness and ... Problems on groups PeterJ. Cameron Associated with Introduction to Algebra , OUP 2008 1. The purpose of this exercise is to constructafamily of groups known as free ... Mathematics Enhancement Programme Activity Notes Codes and Ciphers 1 Introduction T: In this first lesson weu0027ll look at the principles of the Lorenz cipher; in the ... KGVu0027s first CAS/Co-curricular Fair It is the first KGV CAS/Co-curricular Fair during tutor-time from Sept 5th to 9th. Each House/College will visit the fair during ... 5 Education Reform The HK Government adopted the u0022 Reform Proposal for the Education System u0022of Education Commission in 2000 Goal: u0022for ALL students to ... Fortran90 Course, Examples A1 1) Using your favouriteeditor, write a program which prints out your name. 2) The following program contains an umber of errors. Basic Mathematics Functions RHoranu0026MLavelle The aim of this document is to provide a short, self assessment programme for students who wish to acquire a basic ... 1 Appendix IV Preliminary Curriculum Framework of Different Key Learning Areas and Liberal Studies for the New Senior Secondary Curriculum During the period for the ...
{"currencyCode":"USD","itemData":[{"},{"priceBreaksMAP":null,"buyingPrice":13.9,"ASIN":"1412960800","isPreorder":0}],"shippingId":"0803958757::WBwaOp37QiMam%2Fy96EioAP9IIXR9%2BDFR2Bibxs0%2B9b6uLDprcRpmst8MKOLijU9lWDHhAl3d8Gjt13v%2BBEZe3XxXKYIlqvy%2B2OAn9SV8eZ8%3D,0803972857::qqQQGlsd%2FYjYfXCz%2Bb2c73GaL4%2F3D5Bi%2FvwLge0eglJ%2BwNhKeZDN4oSpveF1PeK4qEOAqoVLn3FCSLJCdprrUH4QzwfYzvOGUUk7o6XyQ5s%3D,1412960800::qqQQGlsd%2FYju%2BbdSyR9EmBYxHjqBiSHYhewZc4%2Fx9WO53Vj8A%2BAMQtCCg4ELT85OY%2FG%2FluI%2F37A2RuiKqkpkPMxoVz5fKEpVknmiYom6LE concise distillation of the basics. Reviews basic algebra, sets, permutations and combinations, limits, derivatives, integrals, and matrix algebra. Good for a refresher, for introduction, or for filling in gaps in one's basic knowledge. Won't make you a Ph.D. mathematician, but it doesn't pull punches. It's all stated here very simply: terms, equations, and examples. No fluff. Nice slim portable volume. Weighs less than a good sandwich. A friend recommended this book to me when I told him I was going to be doing Mathematics for Social Sciences this Semester. However it is more of a refresher for persons who are familiar with mathematics and not now learning. It did not help me, so if you are a student now learning college mathematics, a textbook is what you need. This book is for those who already know the concepts but just need a refresher. If you've been using stat software to analyze data so long you've forgotten what it's really doing, this book may be for you. Timothy Hagle "... lays bare the basic math underlying the leading statistical procedures of social and behavioral research." Successive chapters review the basic concepts and equations of algebra, limits, differential calculus, multivariate functions, integral calculus and matrix algebra. The concise matrix algebra review seems easier to follow than the longer treatment of the same material in Namboodiri's Matrix Algebra: An Introduction. The book is recommended for a quick review of basic math. The author's discussion of the importance this math has for analysis of social science data is instructive and motivating to applied researchers. The book is well-supplemented by the example calculations in the author's companion volume, Basic Math for Social Scientists: Problems and Solutions. I would suggest this more for a Social Scientist than a math major. I say this because its essentially a quick review of statistics, calculus, etc... which a math major would find redundant. If your a Social Scientist and need to know some math for what ever project your on, I would suggest it.
Prealgebra -With CD (Custom) - 3rd edition Summary: Learn to read, write, and think mathematically with Tussy and Gustafson's PREALGEBRA and its accompanying technology tools designed to help you save time studying and improve your grade. With this prealgebra textbook, you'll develop your study skills, problem solving, and critical thinking as you master mathematical concepts. A pretest gauges your understanding of prerequisite concepts; problems that make correlations between your daily life and the mathematical conc...show moreepts; and study skills information to give you the best chance to succeed in the course. The accompanying CD-ROM and access to iLrn Tutorials, MathNOW personalized study system, and live online tutoring with a math expert who has a copy of your textbook help you every step of the way to success. ...show less 05346185538.25 +$3.99 s/h Good Bookmans AZ Tucson, AZ 2006 Paperback Good Some shelf wear. Satisfaction 100% guaranteed. $156.90 +$3.99 s/h New Textbookcenter.com Columbia, MO Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book $156
Mathematics for aviationDocument Transcript Gall No. Tliis OSMAN1A UNIVERSITY LIBRARY r>' X Dookshould be returned on or before the date last marked below MATHEMATICS FOR THE AVIATION TRADES MATHEMATICS FOR THE AVIATION TRADES by JAMES NAIDICH Chairman, Department of Mafhe mati r.v, Manhattan High School of Aviation Trades MrGKAW-IIILL HOOK COMPANY, N JO W YOK K AND LONDON INC. MATHEMATICS FOR THK AVI VTION TRADES COPYRIGHT, 19I2, BY THK BOOK TOMPVNY, INC. PRINTED IX THE UNITED STATES OF AMERICA AIL rights referred. Tin a book, or parts thereof, in may not be reproduced any form without perm 'nation of the publishers. PREFACE This book has been written for students in trade and who intend to become aviation mechanics. The text has been planned to satisfy the demand on the part of instructors and employers that mechanics engaged in precision work have a thorough knowledge of the fundamentals of arithmetic applied to their trade. No mechanic can work intelligently from blueprints or use measuring tools, such as the steel rule or micrometer, without a knowledge of these fundamentals. Each new topic is presented as a job, thus stressing the practical aspect of the text. Most jobs can be covered in one lesson. However, the interests and ability of the group will in the last analysis determine the rate of progress. Part I is entitled "A Review of Fundamentals for the Airplane Mechanic." The author has found through actual experience that mechanics and trade-school students often have an inadequate knowledge of a great many of the points covered in this part of the book. This review will serve to consolidate the student's information, to reteach what he may have forgotten, to review what he knows, and to technical schools order to establish firmly the basic essentials. Fractions, decimals, perimeter, area, angles, construction, and graphic representation are covered rapidly but provide drill in systematically. For the work in this section two tools are needed. First, a steel rule graduated in thirty-seconds and sixty -fourths is indispensable. It is advisable to have, in addition, an ordinary ruler graduated in eighths and sixteenths. Second, measurement of angles makes a protractor necessary. Preface vi Parts II, III, and IV deal with specific aspects of the work that an aviation mechanic may encounter. The airplane and its wing, the strength of aircraft materials, and the mathematics associated with the aircraft engine are treated as separate units. All the mathematical background required for this Part work is covered in the first part of the book. 100 review examples taken from airplane V contains shop blueprints, aircraft-engine instruction booklets, airplane supply catalogues, aircraft directories, and other trade literature. The airplane and its engine are treated as a unit, and various items learned in other parts of the text are coordinated here. Related trade information is closely interwoven with the mathematics involved. Throughout the text real aircraft data are used. Wherever possible, photographs and tracings of the airplanes mentioned are shown so that the student realizes he is dealing with subject matter valuable not only as drill but worth remembering as trade information in his elected vocation. This book obviously does not present all the mathematics required by future aeronautical engineers. All mathematical material which could not be adequately handled by elementary arithmetic was omitted. The author believes, student who masters the material included in this text will have a solid foundation of the type of mathematics needed by the aviation mechanic. Grateful acknowledgment is made to Elliot V. Noska, principal of the Manhattan High School of Aviation Trades for his encouragement and many constructive suggestions, and to the members of the faculty for their assistance in the preparation of this text. The author is also especially indebted to Aviation magazine for permission to use however, that the numerous photographs throughout of airplanes and airplane parts the text. JAMES NAIDICH. NEW YORK. FOREWORD fascinating. Our young men and our young women will never lose their enthusiasm for wanting to know more and more about the world's fastest growing Aviation is and most rapidly changing industry. We are an air-conscious nation. Local, state, and federal agencies have joined industry in the vocational training of our youth. This is the best guarantee of America's continued progress in the air. Yes, aviation is fascinating in its every phase, but it is not all glamour. Behind the glamour stands the training and work of the engineer, the draftsman, the research worker, the inspector, the pilot, and most important of all, the training and hard work of the aviation mechanic. Public and private schools, army and navy training centers have contributed greatly to the national defense by training and graduating thousands of aviation mechanics. These young men have found their place in airplane factories, in approved repair stations, and with the air lines throughout the country. The material in Mathematics for the Aviation Trades has been gathered over a period of years. It has been tried out in the classroom and in the shop. For the instructor, it solves the problem of what to teach and how to teach it. The author has presented to the student mechanic effort in the aviation trades, the necessary mathematics which will help him while receiving his training in the school home on his own, and while actually work in industry. performing The mechanic who is seeking advancement will find here broad background of principles of mathematics relating a shop, while studying at his to his trade. IX Foreword x The a real need. I firmly believe that the use of this book will help solve some of the aviation text therefore fills help him to do his work more intelligently and will enable him to progress toward the goal he has set for himself. mechanic's problems. It will ELLIOT V. NOSKA, NEW YORK, December, 1941. Principal, Manhattan High School of Aviation Trades Chapter I THE STEEL RULE Since the steel rule chanic's tools, it is one of the it is very important quickly and accurately. Job 1 : most useful for him of a me- to learn to use Learning to Use the Rule Skill in using the rule depends almost entirely on the of practice obtained in measuring and in drawing amount a definite length. The purpose of this job is to give the student some very simple practice work and to stress the idea that accuracy of measurement is essential. There lines of should be no guesswork on any job; there must be no guess- work in aviation. Fig. 1 In Fig. . Steel rule. a diagram of a steel rule graduated in 3 c2nds and G4ths. The graduations (divisions of each inch) are extremely close together, but the aircraft mechanic is often expected to work to the nearest 64th or closer. 1 is Examples: 1. How are the rules in Figs. 2a and 26 graduated? T Fig. 2a. I I M Fig. 2b. The Steel Rule nearest graduation (a) using a rule graduated in 16ths, (b) using a rule graduated in 64ths. Estimate the length of the lines in Fig. 6; then measure them with a rule graduated in 64ths. See how well you can judge the length of a line. 8. Write the answers in your own notebook. Do NOT write in your textbook. s. 6. Job 2: Accuracy of Measurement mechanics find it difficult to understand that can ever be measured exactly. For instance, a nothing piece of metal is measured with three different rules, as Many Fi 9 . 7. shown Notice that there is a considerable differanswers for the length, when measured to the in Fig. 7. ence in the nearest graduation. 1 . The rule graduated in 4ths gives the answer f in. 6 Mathematics The The 2. 3. for the Aviation Trades answer \|- in. answer y-f in. 4ths, it can be used to rule graduated in 8ths gives the rule graduated in IGths give the Since the first rule is graduated in measure to the nearest quarter of an inch. Therefore, f- in. is the correct answer for the length to the nearest quarter. 1 The second rule measures to the nearest 8th (because it is graduated in Hths) and \|- in. is the correct answer to the nearest 8th of an inch. Similarly, the answer y\|- in. is correct to the nearest I6th. If it were required to measure to the nearest 32nd, none of these answers would be correct, because a rule graduated in 32nds would be required. What rule would be required to measure to the nearest 64th of an inch? To obtain the exact length of the metal shown in the figure, a rule (or other measuring instrument) with an infinite number of graduations per inch would be needed. No such rule can be made. No such rule could be read. The micrometer can be used to measure to the nearest thousandth or ten-thousandth of an inch. Although special devices can be used to measure almost to the nearest millionth of an inch, not even these give more than a very, very, close approximation of the exact measurement. The mechanic, therefore, should learn the degree of accuracy required for each job in order to know how to make and measure his work. This information is generally given in blueprints. Sometimes it is left to the best judgment of the mechanic. Time, price, purpose of the job, and measuring tools available should be considered. The mechanic who carefully works to a greater than necessary degree of accuracy The mechanic who less carelessly is wasting time and money. works to a degree of accuracy than that which the job requires, often wastes material, time, and money. When measured by reading the nearest ruler graduation, the possible between graduations. Thus $ in. is the correct length within J in. See ('hap. II, Job 7, for further information on accuracy of measurements. 1 a line is error cannot be greater than half the interval the Steel Rule Examples: 1. What kind nearest 16th? 2. Does it of rule (6) would you use to measure (a) to the to the nearest 32nd? make any difference whether a mechanic works to the nearest 16th or to the nearest 64th? Give reasons for your answer. To what degree of accuracy is work generally done in a woodworking shop? (6) a sheet metal shop? (c) a (a) 3. machine shop? 4. Measure the distance between the points in Fig. 8 to the indicated degree of accuracy. Note: A point is indicated by the intersection of two lines as shown in the figure. What students sometimes call a point is more correctly known as a blot. Fig. 8. In aeronautics, the airfoil section is the outline of the wing rib of an airplane. Measure the thickness of the airfoil section at each station in Fig. 9, to the nearest 64th. 5. Station 1 Fig. 9. Airfoil section. Mathematics 8 6. What is for the Aviation Trades the distance between station 5 and station 9 (Fig. 9)? How well can you estimate the length of the lines in 10? Write down your estimate in your own notebook; Fig. then measure each line to the nearest 32nd. 7. H K Fi g . 10. In your notebook, try to place two points exactly 1 in. apart without using your rule. Now measure the distance between the points. How close to an inch did you 8. come ? Job 3: Reducing Fractions to Lowest Terms Two Important Words: Numerator, Denominator. You probably know that your ruler is graduated in fractions or parts of an inch, such as f ^, j/V, etc. Name any other fractions found on it. Name 5 fractions not found on it. These fractions consist of two parts separated by a bar or fraction line. Remember these two words: A. , Numerator is the number above the fraction line. Denominator is the number below the fraction line. For example, in the fraction \|- 5 is the numerator and 8 the denominator. , is Examples: 1. Name the numerator and the denominator in each of these fractions : f 7 8"> 16 ~3~> 13 5 > T8~> 1 16 9 The Steel Rule 2. Name 5 fractions in which the numerator is smaller than the denominator. 3. Name 5 fractions in which the numerator is larger than the denominator. 4. If the numerator of a fraction what nator, is equal to the denomithe value of the fraction? is 5. What part of the fraction -g^- shows that the measurement was probably made to the nearest 64th? B. Fractions Have Many Names. It may have been noticed that it is possible to call the same graduation on a ... rule bv several different names. _, This con be ecu led , Students sometimes " which of these a fraction ways ask, $ or $ or jj , or j$ , etc* of calling moat correct?" All is t * them are "equally" correct. However, it is very useful to be of .. I graduation IS ' able to change a fraction into an equivalent fraction with a different numerator and denominator. Examples: Answer these questions with the help 3 1- _ = how many? HT^" 4 3 qoL ^8 d - 9 ~ - h w many? ~ 32 4 3 8 ? ~~ o._l_ of Fig. 11: 4 ^1() = ? 2 32 Hint: Multiplying the numerator and denominator of any fraction by the same number will not change the value of the fraction. - 7 ' i_ ~ 3 e 8 (>4 Q^_' ~ Te 8 9> =-1 = fi A__L ~ .'52 16 1 4 2. 8 ? 2 K V ? ~ _ 4 8 1 1 IA a i 10> 2 - ? 9 1 * 4 -^ 32 Mathematics 1 U 11 " <J 4 ' "' 13 12 19 12 32 ?=J 84 1K 15 < J for the Aviation Trades - 4 16 ?8 32 17 ? 1C lb 8 = A ' ~ 8 A ' 14 ? ? _ ~~ i ' 16 = i T 64 ^ 2 _ ~ ? _ ~ 2 ? 4 ? - 32 _?_ 64 8 Hint: Dividing the numerator and denominator of a by the same number will not change the value fraction of the fraction. When a fraction with which it is expressed by the smallest numbers can be written, it is said to be in its "lowest terms." Reduce to lowest terms: A 19. A 20. M 21. Iff 18. 22. 23. 2ff 24. 2ff 25. Which ff 8ff fraction in each of the following groups is the larger? A or i f or H \| or M 26. 29. 32. Job or i 28. or J 33. \| or ff 31. 27. 30. T& A 4: >4n Important Word: ^ ,% or or f-f A Mixed Number Numbers such as 5, 12, 3, 1, 24, etc., are called whole numbers; numbers such as ^, f, j^, etc. are called fractions. Very often the mechanic meets numbers, such as 5^, 12fV, which is a combination of a whole number and a fraction. Such numbers are called mixed numbers. or 1^, each of Definition: A mixed number consists of a whole number and a For example, 2f 3-J, if are mixed numbers. fraction. . The Steel Rule 11 Write 5 whole numbers. Write 5 fractions. Write 5 mixed numbers. Is this statement true: Every graduation on a rule, beyond the 1-in. mark, corresponds to a mixed number? Find the fraction f on a rule? The fraction / is {he same -XT i i j.i_ 8 Notice that it is beyond the asfhemfxed number j." 1-in. ! . . ^ graduation, and by actual ' 1 ' I I I ! I count is equal to 1-g- in. t A. Changing Improper Fractions to Mixed Numbers. Any improper fraction (numerator larger than the denominator) can be changed to a mixed number by dividing the numerator by the denominator. ILLUSTRATIVE Change J = \| to 9 -5- EXAMPLE a mixed number. 8 = Ans. IS Examples: Change these 1 2 -r- 6. 11. V ^ 16. Can ?! mixed numbers: 4. 3. \|f 8.' \|f 9.' f\| 14. 12. ^ all fractions be Explain. B. Changing A fractions to 13. ff 10.' W- 15. ff changed to mixed numbers? Mixed Numbers to mixed number may be changed to a ILLUSTRATIVE Change 2\| W 5. ^-f Improper Fractions. fraction. EXAMPLE to a fraction. 44 44 Check your answer by changing the number. fraction back to a mixed T/ie Steel Job 15 Rule Multiplication of Fractions 7: The multiplication of fractions has many very important applications and is almost as easy as multiplication of whole numbers. ILLUSTRATIVE EXAMPLES 35 88 Multiply 4 \|. 15 04 I < iy ' Take X X 8~X~8 VX 1 2 3X5 15 _ of IS. } X 1 15 _ 4X8" 8 n " 15 ' 3 Method: a. Multiply the numerators, then the denominators. b. Change all mixed numbers to fractions Cancellation can often be used to make first, if necessary. the job of multiplication easier. Examples: 1. 4. 7. X 4 of 18 33 8. 7$ 2. 3\| X I 5. X V X -f X 3f X 2i X 3. of 8;V X M 16 6. 4 1 Find the total length in. An Airplane rib of 12 pieces of round stock, each long. 9. of -?- weighs 1 jf Ib. What is the total weight 24 ribs? 10. The fuel tanks of the Bellanca Cruisair hold 22 gal. of gasoline. What would it cost to fill this per gallon? 11. If 3 Cruisairs were placed wing tip to much room would they need? (Sec Fig. 19.) 12. If tank at 25^<i w ing T tip, they were lined up propeller hub to rudder, 5 of these planes need (Fig. 19) ? much room would how how Mathematics 16 for the Aviation Trades ~2" 34 f Fig. 19. Job Bellanca Cruisair low-wins monoplane. (Courtesy of Aviation.) of Fractions 8: Division A. Division by Whole Numbers. Suppose that, while working on some job, a mechanic had to shear the piece of 20 into 4 equal parts. The easiest way of doing this would be to divide J) T by 4, and then mark the points with the help of a rule. metal shown in Fig. 4- -h Fig. 20. ILLUSTRATIVE 91- + 4 EXAMPLE Divide 9 j by 4. -V X t = = 91 X \| = H = 2& Ans. Method: To divide any fraction by a whole number, multiply by the whole number. Examples: How 1. 4. 4i f quickly can you get the correct answer? + 3 2. H- S 5. 2f -r 4 4-5-5 3. 7\| -s- 9 6. A + 6 1 over The Steel Rule 7. The metal strip in Fig. 21 is 17 to be divided into 4 equal Find the missing dimensions. parts. 7 " 3% + g. 8. 21. Find the wall thickness of the tubes in Fig. 22. Fig. 22. B. Division by Other Fractions. ILLUSTRATIVE 3g - EXAMPLE Divide 3f by \|. 1 = S\| X \| = ^X \| = this example. can the answer be checked? Complete How Method: To divide any and multiply. fraction by a fraction, invert the second fraction Examples: 1. 4. I 12f 7. is - i% 8. I - i A 5. pile of aircraft in. thick. A 2. - \| 14f - If li plywood is 3. 6. 7^ Of l(>i in. -5- high. - \| 7f Each piece pieces are there altogether? stock 12f in. long is to be cut into How many piece of round 8 equal pieces allowing Y$ in. for each cut. What is the 18 Mathematics length of each piece? this distance? Why? for the Aviation Trades Can you use a steel rule to measure How many pieces of streamline tubing each 4-f in. long can be cut from a 72-in. length? Allow ^2 in. for each cut. What is the length of the last piece? 9. Find the distance between centers 10. of the equally spaced holes in Fig. 23. Fi 3 . Job 1. 9: 23. Review Test Find the over-all lengths' in Fig. 24. . '64 2. Find the missing dimensions in Fig. 25. (a) Fig. 25. 3. One of the dimensions Can you find it? of Fig. 26 has been omitted. Chapter DECIMALS II AVIATION IN The ruler is an excellent tool for measuring the length most things but its accuracy is limited to -$% in. or less. For jobs requiring a high degree of accuracy the micrometer of caliper should be used, because thousandth of an inch or closer. it measures to the nearest Spindle, HmlniE Thimble Sleeve Frame Fig. 30. Job 1 : Micrometer caliper. Reading Decimals When is used to measure length, the answer is as a ruler fraction, such as \|, 3^V, or 5^. When expressed a micrometer is used to measure length, the answer is a rule A decimal fraction is a kind of fraction whose denominator is either 10, 100, special 1,000, etc. For example, yV is a decimal fraction; so are expressed as a decimal fraction. T^ and 175/1,000. For convenience, these special fractions are written this way: = 10 0.7, read as seven tenths 20 in Decimals 35 ~ in Aviatior? 21 = = - 0.005 read as five thousandths = 5 0.35, read as thirty -five hundredths 0.0045, 1,000 45 10,000 read as forty-five ten-thousandths, or four and one-half thousandths Examples: 1. Read 2. Write these decimals: these decimals: (a) 45 hundredths (b) (e) 3 and 6 tenths (rf) (e) 35 ten-thousandths Most mechanics will five thousandths seventy-five thousandths (/) one and three thousandths not find much use for decimals beyond the nearest thousandth. When a decimal is given in places, as in the table of decimal equivalents, not these places should or even can be used. The type of (> the mechanic is doing will all of work determine the degree of accuracy required. ILLUSTRATIVE EXAMPLE Express 3.72648: (a) to the nearest thousandth (b) to the nearest hundredth (c) to the nearest tenth 3.7 C2(> 3.73 Ans. Ans. 3.7 Aus. Method: a. Decide how b. If c. Drop the all decimal places your answer should have. following the last place is 5 or larger, add 1. many number other numbers following the last decimal place. Decimals 25 Aviation in ILLUSTRATIVE EXAMPLE Find the total height of 12 sheets of aircraft sheet aluminum, B. and S. gage No. 20 (0.032 in.). I2$heefs ofB.aS Fi g . #20 36. Multiply 0.032 by 12. 0.032 in. _X12 064 J32 0.384 Am. in. Method: a. Multiply as usual. 6. Count the number of decimal places in the numbers being multiplied. c. Count off the same number of decimal places in the answer, starting at the extreme right. Examples: answers to the nearest hundredth Express all X X 2.3 2. 1.2 4. 1. 0.35 3. 8.75 6. 3.1416 7. A 8. The X 0.25 6. : X 14.0 5.875 X 0.25 3.1416 X 4 X 1.35 4 1 dural sheet of a certain thickness weighs 0.174 Ib. per sq. ft. What is the weight of a sheet whose area is 16.50 sq. ft.? tubing 9. is price per foot of a certain size of seamless steel $1.02. What is the cost of 145 ft. of this tubing? The Grumman G-21-A has a wing area of 375.0 the wing can carry an average sq. ft. If each square foot of 1 The word dural is a shortened form the aircraft trades. of duralumin and is commonly used in 26 Mathematics weight of 21.3 lb., for the Aviation Trades how many pounds can the whole plane carry ? Fis. 37. Job Grumman G-21-A, 4: Division of an amphibian monoplane. (Courtesy of Aviation.) Decimals A piece of flat stock exactly 74.325 in. long is to be sheared into 15 equal parts. What is the length of each part to the nearest thousandth of an inch ? 74.325" Fi 9 . 38. ILLUSTRATIVE EXAMPLE Divide 74.325 by 4.9550 15. 15)74.325^ 60 14~3 13 5 75 75~ 75 Each piece will be 4.955 in. long. Ans. Decimals in 27 Aviation Examples: Express answers to the nearest thousandth: all ^9 1. 9.283 -T- 6 2. 7.1462 4. 40.03 -T- 22 5. 1.005 -5-7 3. 2G5.5 6. 103.05 18 -r ~- 37 Express answers to the nearest hundredth: ~ 46.2 7. 2.5 8. 10. 0.692 4- 0.35 A 13. f-in. rivet there in 50 Ib. 42 -5- -r- 0.8 0.5 weighs 0.375 12. 125 Ib. ~ 0.483 9. -f- How many 4.45 3.14 rivets are ? Find the wall thickness 14. / 11. 7.36 of the tubes in Fig. 39. 15. strip of metal 16 in. A long to be cut into 5 equal is parts. What the length of to the nearest is (b) (ct) ' 9> each part thousandth of an inch, allowing nothing for each cut of the shears ? Job 5: Any Changing Fractions to Decimals fraction can be changed into a decimal by dividing the numerator by the denominator. ILLUSTRATIVE EXAMPLES Change to a decimal. 4 6 ~ = 0.8333-f An. 6)5.0000""" The number of decimal places in the answer depends on number of zeros added after the decimal point. Hint: the Change f to a decimal accurate to the nearest thousandth. 0.4285+ = 0.429 f = 7)3.0000"" Arts. 28 Mathematics for the Aviation Trades Examples: Change these 1. fractions to decimals accurate to the nearest thousandth: () f (6) / / Pi ( ra 0) (/) I Q Change these nearest hundredth () f Tff /- (<0 I / 7 UK/ (A) 1 3fe ?i 1 (o fractions to decimals, accurate to the : W -,V (ft) / (sO ifl (j) 2. (<0 A T (rf) ii (^) (/) I- i Convert to decimals accurate to the nearest thousandth 3. : () I (^) i (/>) (/) 1 (C) ^T (rf) (.</) 2 i 5- (/O T'O 4. Convert each of the dimensions in Fig. 40 to decimals accurate to the nearest thousandth of an inch. 3 Drill //sOn Fig. assembly 40. 5. Find the missing dimension 6. What Job 6: is of the fitting in Fig. 40. the over-all length of the fitting? The Decimal Equivalent Chart Changing ruler fractions to decimals ruler fractions is made much easier and decimals to by the use of a chart similar to the one in Fig. 41. A. Changing Fractions to Decimals. Special instructions on how to change a ruler fraction to a decimal by means of the chart are hardly necessary. Speed and accuracy are 30 Mathematics for the Aviation Trades to decimals accurate to the nearest Change these fractions thousandth 21. : & 22. M. 23. -& Change these mixed numbers 24. to decimals accurate to the nearest thousandth: Hint: Change the fraction only, not the whole number. 3H 25. 8H 26. 9& 27. 28. 3ft Certain fractions are changed to decimals so often that it is worth remembering their decimal equivalents. Memorize the following and fractions their decimal equivalents to the nearest thousandth: = = = i i -iV 0.500 0.125 i \| 0.063 jfe = = = 0.250 0.375 0.031 f = 0.750 f = 0.625 ^f = 0.016 % = 0.875 B. Changing Decimals to Ruler Fractions. The decimal equivalent chart can also be used to change any decimal to its nearest ruler fraction. This is extremely important metal work and in in the machine shop, as well as in many other jobs. ILLUSTRATIVE Change 0.715 to the nearest From ruler fraction. the decimal equivalent chart If 0.715 EXAMPLE lies = between f\| .703125, f and ff but , we can - it is see that .71875 nearer to -f-g-. Ans. Examples: 1. (a) 2. (a) 3. (a) Change these decimals 0.315 (b) 0.516 (c) Change these decimals 0.842 (6) 0.103 Change these 0.309 (b) (c) to the nearest ruler fraction: 0.218 (rf) (c) (e) 0.832 to the nearest ruler fraction: 0.056 (d) to the nearest 64th 0.162 0.716 0.768 0.9032 (e) 0.621 0.980 (e) 0.092 : (d) Decimals 4. Fig. in As a mechanic you are 42, but all you have is a Convert all Aviation to 31 work from the drawing steel rule in in 64ths. graduated dimensions to fractions accurate to the nearest 64th. ^44- -0< _2 Fig. 5. 42. Find the over-all dimensions in Fig. 42 (a) in decimals; (6) in fractions. Fig. 43. Airplane turnbuclde. Here is a table from an airplane supply catalogue the dimensions of aircraft turnbuckles. Notice how giving the letters L, A, D, ./, and G tell exactly what dimension is 6. referred to. Convert to the nearest 64th. all decimals to ruler fractions accurate 32 Mathematics A 7. What for the Aviation Trades is to be sheared into 3 equal parts. the length of each part to the nearest 64th of an line 5 in. long is inch ? Job 7: Tolerance and Limits A group of apprentice mechanics were given the job of cutting a round rod 2^ in. long. They had all worked from the drawing shown in Fig. 44. The inspector work found these measurements their who checked : H (6) Should all (c) 2f pieces except e be thrown 2-'" . Fig. 44. Since (d) >\| Round away? Tolero,nce'/ 2 3 rod. impossible ever to get the exact size that a blueprint calls for, the mechanic should be given a certain permissible leeway. This leeway is called the tolerance. it is Definitions: Basic dimension the exact size called for in a blueprint or working drawing. For example, 2-g- in. is the basic is dimension in Fig. 44. Tolerance is the permissible variation from the basic dimension. of Tolerances are always marked on blueprints. A tolerance means that the finished product will be acceptable even ^ if it is as much basic dimension. A 1 as y ^ in. greater or tolerance of 0.001 missible variations of more and acceptable providing they dimension. A tolerance of part will be acceptable even fall in. less than the means that per- than the basic size are with 0.001 of the basic less A'QA.I if it is means that the as much finished as 0.003 greater Decimals in 33 Aviation than the basic dimension; however, it may only be 0.001 less. Questions: 1. 2. What does a tolerance of -gV mean? What do these tolerances mean? 0.002 (a) W , I* (6) +0.0005 -0.0010 , , (e) 0.015 , . (C) +0.002 -0.000 +0.005 -0.001 What is meant by a basic dimension of 3.450 in. ? In checking the round rods referred to in Fig. 44, the inspector can determine the dimensions of acceptable pieces 3. work by adding the plus tolerance to the basic dimension and by subtracting the minus tolerance from the basic dimension. This would give him an upper limit and a lower limit as shown in Fig. 44a. Therefore, pieces measuring less of than 2^\| in. are not acceptable; neither are pieces measur- +Basf'c size = 1 Z //- 2 -Upper limii-:2^ =2j 2 Fig. more than 2^-J in. As a is rejected. passed, and ing > 44a. result pieces a, 6, d, and e are c. There another way of settling the inspector's problem. All pieces varying from the basic dimension by more than 3V in. will be rejected. Using this standard we find that pieces a and 6 vary by only -fa; piece c varies by -g-; piece d is by 3^; piece e varies not at all. All pieces except are therefore acceptable. The inspector knew that the tolerance was -&$ in. because it was printed on the varies c drawing. 34 Mathematics for the >Av/at/on Tracfes Examples: 1. The basic dimension of a piece of work is 3 in. and is in. Which of these pieces are not ^ the tolerance acceptable ? (a) %V (b) ff (c) 2-J ((/) Si (e) Sg^s- A blueprint gives a basic dimension of 2\| in. arid tolerance of &$ in. Which of these pieces should be 2. rejected? (a) 2\|i (b) 3. What (c) 2 Vf (d) : 2.718 (e) 2.645 What 4. 2-$\| are the upper and lower limits of a job whose basic dimension is 4 in., if the tolerance is 0.003 in.? n nm U.Uul 6. ; tf What are the limits of a job where the tolerance ^ e basic dimension is is 2.375? are the limits on the length and width of the job in Fig. Fis. Job 1. 8: Review Test Express answers to the nearest hundredth: (a) 3.1416 X (c) 4.7625 + 2. 44b. 2.5 X 0.325 2.5 + 42 - (6) 20.635 Convert these gages to nearest 64th: 4.75 - - 0.7854 0.0072 fractions, accurate to the Decimals in Aviation 35 Often the relation between the parts of a fastening is given in terms of one item. For example, in the rivet in 3, Fis. Fig. 45, all parts follows: 45. depend on the diameter R = C = B = 0.885 0.75 1.75 XA XA XA of the shank, as 36 Mathematics for the Aviation Trades Complete the following 4. A 20-ft. table: length of tubing is to be cut into 7\|-in. lengths. Allowing jV in. for each cut, how many pieces of tubing would result? What would be the length of the last piece ? 5. Measure each of the lines in Fig. 64th. Divide each line into the indicated. What is 45a to the nearest number of equal parts the length of each part as a ruler fraction ? H 3 Equal paris (a) H 5 Equal parts (c) h -I 6 Equal parts 4 Equal parts (d) Fig. 45a. Chapter III MEASURING LENGTH The work in the preceding chapter dealt with measuring lengths with the steel rule or the micrometer. The answers to the Examples have been given as fractions or as decimal parts of an inch or inches. units of length. Job 1 However, there are many other Units of Length : Would it be reasonable to measure the distance from New York to Chicago in inches? in feet? in yards? What unit is generally used? If we had only one unit of length, could it be used very conveniently for all kinds of jobs? In his work, a mechanic will frequently meet measurements in various units of length. Memorize Table 1. TABLE 12 inches 8 feet 5,280 feet 1 meter (in. LENGTH 1. or ") = = = = or 1 foot 1 yard (yd.) 1 mile (mi.) (ft. ') 89 inches (approx.) Examples: How many inches are there in 5 ft.? in How many feet are there in 1. 1 yd.? in S^ft.? 2. 3^ yd.? in 48 in.? in Similes? yards are there in 4. How many How many 6. Round rod of a certain 3. 1 mile? yV mile? diameter can be purchased at $.38 per foot of length. What is the cost of 150 in. of this rod? inches are there in 37 39 Measuring Length r Job 2: Perimeter Perimeter simply means the distance around as shown Fig. 48. in Fig. 48. To find the perimeter of a figure of of sides, add the length of all the sides. EXAMPLE ILLUSTRATIVE Find the perimeter of the triangle in Fig. 49. Fig. Perimeter Perimeter any number = = 2 49. + 5f IT in. + 2^ Anx. in. Examples: Find the perimeter of a triangle whose sides are 3-g- in., (>rg in., 2j in. 2. Find the perimeter of each of the figures in Fig. 50. 1. All dimensions are in inches. (a) (b) Fig. 50. 40 Mat/iematics for the Aviation Trades 3. Find the perimeter of the figure in Fig. 51. Measure accurately to the nearest 32nd. Fis. 51. 4. A regular hexagon (six-sided figure in which all sides are of equal length) measures 8^ in. on a side. What is its perimeter in inches? in feet? Job 3: Nonruler Fractions be noticed that heretofore we have added fractions whose denominators were always 2, 4, 6, H, 16, 32, or 64. These are the denominators of the mechanic's most useful fractions. Since they are found on the rule, these It should have been called ruler fractions. There are, however, many occasions where it is useful to be able to add or subtract nonruler fractions, fractions that are not found on the ruler. fractions ILLUSTRATIVE Find the perimeter Perimeter EXAMPLE of the triangle in Fig. 52. = Sum = 15U ft. Ans. 41 Measuring Length Notice that the method used in the addition or subtraction of these fractions is identical to the method already learned for the addition of ruler fractions. It is sometimes harder, however, to find the denominator of the equivalent fractions. This denominator is called the least common denominator. Definition: The common denominator (L.C.D.) of a group of the smallest number that can be divided exactly least fractions is by each of the denominators of all the fractions. For instance, 10 is the L.C.D. of fractions -^ and because 10 can be divided exactly both by 2 and by 5. Similarly 15 is the L.C.D. of f and . Why? There are various methods of finding the L.C.D. easiest one L.C.D. of 2 and 3, (> -g- is the L.C.D. Examples: 1. Find the L.C.D. Of i and i Of i f Of i, i, of f (a) (6) , (<) (<) 2. () , k, 3. (a) Add ! 4. i , A A these fractions: (&) i i, ro (c) i i, i % Solve the following: -f The by inspection or trial and error. What is the and ^? Since 6 can be divided exactly by both is (t) f Find the sum +f of 4i A ft., 5 TV ft., li ft. M- Fi g . 53. 42 Mathematics for the Aviation Trades Find the total length in feet of the form in Fig. 53. 6. Find the total length in feet of 3 boards which are ft., 8f ft., and 12f ft. long. 7. Find the perimeter of the figure in Fig. 54. 5. -b f _ O ; Fi g . = W l/l2 , c = Ct = /3 /4 1 54. 8. Find the perimeter of the plate in Fig. 55. Express the answer in feet accurate to the nearest hundredth of a foot. 9. The perimeter is 4-g- ft. of a triangle is 12y^ ft. If the first side side is 2f ft., what is the length of and the second the third side? 10. Find the total length in feet of a fence needed to enclose the plot of ground shown in Fig. 56. Pis. 56. 43 Measuring Length Job 4: The Circumference of a Circle Circumference is a special word which means the distance around or the perimeter of a circle. There is absolutely no reason why the word perimeter could not be used, but it never is. A 1. Any line Few Facts about the Circle from the center to the circumference is called a radiux. 2. Any line drawn through the center and meeting the circumference at each end is called a diameter. 3. The diameter is twice as long as the radius. 4. All radii of the equal; all same circle are diameters of the same circle are equal. Finding the circumference of a is a little harder than finding: the distance around figures with straight sides. The following formula circle Formula: = = D where C C= 3.14 _ "V" V"7 Circle, Fig. 57. is used: X D circumference. diameter. The "key number" is used in finding the circummatter what the diameter of the circle is, to find its circumference, multiply the diameter by the "key number/' 3.14. This is only an approximation of the exact number 3.1415926+ which has a special name, TT (pronounced pie). Instead of writing the long number 3. 1415926 +, it is easier to write TT. The circumference of a circle can therefore be written ference of circles. 3.14 No C= X D 44 Mathematics for the Aviation Trades If a greater degree of accuracy is required, 3.1416 can be used instead of 3.14 in the formula. The mechanic should practically never have any need to go beyond ILLUSTRATIVE Find the circumference this. EXAMPLE of a circle whose diameter is 3.5 in. Fig. 58. Given: D = 3.5 in. Find Circumference : C = C = C = 3.14 3.14 X D X 3.5 10.99 in. Ans. Examples: 1. Find the circumference of a circle whose diameter is 4 in. 2. What diameter 3. A is the distance around a pipe whose outside is 2 in. ? circular tank has a diameter of 5 ft. What is its circumference ? Measure the diameter of the circles in Fig. 59 to the nearest 32nd, and find the circumference of each. 4. (C) 45 Measuring Length Estimate the circumference of the 5. circle in Fig. 60a. Calculate the exact length after measuring the diameter. How close was your estimate? Find the circumference 6. of a circle whose radius is 3 in. Hint: First find the diameter. What the total length in feet of 3 steel bands which must be butt- welded around the barrel, as shown in Fig. 60& ? 7. is Fi9. What 8. radius is 15-g- What 9. is diameter is 60a. Fig. 60b. the circumference in feet of a steel plate whose in.? is the circumference of a round disk whose 1.5000 in.? Use TT = 3.1416 and express the answer to the nearest thousandth. Job 5: 1. 246.5 Review Test The Monocoupe shown in. Fig. What 61 . is its in Fig. 61 has a length of length in feet? Monocoupe high-wing monoplane. (Courtesy of Aviation.) Copter IV THE AREA OF SIMPLE FIGURES The length of any object can be measured with a rule however, to measure area so directly and It is impossible, simply as that. In the following pages, you will meet geometrical shapes like those in Fig. 65. Circle Square Rectangle Trapezoi'd Triangle Fi g . Each of these shapes some arithmetic before 65. separate formula and area can be found. You should will require a its know A these formulas as well as you know how to use a rule. mechanic should also know that these are the cross- sectional shapes of most common beams, rivets, sheet metal, etc. Job Units of objects, such as nails, 1 : Area Would you measure the area of a small piece of metal in square miles? Would you measure the area of a field in square inches? The unit used in measuring area depends on the kind of work being done. Memorize 47 this table: 50 Mathematics for the Aviation Trades nearest 16th) of the rectangles in Fig. 67. the area of each. 8. Find the area Then in square feet of the airplane calculate wing shown Fig/ 68. in Trailin A/Te ran I '- Fig. 9. 68. Aileron V Leading edge Airplane wing, top view. Calculate the area and perimeter of the plate shown in Fig. 69. i Fig. B. Length and Width. To 69. find the length or the width, use one of the following formulas: Formulas: L w A L where L = A W = length. area. width. ILLUSTRATIVE The its EXAMPLE area of a rectangular piece of sheet metal width Given: ^ ft. What A = 20 W sq. ft. 2i ft. is = is its length? is 20 sq. ft.; The Area of Simple Figures 51 Find: Length L = y IF 20 = Check: yt=L H' 2i 20 = X = \| 8X 2 Ann. Hft. - 20 sq. ft. Examples: 1. The area of a rectangular floor length of the floor 2-5. Complete if its width this table is 7 is ft. 75 sq. ft. What is the in. ? by finding the missing dimen- sion of these rectangles: the width of a rectangular beam whose cross-sectional area is 10.375 sq. in., and whose length is 3 in. as shown in Fig. 70? 5, 6. What must be The length (span) of a rectangular wing is 17 ft. 6 in.; its area, including ailerons, is 50 sq. ft. What is the width (chord) of the wing? 7. 54 Mathematics Job Square Root 4: Introduction to The for the Aviation Trades following squares were learned from the last job. TABLE = = = = 2 I & 32 42 52 Find the answer to 8 2 1 (i 4 7 2 82 1(5 92 25 10 2 = = - 86 49 64 81 100 this question in Table 3: What number when multiplied by itself equals 49? which is said to be the square root of 41), written /49. The mathematical shorthand in this case is (read "the square root of ") The entire question can be The answer is 7, V . written What V49? The answer is Check: 7X7= is 7. 49. Examples: 1. What is the number which when multiplied by equals 64? This answer is 8. Why? 4. What number multiplied by itself What is the square root of 100? What is V36? 5. Find 2. 3. * (a) (e) itself V_ V400 6. How (6) VsT (/) VT (r) (g) equals 2.5 ? (</) /49 (A) VlO_ Vl44 Vil can the answers to the above questions be checked ? 7. 8. 9. (a) (g) Between what two numbers does VI 7 ? Between what two numbers is Between what two numbers are V7 V V4 10. lie? (6) (/) VS^ VTS From Table than 75? 1 what (r) (0) is V4S Viw (rf) (/O the nearest perfect square less The Area of Simple Figures Job 5: 55 The Square Root of a Whole Number So far the square roots of a few simple numbers have been found. There is, however, a definite method of finding the square root of any whole number. ILLUSTRATIVE What EXAMPLE the square root of 1,156? 3 Am. is 9 64) 256 256 Check: 34 X 34 = 1,156 Method: a. Separate the number into pairs starting from the b. /H lies smaller c. A/11 56 right: between 3 and 4. Write the 3, above the 11: 56 number Write 3 2 or 9 below the 11: 3 Xli~56 9 d. Subtract and bring down the next pair, 56 _ A/11 56 9 __ 2 56 e. Double the answer (3 X Write 6 as shown /. * so far obtained = 6). : Using the 6 just obtained as a trial divisor, it into the 25. Write the answer, 4, divide as shown: 3_ XlF56 3 __ A/11 56 9 56 Mathematics g. for the Aviation Trades Multiply the 64 by the 4 just obtained and write the product, 256, as shown: 4 Ans. 3 56 9 64) 2 56 2 56 h. Since there no remainder, the square root is of 1,156 is exactly 34. Check: 34 X 34 - 1,156. Examples: Find the exact square root 1. of 2,025. Find the exact square root of 2. What 6. 3. 4,225 4. 1,089 625 5. 5,184 the exact answer? is V529 7. V367 8. /8,4(>4 9. Vl~849 Find the approximate square root of 1,240. Check answer. your Hint: Work as explained and ignore the remainder. To check, square your answer and add the remainder. 10. ! Find the approximate square root of I , 11. 4,372 12. 9,164 13. 3,092 5 14. 4,708 15. 9,001 16. 1,050 17. Fi9 ' connection 73 is 300 18. 8,000 ' 19. Study Fig-. 73 carefully. What there between the area of this square and the length of its sides? Job 6: The Square Root of Decimals Finding the square root of a decimal is very much like finding the square root of a whole number. Here are two rules: The Area of Simple Figures Rule The grouping 57 numbers into pairs should always be started from the decimal point. For instance, 1. 362.53 is 893.4 is 15.5 paired as 3 62. 53 paired as 71. 37 83 is 71.3783 of is paired as 8 93. 40 paired as 15. 50 is added to complete any incomplete, on the right-hand side of the decimal point. pair Rule 2, The decimal point of the answer is directly above the decimal point of the original number. Notice that a zero Two examples are given below. Study them carefully. ILLUSTRATIVE EXAMPLES Find V83.72 9.1 Ans. -V/83.72 81 181) 2 72 1JU " 91 Check: 9.1 X 9.1 Remainder = 82.81 = +.91 83 72 . Find the square root of 7.453 Ans. 2.7 3 V7.45 30 J 47)^45 329 543) 16 30 16 29 _ Check: 2.73 X 2.73 = 7.4529 Remainder = +.0001 7.4530 58 Mathematics for the Aviation Trades Examples: 1. What the square root of 34.92? Check your is answer. * What is the square root of 15.32 2. What 3. 80.39 4. 342.35 5. is 7. 10. 75.03 VT91.40 7720 /4 1.35 11. A/137.1 27.00 12. 9. 13. V3.452 V3.000 Find the square root to the nearest tenth: 14. 15. 39.7000 462.0000 17. 193.2 16. 4.830 to the nearest tenth. 18. Find the square root of Hint: Change y to a decimal and find the square root of the decimal. jj Find the square root of these fractions to the nearest hundredth: 22. 20. 19. i 23. 1 75.00 24. Job 7: Find the square root to the nearest tenth. of .78 The Square A. The Area of a Square. The square is really a special kind of rectangle where all sides are equal in length. 59 The Area of Simple Figures A Few Facts about the Square 3. have the same length. four angles are right angles. The sum of the angles is 360. 4. A line joining two opposite corners is called a diagonal. 1. All four sides 2. All Formula: where N means the side A- S2 = S X S of the square. ILLUSTRATIVE EXAMPLE Find the area of a square whose side Given S = 5^ in. Find: Area 5^ in. 3. side = is : A A A A = = = = /S 2 (5i) V X -HP V30-i sq. in. Ans. Examples: Find the area 1. = 2i side 4-6. shown in. of these squares: 2. side = 5i Measure the length ft. 3.25 in. of the sides of the squares and find the area in Fig. 75 to the nearest 32nd, of each: Ex. 4 Ex. 5 Fi 9 . Ex. 6 75. 7-8. Find the surface area of the cap-strip gages in Fig. 76. shown 60 Mathematics for the Aviation Trades Efe S T~ L /" r- _t . 2 --~,\| ^--4 \|< Ex. 8 Ex. 7 Fig. 76. 9. a side. Cap-strip sages. A square piece of sheet metal measures 4 ft. 6 in. on Find the surface area in (a) square inches; (6) square feet. A family decides to buy linoleum at $.55 a square yard. What would it cost to cover a square floor measuring 12 ft. on a side? 10. B. The Side of a Square. the following formula. To find Formula: S = A = where 8 = /A side. area of the square. ILLUSTRATIVE A the side of a square, use EXAMPLE mechanic has been told that he needs a square beam whose cross-sectional area 5 sq. in. is 6. What are the dimensions of this beam? Given: A = Find: Side 6.25 sq. in. = 8 = /25 S = 2.5 in. s Check: A = 8 = 2 2.5 X 2.5 = Ans. 6.25 sq. in. The Area of Simple Figures 61 Method: Find the square root of the area. Examples: Find to the nearest tenth, the side of a square whose area is 1. 3. 47.50 sq. 8.750 sq. in. 2. in. 4. 24.80 sq. ft. 34.750 sq. yd. 5-8. Complete the following table by finding the sides in both feet and inches of the squares whose areas are given: Job A. 8: The Circle The Area of a Circle. The circle is the cross-sectional shape of wires, round rods, bolts, Fig. 77. Formula: where A /) 2 D area of a D XD. diameter. A= circle. rivets, etc. Circle. 0.7854 X D2 62 Mathematics for the Aviation Trades EXAMPLE ILLUSTRATIVE Find the area of a Given: D = 3 in. circle whose diameter 3 is in. A Find: A = A = A = = ^4 0.7854 0.7854 0.7854 X D X3 X X 9 2 7.0686 sq. in. 3 Ans. Examples: Find the area 1. 4 4. S of the circle 2. ft. i 5. in. 7-11. whose diameter is 3. 5 in. li yd. ft. 6. Measure the diameters 2\| mi. of the circles shown in Fig. 78 to the nearest Kith. Calculate the area of each circle. Ex.8 Ex.7 4 -- Ex.10 Ex.11 What is the area of the top of a piston whose diameter 12. is Ex.9 in. ? 13. What is the cross-sectional area of a \|-in. aluminum rivet ? 14. radius 16. Find the area in square inches of a in. Find whose circle is 1 ft. A circular plate has a radius of 2 ft. (> (a) the area in square feet, (b) the circumference in inches. B. Diameter and Radius. The diameter of a circle can be found if the area is known, by using this formula: The Area of Simple Figures Formula: D = D diameter. A = where 63 area. ILLUSTRATIVE Find the diameter 3.750 sq. EXAMPLE a round bar whose cross-sectional area of is in. Given: A = 3.750 Find Diameter sq. in. : - 0.7854 /A0 /) " J) = V4.7746 0.7854 D = A = Check: 0.7854 X D = 2 0.7854 X X 2.18 2.18 = 3.73 + sq. in. Why doesn't the answer check perfectly? Method: a. 6. Divide the area by 0.7S54. Find the square root of the result. Examples: 1. Find the diameter 2. What sq.ft.? 3. is whose area is 78.54 ft. a circle whose area is 45.00 of a circle the radius of . The area of a piston is 4.625 sq. diameter ? 4. A What is (6) What 6. A area of 0.263 sq. in. () the diameter of the wire? is its HJ f y Section A -A ' has a cross-sec- tional area of 1.025 sq. in. its Are1.02Ssq.in. radius? steel rivet is / , copper wire has a cross- sectional What in. What is its diameter 9 * (see Fig. 71)) ? 64 Mathematics 6-9. Complete for the Aviation Trades this table: Find the area of one side 10. of the washers shown in Fig. 80. Fig. 80. Job The Triangle 9: So far we have studied the rectangle, the square, and the circle. The met on the triangle is another simple geometric figure often job. A 1. A 2. Few Facts about the Triangle The sum triangle has only three sides. of the angles of a triangle Base Fig. 81. Triangle. is 180. Area of Simple Figures Tfie 3. A triangle having one right angle is 65 called a right triangle. A 4. an triangle having all sides of the same length is called equilateral triangle. A 5. two equal triangle having an sides is called isosceles triangle. Right The area Isosceles Equilaferal Fig. 81 a. of any Types of triangles. triangle can be found by using this formula: A= Formula: where = = b a l/2 X b the base. EXAMPLE Find the area of a triangle whose base is Given: 6 a Find: a the altitude. ILLUSTRATIVE altitude X 3 is 7 in. long and whose in. = = 7 3 Area A = A = A = \| i- X X -TT = b 7 X X 10-g- a 3 sq. in. Ans. Examples: 1. Find the area whose altitude 5 whose base is 8 in. and in. A triangular piece of sheet metal has a base of 16 a height of 5^ in. What is its area? 2. and is of a triangle What the area of a triangle whose base whose altitude is 2 ft. 3 in.? 3. is is 8-5- ft. in. and 66 Mathematics for the Aviation Trades 4-6. Find the area of the following triangles : Measure the base and the altitude of each triangle in Fig. 82 to the nearest 64th. Calculate the area of each. 7-9. Ex.9 Ex.8 Ex.7 Fig. 82. 10. Measure and calculate the area the three different ways shown to the nearest 04th of the triangle in Fig. 83 in c Fig. in the table. Does it 83. make any difference which side is called the base? Job 10: The Trapezoid The trapezoid often appears as the shape of various parts of sheet-metal jobs, as the top view of an airplane wing, as the cross section of spars, and in many other connections. Tfie A 1. A Area of Simple Figures Few Facts about the Trapezoid trapezoid has four sides. 2. Only one pair of opposite sides are called the bases. is Base (bj) \|< parallel. Fig. 84. The perpendicular These sides * Base (b2 ) k 3. 67 ->\| Trapezoid. between the bases distance is called the altitude. Notice how closely the formula for the area of a trapezoid resembles one of the other formulas already studied. Formula: where a 61 62 = = = A- l/2 X 61 62 Find: 2) one base. the other base. Find the area Given: a +b (b t the altitude. ILLUSTRATIVE bases are 9 X a in. = = = of a trapezoid and 7 6 is 6 in. and whose in. 9 whose altitude in. in. 7 EXAMPLE in. Area A = A = A = A i 1 i X X x 48 a 6 6 X X X sq. in. (fci (7 +6 + 9) 2) 16 Atts. Examples: Find the area of a trapezoid whose altitude and whose bases are 15 in. and 12 in. 1. is 10 in, 68 Mathematics for the Aviation Trades Find the area of a trapezoid whose parallel sides are 1 ft. 3 in. and 2 ft. 6 in. and whose altitude is J) in. Express your answer in (a) square feet (&) square inches. 3. Find the area of the figure in Fig. S4a, after making all necessary measurements with a rule graduated in 3nds. Estimate the area first. 2. Fig. 84a. Find the area in square feet of the airplane wing, the ailerons, shown in Fig. Mb. including 4. ^^ / Leading edge /5 -'6">\| A Heron Aileron IQ'-9'L Trailing Fi s . 5. Find the area edge 84b. of the figures in Fig. 84c. . / r' (a) Fis. Job 11 Review : 1. by 84c. Test Measure to the nearest 32nd letters in Fig. 85. all dimensions indicated 69 The, Area of Simple Figures E 4-- Fig. 2* shown F H* K <?- Box beam. 85. Calculate the cross-sectional area of the box beam in Fig. 85. 3-4. The advertisement shown in the real estate section of a large in Fig. 86 appeared newspaper. Note: Jackson Are. crosses at right angles to Argyle Rd. 1_ - 220' - JACKSON AVENUE Fi 9 . 86. Find the number of square feet in each of the four lots. of putting a fence completely (6) What would be the cost around lot 4, if the cost of fencing is &l per foot? Find to the nearest tenth the square root of (a) 5. 7S.62 6. 10,009 7. 0.398 the diameter of a piston whose area is 23.2753 sq. in.? Express your answer as a decimal accurate to the nearest hundredth of an inch. 9. A rectangular board is 14 ft. long. Find its width if 8. its What is surface area 10. What is 38.50 sq. in.? is 10.5 sq. ft. the circumference of a circle whose area is Gapter V VOLUME AND WEIGHT A the technical term for anything that occupies space. For example, a penny, a hammer, and a steel rule are all solids because they occupy a definite space. Volume is solid is the amount of space occupied by any object. Job 1 : Units of Volume too bad that there no single unit for measuring all kinds of volume. The volume of liquids such as gasoline is generally measured in gallons; the contents of a box is measured in cubic inches or cubic feet. In most foreign countries, the liter, which is about 1 quart, is used as the unit of volume. However, all units of volume are interchangeable, and any one of them can be used in place of any other. MemoIt is is rize the following table: TABLE 1,728 cubic inches 27 cubic feet 2 pints 4 quarts 281 cubic inches 1 cubic foot 1 liter VOLUME 4. = 1 1 = = = = = cubic foot (cu. ft.) cubic yard (cu. yd.) 1 quart 1 gallon (gal.) 1 gallon (approx.) (qt.) 1 gallons (approx.) 1 auart faoorox.) Volume and Weight 71 Fig. 87. Examples: 1. 1 How many cu. yd.? in 2. 3. 4. Job ^ How many How many How many 2: cubic Inches are there in 5 cu. ft.? in cu. ft.? in 3-j- cu. yd.? pints are there in (> qt.? in 15 gal.? cubic inches are there in C2 qt. ? in ^ gal. ? gallons are there in 15 cu. ft. ? in 1 cu. ft. The Formula for ? Volume Figure 88 below shows three of the most common geometrical solids, as well as the shape of the base of each. Solid ~ TTfjl h !'! Box Cylinder Cube Boise Circfe Rectangle Fi 3 . Square 88. The same formula can be used to find the cylinder, a rectangular box, or a cube. volume of a 72 Mathematics for the Aviation Trades V Formula: where V = A = h =AX h volume. area of the base. = height. Notice that it will be necessary to remember the formulas for the area of plane figures, in order to be able to find the volume of solids. ILLUSTRATIVE EXAMPLE Find the volume in cubic inches of a rectangular box whose is 4 by 7 in. and whose height is 9 in. Given base : Base: rectangle, L = = h = 7 in. 4 H' in. 9 in. Find: a. Area 6. Volume of base a. b. Area Area Area W ^7X4 = L X = 28 sq. in. Volume = A X h Volume = 28 X 9 Volume = 252 cu. in. Arts. Examples: Find the volume in cubic inches of a box whose height is 15 in. arid whose base is 3 by 4-^ in. 2. Find the volume of a cube whose side measures 1. 3^ in. 3. its A cylinder has a base whose diameter volume, 4. if it is What is is 2 the volume of a cylindrical base has a diameter of 15 in. oil tank whose and whose height Express the answer in gallons. 6. How many cubic feet of air does a room 12 by 15 ft. by 10 ft. Find in. 3.25 in. high. 3 in. contain? is 2 ft.? ft. 6 in. Volume and Weight 73 6. Approximately how many cubic feet of baggage can be stored in the plane wing compartment shown in Fig. 89? 7. How many gallons of tangular tank 3 by 3 by 5 oil can be contained in a rec- ft. ? Hint: Change cubic feet to gallons. 8. What is the cost, at $.19 per gal., of enough gasoline to fill a circular tank the diameter of whose base is 8 in. and whose height 9. tank 12 3 ft. 6 in. ft. 10. is 15 in.? How many An 3 in. quarts of oil can be stored in a circular long if the diameter of the circular end is ? airplane has 2 gasoline tanks, each with the specifications shown in Fig. can this plane hold? [<_ Job 3: !)0. How many gallons of fuel j' TheWeight of Materials In comparing the weights of different materials a standard unit of volume must be used. Why? In the table below, the unit of volume used as a basis for the comparison of the weights of different materials is 1 cu. ft. Volume and Weight b. Weight = V X unit weight Weight = 44- X 52 Weight - 234 11). A ?iff. Notice that the volume (cubic feet) essential 75 is calculated in the the table of unit weights. as same units Why is this ? Examples: 1. Draw up a table of weights per cubic inch for all the 5. Use this table in the following materials given in Table examples. Find the weight of each of these materials: 2. 1 round aluminum rod 12 ft. long and with a diameter of in. 3. square aluminum rods, l\| by 5 1-J- in in., 12-ft. lengths. 4. 100 square hard-drawn copper rods in 12-ft. lengths each J by J in5. 75 steel strips each 4 by f in. in 25-ft. lengths. Find the weight of 6. 7. 8. 1 A spruce beam 1^ by 3 in. by 18 ft. in. by 15 6 oak beams each 3 by 4 - ft. 500 pieces of ^-in. square white pine cap strips each yd. long. 9. and A solid \|- in. mahogany table top which The wood required for a floor 25 thick white pine is used. f-in. 10. 11. By means metals in is (j ft. in diameter thick. Table of ft, by 15 ft. 6 in., if a bar graph compare the weights of the 5. Represent by a bar graph the weights of the wood given in Table 5. 13. 50 round aluminum rods each 15 ft. long and f in. in 12. diameter. 14. Find the weight Fig. 92. of the spruce I beam, shown in Mathematics 76 for the Aviation Trades -U LJ _ k- ' /2 Fis. 15. shown Find the weight U'"4 -I 92. I beam. of 1,000 of each of the steel items in Fig. 93. (b) Fig. Job 93. Board Feet 4: Every mechanic sooner or later finds himself ready to purchase some lumber. In the lumberyard he must know I Booird foot I Fig. Board fool 94. the meaning of "board feet," because that lumber is sold. is how most Volume and Weight 77 Definition: A board foot is a unit of measure used in lumber work. A board having a surface area of 1 sq. ft. and a thickness of 1 in. or less is equal to 1 board foot (bd. ft.). ILLUSTRATIVE Find the number 2 of board feet EXAMPLE in a piece of lumber 5 by 2 ft. by in. thick. Given : Find: L = W= 5 f t. 2ft. / = 2 in. Number of board A feet =LX v .1-5X2 Board Board A = = feet feet 10 sq. X 10 X .1 ft. t 2 - 20 bd. ft. Ans. Method: a. /;. Find the surface area in square feet. Multiply by the thickness in inches. Examples: number of board rough stock shown in F'ig. 9.5. 1-3. Find the of Example 2 feet in each of the pieces Example 3 Fig. 95. Mathematics 78 for the Aviation Trades 4. 5. 9 Find the weight Calculate the cost of 5 pieces of pine 8 ft. long by wide by 2 in. thick, at 11^ per board foot. in. 6. Job 1. of each of the boards Calculate the cost of this 5: bill of in Examples 1-3. materials: Review Test Measure all dimensions on the airplane tail in Fig. 9(5, to the nearest 8 Fig. 2. 96. Horizontal stabilizers and elevators. Find the over-all length and height of the crankshaft in Fig. 97. Fig. 3. Find the weight 97. Crankshaft. of the steel crankshaft in Fig. 97. Volume and Weight 4. Find the area of the airplane wing 79 in Fig. 98. 49-3- ^ 65-10"Fi g . -s 98. Find the number of board feet and the weight spruce board 2 by 9 in. by 14 ft. long. 5. of a Chapter VI ANGLES AND CONSTRUCTION has been shown that the length of lines can be measured by rulers, and that area and volume can be calculated with the help of definite formulas. Angles are measured with the It Fig. 99. Protractor. help of an instrument called a protractor (Fig. 99). It will be necessary to have a protractor in order to be able to do any of the jobs in this chapter. (a) Fi 9 . is In Fig. 100(a), called the vertex. is and The angle BC is are sides of the angle. B known as LAEC or Z.CBA, always the middle letter. The symbol mathematical shorthand for the word angle. Name since the vertex Z AB 100. is Angles and Construction 81 the sides and vertex in Z.DEF', in Z.XOY. Although the sides of these three angles differ in length, yet Definition: An line the is angle from an amount of rotation necessary to bring a a final position. The length nothing to do with the size initial position to of the sides of the angle has of the angle. Job 1: How to Use the Protractor ILLUSTRATIVE How many degrees does EXAMPLE /.ABC contain? A I 4 ABC =70 B Fi 3 . 101. Method: a. Place the protractor so that the straight edge coincides with the line BC (see Fig. 101). mark of the protractor on the vertex. number of degrees at the point where line A B 6. Place the center c. Read the cuts across the protractor. d. Since /.ABC is less than a right The answer is 70. smaller number. angle, we must read the 82 Mathematics for the Aviation Trades Examples: Measure the angles 1. in Fig. 102. s -c Fi 9 . 102. Measure the angles between the center 2. parts of the truss member lines of of the airplane rib shown the in E Fig. 1015. Fig. How many (d) Z.AOB /.COA Job 2: (a) How degrees are there in (I)} (e) to 103. LEOC LEO A (c) LCOD (/) Draw an Angle The protractor can also be used to draw angles of a definite number of degrees, just as a ruler can be used to draw lines of a definite length. Angles and Construction ILLUSTRATIVE Draw an angle of 30 with A 83 EXAMPLE as vertex and with AB as one side. Method: a. is Plaee the protraetor as at A if (see Fig. 104). A 4 ABC = 30 Fig. b. measuring an angle whose vertex Mark * 104. a point such as (^ at the 80 graduation on the protractor. c. Aline from A to this point will make /.MAC = 30. Examples: Draw angles of 40 90 2. 60 3. 45 4. 37 7. 110 8. 145 9. 135 11. 1. 6. With the help each of the angles 12. 13. in 6. 10 10. 175 of a protractor bisect (cut in half) Examples, 1 -5 above. Draw an angle of 0; of 180. Draw angles equal to each of Y C the angles in Fig. 105. O Fig. 105. 84 Mathematics Draw angles equal 14. for the Aviation Trades to one-half of each of the angles in Fig. 105. Job 3: Units of Angle Measure So far only degrees have been mentioned in the measureof angles. There are, however, smaller divisions than ment the degree, although only very skilled mechanics will have much occasion to work with such small Memorize the following units. table: TABLE 6. ANGLE MEASURE 60 seconds (") GO minutes 90 degrees Fig. = = = = 18 106. degrees 360 degrees 1 minute 1 degree () (') 1 right angle 1 straight angle 1 circle Questions: How many 1. right angles are there (a) in 1 straight angle? (ft) in a circle? How many 2. in5? (a) (6) How many 3. minutes are there in 45? (c) in 90? seconds are there in 1 degree? (6) in 1 right angle? (a) Figure 107 shows the position of rivets on a circular patch. Calculate the number of degrees in 4. (a) /.DEC (c) ^FBC (d) (e) LAEF (f) (I)} Definition: An angle whose vertex is the center of a circle is called a central angle. For instance, Z.DBC in the 85 Angles and Construction circular patch in Fig. 107 is a central angle. central angles in the same diagram. Name any other Examples: 1. 108. In your notebook draw four triangles as shown in Fig. as accurately as you can each of the angles in Measure B B A each triangle. What of the angles of 2. C conclusion do you draw as to the sum any triangle? Measure each angle plane figures) B in Fig. in the 109, after quadilaterals (4-sided drawing similar figures n Rectangle Parallelogram Irregular Trapezoid quadrilateral Fi g . 109. Square 86 Mathematics in your own notebook. for the Aviation Trades Find the sum of the angles of a quadilateral. Point Point 2 1 Fig. Measure each angle in angles around each point. 110. Fig. 110. 3. Find the sum of the Memorize: 1. 2. 3. Job The sum The sum The sum of the angles of a triangle is 180. of the angles of a quadilateral is 300. of the angles around a point C is , H)0. 4: Angles in Aviation This job will present just two of the many ways in which angles are used in aviation. A. Angle of Attack. The angle of attack is the angle between the wind stream and the chord line of the airfoil. In Fig. Ill, AOB is is the angle of attack. the angle of attack Fig. Wind Chord fine of airfoil 111. The lift of an airplane increases as the angle of attack is increased up to the stalling point, called the critical angle. Examples: 1-4. Estimate the angle of attack of the airfoils in Fig. 112. Consider the chord line to run from the leading edge Angles and Construction to the trailing edge. The direction of the wind 87 is shown by W. What wind condition might cause a situation like the one shown in Example 4 Fig. 112? 5. 3. Fig. 112. B. Angle of Sweepback. Figure 113 shows clearly that the angle of sweepback is the angle between the leading edge and a line drawn perpendicular to the center line of the airplane. In the figure, Z.AOB is the angle of sweepback. The angle of sweepback is Fi 9 . Most planes now being sweepback Sweepback in is 4.AOB 113. built have a certain amount of order to help establish greater stability. in giving the pilot an even more important increased field of vision. Examples: Estimate the angle Figs. 114 and 115. of sweepback of the airplanes in Mathematics Fig. for the Aviation Trades Vultee Transport. (Courtesy of Aviation.) 114. v^x- Fig. 1 Job 5: 1 5. Douglas DC-3. (Courtesy of Aviation.) To Bisect an Angle This example has already been done with the help of a protractor. However, it is possible to bisect an angle with a ruler and a compass more accurately than with the protractor. Why? Perform the following construction in your notebook. ILLUSTRATIVE CONSTRUCTION Given /.A : Required To : BC bisect , Angles and Construction 89 Method: a. Place the point of the compass at B (see Fig. 116). arc intersecting BA at D, and BC at E. b. Draw an c. Now with D and Do not change the d. Draw line E radius as centers, draw arcs when moving intersecting at 0. to E. the compass from D BO. Check the construction by measuring /.ABO with the protractor. Is bisected. it equal to ^CBO? If it is, the angle has been Examples: In your notebook draw two angles as shown in Fig. 117. 1. Bisect ^AOB and Z.CDE. Check the work with a protractor. 2. Divide /.CDE in Fig. 117 into four equal parts. Check the results. C 3. Is it possible to construct straight angle? Job 6: Try a right angle by bisecting a it. To Bisect a Line This example has already been done with the help of a Accuracy, however, was limited by the limitations of the measuring instruments used. By means of the following method, any line can be bisected accurately without rule. first measuring its length. 90 Mathematics for t/ie Aviation Tracks ILLUSTRATIVE CONSTRUCTION Given: Line Required: To AB bisect AB Method: a. Open a compass a distance which you estimate to be greater than one-half of AB (see Fig. 118). 6. First with A as center then with B as i Do h- ^ AO arcs intersecting at c. ' ' Draw Check with a steel C and D. not change the radius when moving the compass from ls draw center \| to K. CD cutting line this construction AO rule. Is A line equal to OB ? AB at 0. by measuring If it is, line AB has been bisected. Definitions: Line CD is called the perpendicular bisector of the line AB. Now measure Z.COA. Measure Z.BOC with a protractor. Two lines are said to they meet at right be perpendicular to each other when angles. Examples: 1. Bisect the lines in Fig. 119 after drawing notebook. Check with a them in your rule. <t (a) (c) (ci) Fig. 2. Draw any 3. Lay a. What 119. line. Divide it into 4 equal parts. a line 4f in. long. Divide it into 4 equal parts. is the length of each part by direct measurement off to the nearest 64th? b. What should be the exact length of each part by arithmetical calculation ? Angles anc/ Construction 91 a line 9 T -# in. long. Divide it into 8 equal parts. the length of each part ? 5. Holes are to be drilled on the fitting shown in Fig. 120 so that all distances marked 4. Lay What A off is are equal. Draw a line -4 and locate the long, centers of the holes. Check in. A -)l(--A ~4-A-->-A -* " the results with a rule. Draw the perpendicular 6. bisectors of the sides of point Job A. "' any l9 ' triangle. Do they meet one in ? 7: To Construct a Perpendicular To Erect a Perpendicular at Any Point on a Line. ILLUSTRATIVE CONSTRUCTION Given: Line AB, and point P on line AB. Required: To construct a line perpendicular to AB at point P. Method: a. With P any convenient as center, using radius, draw an arc D cutting AB at C and (see Fig. 121). b. First with C as center, then with D as center and with any convenient radius, draw A C D P Fig. c. I arcs intersecting at 0. Draw line OP. B Check: 121. LOPE with a protractor. Is it a right angle? OP is perpendicular to AB. What other angle is 90? Measure then B. To Drop a Perpendicular Not on the Line. to a Line If it is, from Any Point ILLUSTRATIVE CONSTRUCTION Given: Line Required: through P. AB and point To construct P not on line AB. a line perpendicular to AB and passing 92 Mathematics for the Aviation Trades Method: With P as center, draw an arc intersecting line Complete the construction with the help AB at C and D. of Fig. 122. D Fig. 122. Examples: xamples: 1. ] In your notebook draw any diagram Fig. Fig. 123. Construct a perpendicular to line similar AB to at point P. +C Al Fi 9 . 2. 123. Drop a perpendicular from point C (Fig. 123) to line AB. Construct an angle of 3. 90 7. Draw line pendiculars to 8. 9. 45 5. AB equal to 2 4. 2230 in. At / 6. A and B (>7i erect per- AB. Construct a square whose side is 1^ in. Construct a right triangle in which the angles are 90, 45, and 45. 10-11. Make full-scale drawings of the layout of the and 125. airplane wing 12. Find the over-all dimensions of each of the spars in spars, in Figs. 124 Figs. 124 and 125. Angles and Construction Job 8: This To Draw an Angle Equal to a Given Angle is an important job, and serves as a basis for many other constructions. Follow this construction in your notebook. ILLUSTRATIVE CONSTRUCTION Given: /.A. Required: To construct an angle equal to Z. ( with vertex at A'. Method: a. KC With A as center and with any convenient radius, draw arc (see Fig. 126a). A' (b) b. With the same B'C' (see Fig. 1266). radius, but with A f as the center, draw arc 94 Mathematics c. for the Aviation Trades With B as center, measure the distance EC. With B as center, and with the radius obtained f d. in (c), intersect arc B'C' at C'. e. Line A'C' will Check make /.C' A' B' this construction equal to /.CAB. by the use of the protractor. Examples: 1. With the help of a protractor draw /.ABD and Z.EDB (Fig. 127) in your notebook, (a) Construct an angle equal to /.ABD. (V) Construct an angle equal to /.EDB. E Fig. 127. 2. In your notebook draw any figures similar to Fig. 128. Construct triangle A'B'C'y each angle of which is equal to a corresponding angle of triangle ABC. 3. Construct a quadrilateral A'B'C'D' equal angle for ABCD. angle to quadrilateral A A C FiS. Job 9: D 128. To Draw a Line Parallel to a Given Line Two lines are said to be parallel when they never meet, no matter how far they are extended. Three pairs of parallel lines are shown in Fig. 129. B L ^ N E AC Fig. 129. AB is H G parallel to CD. EF is parallel to GH. M LM is P parallel to NP. Angles and Construction ILLUSTRATIVE CONSTRUCTION Given: Line AH. To Required: construct a line parallel to AB and passing through point P. Method: a. Draw any line PD through P cutting line AB at C (see Fig. 130). _-jrvi b. With P ^ 4' Fig. 130. as vertex, construct an angle equal to /.DCB, as shown. c. PE is parallel to AB. Examples: 1. your notebook draw any diagram similar to Fig. 131. a line through C parallel to line A B. lit Draw xC xD Fig. 2. Draw lines through D 131 parallel to line AB, in Fig. 131 to line AB. and E, each in Fig. 131. 3. Draw 4. Given /.ABC a perpendicular from in Fig. 132. A Fig. 132. E 96 Mathematics AD parallel to b. BC. CD Construct Construct a. for the Aviation Trades AB. parallel to What is the name of the resulting 6. Make a full-scale drawing of quadilateral? this fitting shown in Fig. 133. ^ \|<_ Fig. 133. Washer plate with 2 holes drilled Job 10: To Divide a Line into 5/16 in. in Any Number diameter. of Equal Parts N method any line can be divided accurately into any number of equal parts without any actual measure- By this ments being needed. ILLUSTRATIVE CONSTRUCTION Given: Line Required: To AB. divide AB into 5 equal parts. Method: a. Draw any line, such as AIL b. See Fig. 134. With any convenient radius, 5 equal parts on AH. These lay are AC, CD, J)E, EF, FG. parts off c. d. Draw line At F draw cutting line c. Find the other points HP in a similar is AB now BG. a line parallel to at point P. one-fifth of line BG AB. manner. Examples: 1. Divide the drawing them in lines in Fig. 135 into 5 equal parts after your notebook. Check the results with a steel rule. 2. Draw a line 4 in. long. Divide it into 3 equal parts. Angles and Construction 3. Draw 97 long. Divide it into 6 equal parts. At division erect a perpendicular. Are the a line 7 in. each point of perpendicular lines parallel to each other? (b) (CL) H h <w Fig. Job 1 1 : 135. Review Test Construct a square 3J\| in. on a side. What is its area? 2. Construct a rectangle whose length is 4j^ in. and whose width is ij-g- in. Divide this rectangle into 5 equal 1. strips. 3. Make shown a full-scale drawing of the laminated wing spar, in Fig. 136. 4 l'L Fig. 1 36. 4. If it Laminated spar, airplane wing. Find the cross sectional area of the spar in Fig. 136. were 5 ft. long and made of spruce, how much would it weigh ? 6. Draw line AD struct angles of triangle a. b. c. is equal to 2 in. At points A and B con60, by using the protractor, so that a formed. How many What What is degrees are there in the third angle? the length of each of the sides ? is the name of the triangle? ttapterVII GRAPHIC REPRESENTATION OF AIRPLANE DATA Graphic representation is constantly growing in importance not only in aviation but in business and government as well. As a mechanic and as a member of society, you ought to learn how to interpret ordinary graphs. There are many types of graphs: bar graphs, pictographs, broken-line graphs, straight-line graphs, and others. All of them have a common purpose: to show at a glance comparisons that would be more cal data alone. In this case is difficult to make from numeri- we might say that one picture worth a thousand numbers. Origin Horizon tot I ax is Fi 9 . The graph 137. a picture set in a "picture frame/' This frame has two sides: the horizontal axis and the vertical is shown in Fig. 137. These axes meet at a point called the origin. All distances along the axis are measured from the origin as a zero point. axis, as Job 1 : TTie Bar Graph easiest way of learning how to the finished product carefully. study The 98 make a graph is to 99 Graphic Representation of Airplane Data A COMPARISON OF THE LENGTH OF Two AIRPLANES DATA Scoile: I space = 10 feet Airplanes Fig. 1 38. (Photo of St. Louis Transport, courtesy of Curtiss Wright Corp.) The three steps in Fig. 189 show how the graph in Fig. 138 was obtained. Notice that the height of each bar may be approximated after the scale is established. Make a graph of the same data using a scale in which 1 space equals 20 ft. Note how much easier it is to make a STEP 2 STEP 3 a convenient scale on each axis Establish I Airplanes Fig. 1 39. 2 Airplanes Steps in Determine points on the scale from the data I 2 Airplanes the construction of a bar graph. graph on "-graph paper" than on ordinary notebook paper. It would be very difficult to rule all the cross lines before beginning to draw up the graph. 1 00 Mathematics for the Aviation Trades Examples: Construct the graph shown in Fig. 140 in your own notebook and complete the table of data. 1. A COMPARISON OF THE WEIGHTS OF FIVE MONOPLANES DATA Scoile Fig. 2. I space * 1 000 Ib. 140. Construct a bar graph comparing the horsepower of the following aircraft engines: 3. Construct a bar graph of the following data on the production of planes, engines, and spares in the United States 4. : Construct a bar graph of the following data: pilots licensed on Jan. 1, 1940, the ratings Of the 31,264 were as follows: Graphic Representation of Airplane Data 101 1,197 air line 7,292 commercial 988 limited commercial 13,452 private 8,335 solo Job 2: Pictographs Within the last few years, a new kind of bar graph called a piclograph has become popular. The pictograph does not need a scale since each picture represents a convenient unit, taking the place of the cross lines of a graph. Questions: 1. How many airplanes does each figure in Fig. 141 represent ? THE VOLUME OF CERTIFIED AIRCRAFT INCREASES STEADILY EACH FIGURE REPRESENTS 2,000 CERTIFIED AIRPLANES JanJ DATA 1935 1936 1937 1938 1939 1940 Fig. 2. 3. 141. (Courtesy of Aviation.) How many airplanes would half a figure represent? How many airplanes would be represented by 3 figures ? 4. Complete the table of data. 102 5. Mathematics Can such data for the Aviation Trades ever be much more than approximate? Why? Examples: 1. Draw up a table of approximate data from the pictograph, in Fig. 142. To OPERATE Quit CIVIL AIRPLANES WE EACH FIGURE REPRESENTS 1937 II WE A (i ROWING FORCE OF PILOTS 2,000 CERTIFIED PILOTS mum mommmmmf FiS. 2. Do you Try this one. 142. (Courtesy of Aviation.) think you could make a pictograph yourself? Using a picture of a telegraph pole to repre- sent each 2,000 miles of teletype, make a pictograph from the following data on the growth of teletype weather reporting in the United States: 3. Draw up employees in a table of data showing the number of each type of work represented in Fig. 143. Graphic Representation of Airplane Data 103 EMPLOYMENT IN AIRCRAFT MANUFACTURING: 1938 EACH FIGURE REPRESENTS 1,000 EMPLOYEES AIRPLANES DililljllllijyilllilllMjUiyililiJIIlli ENGINES INSTRUMENTS PROPELLERS PARTS & ACCES. III Fig. 143. (Courtesy of Aviation.) Make a pictograph representing the following data on the average monthly pay in the air transport service: 4. Job 3: The Broken-line Graph An examination of the broken-line graph in Fig. 144 will it differs in no essential way from the bar graph. If the top of each bar were joined by a line to the top of the next bar, a broken-line graph would result. a. Construct a table of data for the graph in Fig. 144. b. During November, 1939, 6.5 million dollars' worth of aeronautical products were exported. Find this point on show that the graph. 104 c. Mathematics What was exported for the for the Aviation Trades the total value of aeronautical products 10 months of 1939? first EXPORT OF AMERICXX AEKON UTTICAL PRODUCTS: 1939 DATA Jan. Feb. Mar. Apr May June July Auq.Sepi Fig. Och Scale 1 1 space =$1,000,000 144. Examples: 1. Construct three tables of data from the graph in is really 3 graphs on one set of axes. Not Fig. 145. This only does it show how the number of passengers varied PASSEXGEKS CARRIED BY DOMESTIC AIR LINES Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Fi g 145. . Nov. Dec. from Graphic Representation of Airplane Data month to month, but and 1940 compare it also 1 05 shows how the years 1938, 1939, in this respect. Make a line graph of the following data showing the miles flown by domestic airlines for the first 6 months of 2. 1940. 3. The data Make are in millions of miles. a graph of the accompanying data on the num- ber of pilots and copilots employed by domestic air carriers. Notice that there will have to be 2 graphs on 1 set of axes. Job 4: The Curved-line Graph generally used to show how two quantities vary with relation to each other. For example, the horsepower of an engine varies with r.p.m. The graph The curved-line graph is in Fig. 146 tells the story for one engine. The curved-line graph does not differ very much from the broken-line graph. Great care should be taken in the location of each point from the data. Answer these questions from the graph b. What What c. At what a. is is : the horsepower of the Kinner at 1,200 r.p.m. the horsepower at 1,900 r.p.m. ? r.p.m. would the Kinner develop 290 hp.? ? 106 Mathematics for the Aviation Trades What should the tachometer read develops 250 hp. ? d. when the Kinner CHANGE IN HORSEPOWER WITH R.P.M. KINNER RADIAL ENGINE DATA Vertical axis: f 1000 1400 1800 R. p.m. Fi g . e. Why isn't 2200 space = 25 hp. Horizontal axis' Ispace-ZOOr.p.m. 146. the zero point used as the origin for this how much space would be particular set of data? If it were, needed to make the graph ? Examples: 1. Make a table of data from the graph in Fig. 147. CHANGE IN HORSEPOWER WITH R.P.M. RADIAL AIRPLANE ENGINE DATA 280 R.p.m. B.hp. 1500 I o 240 1600 &200 1700 ^ 1800 1900 I 2000 120 CD < 2100 80 1500 1600 1700 1800 1900 2000 2100 R.p.m Fis. 2. The attack lift of 147. an airplane wing increases as the angle increased until the stalling angle Represent the data graphically. is is of reached, Graphic Representation of Airplane Data Question: At what angle does the lift fall off? 107 This is called the .stalling angle. 3. The drag also increases as the angle of attack is increased. Here are the data for the wing used in Example 2. Represent this data graphically. Could you have represented the data for and 3 on one graph? The lift of an airplane, as well as the drag, depends Question: Examples 2 4. other factors upon the area of the wing. The graph in Fig. 148 shows that the larger the area of the wing, the greater will be the lift and the greater the drag. among Why to are there show wing just area. two how vertical axes? the lift Draw up a table of data and drag change with increased 108 Matfiemat/cs for the Aviation Trades LIFT AND 2 DRAG VARY WITH WING AREA 6 4 8 10 12 14 Wing drea in square feet Fi 9 . Job 1. 5: 16 148. Review Test Make new type a bar graph representing the cost of creating a of aircraft (see Fig. 149). COST OF CREATING NEW OR SPECIAL TYPE AIRCRAFT BEECH AIRCRAFT CORP. ARMY TWIN Cost of First Ship $180,000 Fig. The air large number 2. 149. transport companies know that it takes a on the ground to keep their planes of people Graphic Representation of Airplane Data 1 09 Draw up a table of data showing how many employees of each type were working in 1938 (see Fig. 150). in the air. Am TRANSPORT'S ANNUAL EMPLOYMENT OF NONFLYING PERSONNEL: 1938 EACH FIGURE REPRESENTS 100 EMPLOYEES OVERHAUL AND U < v MAINTENANCE CREWS FIELD AND HANGAR CREWS IAI turn DISPATCHERS STATION PERS. METEOROLOGISTS RADIO OPS. TRAFFIC PERS. OFFICE PERS. Fig. 1 50. (Courtesy of Aviation.) As the angle of attack of a wing is increased, both the and drag change as shown below in the accompanying table. Represent these data on one graph. 3. lift 4. The following graph (Fig. 151) was published by the in a commercial advertisement Chance Vought Corporation 110 Mathematics for the Aviation Trades to describe the properties of the Vought Corsair. read it? Complete two tables of data: a. Time many to altitude, Can you show how in minutes: This table will minutes the plane needs to climb to any altitude. PERFORMANCE OF THE VOUGHT-CORSAIR LANDPLANB Time to altitude, in minutes 400 2400 1600 2000 800 1200 of climb at altitude, in feet per minute 4 Roite Fi 3 . 151. per minute: It is imporcan climb at any altijust tude. Notice that at zero altitude, that is, at sea level, this 6. Rate of climb tant to know at altitude, in feet how fast a plane plane can climb almost 1,600 climb at 20,000ft.? ft. per min. How fast can it CAapterVIII THE WEIGHT OF THE AIRPLANE Everyone has observed that a heavy transport plane has larger wing than a light plane. The reason is fairly simple. There is a direct relation between the area of the wing and the amount of weight the plane can lift. Here are some interesting figures: a much TABLE 7 Draw a broken-line graph of this data, using the gross weight as a vertical axis and the wing area as a horizontal axis. What is the relation between gross weight and wing area ? Job 1: CalculatingWing Area wing is calculated from its plan form. Two typical wing-plan forms are shown in Figs. 152 A and 5%B. The area of these or of any other airplane wing can be found by using the formulas for area that have already been The area of a is particularly easy to find the area of a rectanguas in Fig. 153, if the following technical terms are lar wing, learned. It remembered. 113 115 The Weight of the Airplane Definitions: the length of the wing from wing tip to wing tip. Chord is the width of the wing from leading edge to Span is trailing edge. Formula: Area = ILLUSTRATIVE span X chord EXAMPLE Find the area of a rectangular wing whose span whose chord is 4.5 ft. = 25.5 Chord = 4.5 Wing area Given: Span Find: is 25.5 ft. and ft. ft. Area = span X chord Area - 25.5 X 4.5 Area = 114.75 sq. ft. Ans. Examples: 1. 20 2. in. (> Find the area of a rectangular wing whose span is and whose chord is 4^ ft. A rectangular wing has a span of 36 in. and a chord of What is its area in square inches and in square feet? Find the area of (a) the rectangular wing in Fig. 154, ft. 3. (b) the rectangular wing with semicircular * tips. a I 35 6 (a) (b) Fig. 4. 7 J > . 154. Calculate the area of the wings in Fig. 155. Fig. 155. 116 6. Mathematics Find the area for the Aviation Trades of the tapered wing in Fig. 156. V" Fig. Job 2: Mean of a Tapered Cfcorc/ J 156. Wing From the viewpoint of construction, the rectangular wing form is probably the easiest to build. Why? It was found, however, that other types have better aerodynamical qualities. In a rectangular wing, the chord is the same at all points but in a tapered wing there is a different chord at each point (see Fig. 157). Wing span Fig. 1 57. A tapered wins h many chords. Definition: Mean chord is the average chord of a tapered wing. It found by dividing the wing area by the span. Formula: Mean chord area span EXAMPLE ILLUSTRATIVE Find the mean chord of the Fairchild 45. Given: Area = 248 sq. ft. Span Find: 39.5 ft. Mean chord area Chord = Chord = span 248 39.5 Chord = 6.3 ft. Ans. is 117 The Weight of the Airplane Examples: 1-3. Supply the missing data: Job 3: /Aspect Ratio Figures 158 and 159 show how a wing area of 360 sq. ft. might be arranged: Airplane Span = 90 Chord = 4 1: Area Area Area = 60 ft. Chord - 6 ft, Span = Chord - 30 ft. 12 ft. Airplane 2: Span Airplane 3: Fig. = = = Area Area Area Area ft. ft. = = = span X chord 90 X 4 360 60 X 360 30 360 sq. ft. 6 sq. ft. X 12 sq. ft. 159. would be very difficult to build this wing strong enough to carry the normal weight of a plane. Why? However, it would have good lateral stability, which means it would not roll as shown in Fig. 1(>0. Airplane 2: These are the proportions of an average Airplane plane. 1: It 118 Mathematics An 160. Fig. tor the Aviation trades illustration of lateral roll. Airplane 3: This wing might have certain structural advantages but would lack lateral stability and good flying qualities. Aspect ratio is the relationship between the span and the chord. It has an important effect upon the flying characteristics of the airplane. Formula: Aspect ratio r In a tapered wing, the aspect ratio. -~, , chord mean chord can be used ILLUSTRATIVE to find the EXAMPLE Find the aspect ratio of airplane Given Span = 90 ft. 1 in Fig. 158. : Chord Find: 4 ft. Aspect ratio A Aspect ratio ,. Aspect ratio Aspect ratio = span , T chord = = ^f- 22.5 Ans. Examples: 1. Complete the following table from the data supplied and 159. in Figs. 158 TheWeight of 119 the Airplane 2-5. Find the aspect ratio of these planes : 6. Make a bar graph comparing the aspect ratios of the four airplanes in Examples 2-5. 7. The NA-44 has a wing area of 255f sq. ft. and a span of 43 ft. (Fig. 161). Find the mean chord and the aspect ratio. Fig. 8. its North American 161. A Seversky has a wing area is NA-44. span of 41 246.0 sq. (Courtesy of Aviation.) ft. Find its aspect ratio, if ft. The GrossWeight of an Airplane The aviation mechanic should never forget that the airplane is a "heavier-than-air" machine. In fact, weight is such an important item that all specifications refer not only to the gross weight of the plane but to such terms as Job 4: the empty weight, useful load, pay load, etc. Mathematics IZU tor the Aviation trades Definition: Empty is weight the weight of the finished plane painted, polished, and upholstered, but without gas, oil, pilot, etc. the things that can be Useful load placed in the empty plane without preventing safe This includes pilots, passengers, is Gross weight safely carry off is the weight of maximum the flight. baggage, oil, gasoline, etc. weight that the plane can the ground and in the Formula: Gross weight The all air. empty weight -f- useful load and gross weight are determined by the manufacturer and U.S. Department of Fig. figures for useful load 162. this The gross weight and center of gravity of an airplane can be found by method. (Airplane Maintenance, by Younger, Bonnalie, and Ward.) Commerce inspectors. They should never be exceeded by the pilot or mechanic (see Fig. 162). Fig. 163. The Ryan SC, a low-wing monoplane. (Courtesy of Aviation.) 122 Fig. 164. Mathematics for the Aviation Trades Pay load. United Airlines Mainliner being loaded before one of nightly flights. its (Courtesy of Aviation.) trying to increase the pay load as an inducement to buyers. good method of comparing the pay loads of different A planes is on the basis of the pay load as a per cent of the gross weight (see Fig. 164). EXAMPLE ILLUSTRATIVE The Aeronca model 50 two-place monoplane has a and a pay load the pay load? of 1,130 Ib. weight is Given: Pay load Gross weight Find: = = of 210 210 Ib. What gross weight per cent of the gross Ib. 1,130 Ib. Per cent pay load Method: Per cent = pay load gross weight 100 Per cent Per cent =* 18.5 Arts. X 100 The Weight of the Airplane 123 Examples: The monoplane 1. Punk Fig. is in Fig. B, whose gross weight 165. 210 lb. is the two-place Akron 1,350 lb., and whose pay load 165 is The Akron Funk B two-place monoplane. (Courtesy of Aviation.) What per cent of the gross weight is the pay load? 2-5. Find what per cent the pay load weight in the following examples 6. : Explain the diagram in Fig. 166. Fi 9 . 166. is of the gross 1 24 Job Mathematics 6: for the Aviation Trades Wing Loading The weight of an airplane, sometimes tens of thousands of pounds, is carried on its wings (and auxiliary supporting surfaces) as surely as if they were columns of gross steel anchored into the ground. Just as it would be dangerous to overload a building till its columns bent, so it would be dangerous to overload a plane till the wings could not safely hold Fig. it 167. aloft. Airplane wings under static test. (Courtesy of Aviation.) Figure 167 shows a section of a wing under static test. Tests of this type show just how great a loading the structure can stand. Definition: Wing loading is the number that each square foot of Formula: of pounds of gross weight the wing must support in flight. Wing loading ILLUSTRATIVE = ; ^ wing area EXAMPLE A Stinson Reliant has a gross weight of 3,875 area of 258.5 sq. ft. Find the wing loading. Given: Gross weight = 3,875 Ib. Area = 258.5 sq. ft. Ib. and a wing 125 The Weight of the Airplane Find Wing : loadin g Wing loading Wing loading = Wing loading gross weight = = wing area 3,875 258.5 14.9 Ib. per sq. ft. Ans. Examples: The Abrams Explorer has a gross weight of 3,400 Ib. of 191 sq. ft. What is its wing loading? 2-4. Calculate the wing loading of the Grummans in the 1. and a wing area following table: 5. Represent by means of a bar graph the wing loadings and wing areas table. One Fig. 168. of the Grumman of these planes Grumman G-37 is planes in the preceding shown in Fig. 168. military biplane. (Courtesy of Aviation.) 126 Mathematics for the Aviation Trades The Pasped Skylark has a wing span of 35 ft. 10 in. and a mean chord of 5.2 ft. Find the wing loading if the 6. gross weight Job 7: is 1,900 Ib. Power Loading The gross weight of the plane must not only be held aloft by the lift of the wings but also be carried forward by the thrust of the propeller. A small engine would not provide enough horsepower for a very heavy plane; a large engine might "run away" with a small plane. The balance or ratio between weight and engine the power loading. Formula: Power loading ILLUSTRATIVE A Monocoupe 90A power is expressed by =? horsepower EXAMPLE has a gross weight of 1,610 Ib. and is engine. What is the power loading? powered by a Lambert 90-hp. Given: Gross weight = Horsepower Power loading Find: ~ 1 1,610 Ib. 90 - ,. weight ower loading = gross ^ horsepower , T> jrower loading = i Power loading = 90 17.8 Ib. per hp. Examples: 1-3. Complete the following table: Arts. 128 Mathematics for the Aviation Trades Does the power loading increase with increased gross weight? Look at the specifications for light training planes and heavy transport planes. Which has the higher power 4. loading? Note: The student may find this information in his school or public library, or by obtaining a copy of a welltrade magazine such as Aviation, Aero Digest, etc. known Find the gross weight and the power loading of the Waco model C, powered by a Jacobs L-6 7 cylinder radial engine (see Figs. 169 and 170). 5. Job 8: Review Test The following are the actual of specifications three different types of airplanes: 1. Fig. Fig. Find the wing and power loading of the airplane 171, which has the following specifications: Gross weight = 4,200 Ib. = 296.4 sq. ft. Wing area = Whirlwind, 420 hp. Engine 171. Beech Find Beechcraft D five-place biplane. (Courtesy of in Aviation.) the wing loading; (6) the power loading; (c) the aspect ratio; (d) the mean chord of the airplane in Fig. 172, which has the following specifications: 2. (a) Gross weight Wing = Engines Wing span 24,400 987 area = - Ib. sq. ft. 2 Cyclones, 900 hp. each 95 ft. Chapter IX AND WING AIRFOILS RIBS tunnel has shown how greatly the shape of the can affect the performance of the plane. The airfoil section is therefore very carefully selected by the manufacturer before it is used in the construction of wing ribs. The wind airfoil N.A.C.A.22 N.A.C.A.OOI2 Clark Y Rib shape of Symmetrical rib shape Rib shape of Douglas DC3 Fl g . No 174.- -Three types of mechanic should change 174 shows three Aeronca common this airfoil section. shape in any way. Figure airfoil sections. The process of drawing up the data supplied by the manufacturer or by the government to full rib size is important since any inaccuracy means a change in the plane's performance. The purpose of this chapter is to show how to draw a wing section to any size. Definitions: Datum line is the base line or horizontal axis (see Fig. 175). Upper camber Vertical^ axis Trailing Leading edge edge '" > Lower camber Datum Fig. 175. 130 line and Wing Ribs Airfoils Vertical axis 131 a line running through the leading edge is of the airfoil section perpendicular to the datum line. Stations are points on the datum line from which measure- ments are taken up or down to the upper or lower camber. Upper camber is the curved line running from the leading edge to the trailing edge along the upper surface of the airfoil section. the line from leading edge to trailing edge along the lower surface of the airfoil section. The datum line (horizontal axis) and the vertical axis Lower camber is have already been defined in the chapter on graphic representation. As a matter of fact the layout of an airfoil is identical to the drawing of any curved-line graph from 1 The only point to be kept in mind is that there are really two curved-line graphs needed to complete the airfoil, the upper camber and the lower camber. These will given data. now be Job 1 : considered in that order. The Upper Camber The U.S.A. 35B is a commonly used airfoil. The following data can be used to construct a 5-in. rib. Notice that the last station tells us how long the airfoil will be when finished. AIRFOIL SECTION: U.S.A. 35B Data in inches for upper camber only Airfoil section: U.S.A. 35 B I l'/ 2 2 2'/2 Fis. 3 3'/2 4 4'/2 176. The term "airfoil'* is often substituted for the more awkward phrase "airfoil section" in this chapter. Technically, however, airfoil refers to the shape of the wing as a whole, while airfoil section refers to the wing profile or rib outline. 1 132 Mathematics for the Aviation Trades Directions: Step Draw 1. the datum line and the vertical axis (see Fig. 176). '/ 2 I 2 l'/ 2 Fig. Step Mark Step fe 2. 3. At in. datum 3 3'/2 4 5 4'/2 177. stations as given in the data. station 0, the data shows that the upper all above the datum Step 2'/2 4. At line. Mark Mark station line. in., camber is Mark this point as shown in Fig. 177. in. above the the upper camber is H this point. all points in a similar manner on the upper Step 5. camber. Connect them with a smooth line. The finished upper camber is shown in Fig. 178. 2'/2 Fig. Job 2: 3 3'/2 178. The Lower Camber The data for the lower camber an are always given together with the data for the upper camber, as shown of airfoil AIRFOIL SECTION: U.S.A. 35B Fig. 179. in Fig. 179. In drawing the lower camber, the same and stations are used as for the upper camber. diagram Airfoils and Wing Ribs 133 Directions: Step 1. At station 0, the lower camber is 7^ in. above the datum* line. Notice that this is the same point as that of the upper camber (see Fig. 180). SfepL Step 2'-* Fis. 180. Step 2. At station in., the lower camber on the datum line as shown in Fig. 180. is in. high, that is, flat Step 3. Mark all the other points on the lower camber and connect them with a smooth line. In Fig. 181 is shown the finished wing rib, together with one of the many planes using this airfoil. Notice that the Fig. airfoil 181. The Piper Cub Coupe uses airfoil section U.S.A. 35B. has more stations than you have used in your will be explained in the next few pages. own work. These Questions: 1. Why does station and lower cambers? have the same point on the upper 134 2. Mathematics What for the Aviation Trades other station must have the upper and lower points close together? Examples: 1-2. Draw indicated the airfoils shown in Fig. 182 to the size the stations. All measurements are in inches. by Example 1. AIRFOIL SECTION: N-22 Example 2. AIRFOIL SECTION: N.A.C.A.-CYH N-22 NACA-CYH Fig. The CLARK Y 182. section N-22 is used for a wing rib on the N.A.C.A.-CYH, which resembles the Clark Y Swallow; airfoil very closely, is used on the Grumman G-37. 3. Find the data for the section in Fig. 183 by measuring to the nearest 64th. Fls. 183. Airfoils 4. Draw up the Clark and Wing Ribs Y 135 airfoil section from the data in Fig. 184. AIRFOIL SECTION: CLARK Y The Clark V airfoil section is used in many planes, such as the Aeronca shown here. Note: All dimensions are in inches. Fig. 6. Make your own airfoil section, by measurement with the Job 3: When 184. and find the data for it steel rule. the Data Are Given in Per Cent of Chord data, including stations and upper and lower are given as percentages. This allows the mechanic cambers, to use the data for any rib size he wants; but he must first Here all do some elementary arithmetic. ILLUSTRATIVE EXAMPLE mechanic wants to build a Clark Y rib whose chord length 30 in. Obtain the data for this size rib from the N.A.C.A. data A is given in Fig. 185. In order to keep the work as neat as possible and avoid any error, copy the arrangement shown in Fig. 185. It will be necessary to change every per cent in the N.A.C.A. data to inches. This should be done for all the stations and the upper camber and the lower camber. Airfoils and Wing Ribs Upper Camber: Arrange your work in 137 a manner similar to the foregoing. Rib Size, In. 30 30 30 X X X Upper Camber, Per Cent . 3.50% = 30 X 9.60% = 30 X 11.36% = Upper Camber, In. .0350 .0960 = = 1.050 .880 Calculate the rest of the points on the upper camber. Insert these in the appropriate spaces in Fig. 185. Do the same for the lower camber. AIRFOIL SECTION: ("LARK Y, 30-iN CHORD Fig. The data of the of 186. be the final step before layout a rule graduated in decimal parts providing in decimals wing rib, an inch is available. may 138 Mathematics for the Aviation Trades If however, a rule graduated in ruler fractions is the only instrument available, it will be necessary to change the decimals to ruler fractions, generally speaking, accurate is suggested that the arrangement be used. Notice that the data in decimals are the answers obtained in Fig. 185. It is a good idea, at this time, to review the use of the* to the nearest 64th. It shown in Fig. 186 decimal equivalent chart, Fig. 64. Examples: 1. Calculate the data for a 15-in. rib of draw the SIKORSKY N-22 Fig. 2. N-22, and GS-M 187. chord are given in Fig. 187 for Sikorsky GS-M. Convert these data to inches for a Data airfoil airfoil airfoil section (see Fig. 187). 9-in. rib, in per cent of and draw the airfoil section. Airfoils and Wing 3. Draw a 12-in. diagram from the following data: 139 Ribs of airfoil section Clark Y-18 AIRFOIL SE( TION: CLARK Y-18 I 4. Job Make 4: a 12-in. solid wood model rib of the Clark Y-18. The Mosep/ece and Tail Section It has probably been observed that stations and 10 per cent are S 01.252.5 5 t a t 7.5 i o n s 20 10 O Fig. 188. per cent not sufficient to give all the necessary Datum Stations between line and 10 per cent of the chord. points for rounding out the nosepiece. As a result there are and 10 per cent, several more intergiven, in addition to mediate stations (see Fig. 188). ILLUSTRATIVE EXAMPLE Obtain the data in inches for a nosepiece based on a 30-in. chord. of a Clark Y airfoil Airfoils and Wing Ribs 141 Y airfoil Figure 189 shows the nosepiece of the Clark upon a 30-in. chord. It is not necessary to lay out the entire chord length of 30 in. in order to draw up the section based nosepiece. Notice, that The data and illustration are carried out only to 10 chord or a distance of 3 in. 2. The data are in decimals but the stations and points on the upper and lower cambers of the nosepiece were 1. per cent of the Note All dimensions are : Fis. 190. in inches Jig for buildins nosepiece of Clark V. ruler fractions. Figure 190 is a blueprint used in the layout of a jig board for the construction of located by using the nosepiece of a Clark The tail Y rib. section of a rib can also be drawn independently by using only part of the total airfoil data. out the examples without further instruction. of the entire rib Work Examples: All data are given in per cent of chord. 1-2. Draw the nosepieces of the airfoils in the following tables for a 20-in. chord (see Figs. 191 and 192). 142 Mathematics 20 Fi g . for the Aviation Trades 80 40 60 Per cent of chord 20 191 .Section: N-60. Fig. Draw the tail Section: 100 U.S.A. 35 A. AIRFOIL: U.S.A. 35A AIRFOIL: N-60 3-4. 192. 40 60 80 Per cent of chord sections of the airfoils in the following tables for a 5-ft. chord. AIRFOIL: U.S.A. 35A AIRFOIL: N-60 Job 5: The Thickness of Airfoils has certainly been observed that there are wide variations in the thickness of the airfoils already drawn. It cantilever wing, which is braced internally, is more easily constructed if the thickness of the airfoil permits work to be done inside of it. thick wing section also The A permits additional space for gas tanks, baggage, etc. On the other hand, a thin wing section has considerably less drag and is therefore used in light speedy planes. Airfoils and Wing Ribs 143 very easy to calculate the thickness of an airfoil from either N.A.C.A. data in per cent of chord, or from the data in inches or feet. It is Since the wing rib is not a flat form, there is a different thickness at every station, and a maximum thickness at about one-third of the way back from the leading edge (see Fig. 193). Fig. 193. ILLUSTRATIVE Find the thickness EXAMPLE in inches of the airfoil I.S.A. 695 at all stations given in the data in Fig. 194. Method: To find the thickness of the airfoil at any station simply sub" " "lower" point from the upper point. Complete the table shown in Fig. 194 after copying it in your notebook. tract the Examples: 1. Find the thickness at all stations of the airfoil section in the following table, in fractions of an inch accurate to the nearest 64th. Data are given in inches for a 10-in. chord. AIRFOIL SECTION: U.S.A. 35B 144 2. Mathematics for the Aviation Trades Figure 193 shows an accurate drawing of an airfoil. a table of data accurate to the nearest 64th for this Make could be drawn from the data alone. Find the thickness of the airfoil in Fig. 193 at airfoil, so 3. stations, that by it actual measurement. all Check the answers thus AIRFOIL SECTION: T.S.A. 695 Fig. 194. obtained with the thickness at each station obtained by using the results of Example 2. Job It 6: Airfoils with Negative Numbers may have been shown were however, noticed that thus far entirely many airfoils all the airfoils above the datum line. There are, that have parts of their lower camber Airfoils below the datum negative numbers. line. and Wing This is 145 Ribs indicated by the use of Definition: A negative number indicates a change of direction (see Fig. 195). +2 -2 Fig. 195. Examples: 1. Complete the table in Fig. 196 from the information given in the graph. 23456 01 Fi 9 . 196. Give the approximate positions of all points on both the upper and lower camber of the airfoil in Fig. 197. 2. 20 -10 10 20 30 40 50 60 70 80 90 100 146 Mathematics for the Aviation Trades N.A.C.A. 2212 is a good example of an airfoil whose lower camber falls below the datum line. Every point on the lower camber has a minus ( ) sign in front of it, except per cent which is neither positive nor negative, since it is right on the datum line (see Fig. 198). Notice that there was no sign in front of the positive numbers. A number is considered positive (+) unless a minus ( ) sign appears in front of it. In drawing up the airfoil, it has been stated that these per cents must be changed to decimals, depending upon the rib size wanted, and that sometimes it may be necessary to Airfoil Section. change the decimal fractions to ruler The methods outlined for doing fractions. work when all numbers are positive (+), apply just as well when numbers are negative ( ). The following illustrative example will show how to locate the points on the lower camber only since all other points can be located as shown in previous jobs. ILLUSTRATIVE this EXAMPLE Find the points on the lower camber for a 15-in. rib whose N.A.C.A. 2212. Data are given in Fig. 198. airfoil section is AIRFOIL SECTION: N.A.C.A. 2212 Data in per cent of chord 20 n 20 Fis- 198. The Bell BG-1 40 60 80 Per cent of chord 100 uses this section. (Diagram of plane, courtesy of Aviation.) 148 Job Mathematics 7: for the Aviation Trades Review Test Calculate the data necessary to lay out a 12-in. rib shown in Fig. 200. All data are in per cent of chord. 1. of the airfoil section 40 60 80 Per cent of chord ZO Fig. 200. 100 Airfoil section: Boeing 103. AIRFOIL SECTION: BOEING 103 Construct a table of data in inches for the nosepiece (0-15 per cent) of the airfoil shown in Fig. 201, based on a 2. 6-ft. chord. 20 Fig. 201. 40 60 80 Per cent of chord Airfoil section: Clark V-22. IOO Airfoils and Wing Ribs 149 AIRFOIL SECTION: CLARK Y-22 3. What 4. Construct a 12-in. rib of the the thickness in inches at each station of a Clark Y-22 airfoil (Fig. 201) using a 10-ft. chord? is airfoil section N.A.C.A. 2412, using thin sheet aluminum, or wood, as a material. This airfoil is used in constructing the Luscombe model 90 (Fig. 202). AIRFOIL SECTION: N.A.C.A. 2412 Data in per cent of chord 20 Fig. 5. 202. 40 60 80 Per cent of chord The Luscombe 90 uses 100 this section. Construct a completely solid model airplane wing is 15 in. and whose chord is 3 in., and use the whose span airfoil section Boeing 103, data for which are given in Example 1. 150 Hint: Mathematics Make for the Aviation Trades a metal template of the wing section to use as a guide (see Fig. 203). Fis. 203. Wins-section template. Chapter X STRENGTH OF MATERIALS "A study of handbooks of maintenance of all metal transport airplanes, which are compiled by the manufacturers for maintenance stations of the commercial airline operators, shows that large portions of the handbooks are devoted to detailed descriptions of the structures and to instructions for repair and upkeep of the structure. In handbooks for the larger airplanes, many pages of tables are included, setting forth the material t -^ of every structural part and the strenqth -9 c 7 i i each item used. What does this mean to the airplane mechanic ? characteristic ol * Tension Y////////////////////////A Bearing xn I ^ ~ > ' ^j Bending - Compression Shear ^^<<M r// Fi S ^^ . 204. Torsion Fi g . 205. means that in the repair stations of these airlines the shop personnel' is expected to maintain the structural strength of the It 1 airplanes." When working at a structural job, every mechanic must take into consideration at least three fundamental stresses, tension, compression, and shear (see Fig. 204). In addition there are other stresses which may be analyzed in terms of these three fundamental stresses (see Fig. 205). 1 From YOUNGER, tenance, J. H., A. F. BONNALIE, and N. F. WARD, Airplane MainMcGraw-Hill Book Company, Inc., Chap. I. 153 154 Mathematics for the Aviation Trades The purpose fundamental Job 1 : of this chapter is to explain the elementary, principles of the strength of materials. Tension Take three wires one aluminum, one copper, and one steel all ^V in. in diameter and suspend them as shown in Fig. 207. A. Demonstration 1. Fig. 206. Note that the aluminum wire hold a certain amount breaks; the copper will will of weight, let us say 2 lb., before it hold more than 4 steel wire will and the lb.; hold much more than either of the others. We can, therefore, say that the tensile strength of steel is greater than the tensile strength of either copper or aluminum. Definition: The 8/b. Fig. 207. tensile strength of the amount of an ob- ject is weight it will support in tension before it fails. The American Society for Testing Materials has used elaborate machinery to test most structural materials, and their figures for everybody's benefit. These which are based upon a cross-sectional area of published figures, 1 sq. in., are called the ultimate tensile strengths (U.T.S.). Definition: Ultimate tensile strength is the amount of weight a bar I sq. cross-sectional area will support in tension before it fails. in. in 155 Strength of Materials TABLE 8. ULTIMATE TENSILE STRENGTHS (In Ib. per sq. in.) Aluminum 18,000 Cast iron 20,000 Copper Low-carbon 32,000 steel 50,000 Dural- tempered 55,000 Brass Nickel 60,000 steel 125,000 High-carbon steel 175,000 Examples: Name 1. tion 2. 6 stresses to which materials used in construc- may be subjected. What is the meaning of * ? tensile strength? 3. Define ultimate tensile strength. Draw a bar graph comparing the tensile strength of the materials in Table 8. 4. B. Demonstration three 2. Take aluminum, of these diameters: ^2 i n -> vfr n -> i in., and suspend them as shown wires, all a J2/b. i in Fig. 208. Notice that the greater the 128 Fi g . Ib. 208. diameter of the wire, the greater is its tensile strength. Using the data in Fig. 208, complete the following table in your own notebook. 1 56 Mathematics for the Aviation Trades Questions: 1. How many the second wire greater than the cross-sectional area? (6) strength? first in (a) 2. How many second in 3. (a) What times is times area? (6) connection is the third wire greater than the strength? there between the cross-sectional is area and the tensile strength? 4. Would you say that the strength of a material cross-sectional area ? Why ? depended directly on its 5. Upon what other factor does the tensile strength of a material depend? Many students think that- the length of a wire affects its tensile strength. Some think that the shape of the cross section is important in tension. Make up experiments to prove or disprove these statements. Name several parts of an airplane which are in tension. C. (b) for Tensile Strength. The tensile strength depends on only (a) cross-sectional area (A) Formula of a material ultimate tensile strength (U.T.S.). AX Formula: Tensile strength U.T.S. The ultimate tensile strengths for the more common substances can be found in Table 8, but the areas will in most cases require some calculation. ILLUSTRATIVE Find the tensile strength of a Given: Cross section: circle Diameter = \| in. U.T.S. = 13,000 EXAMPLE f-in. Ib. aluminum per sq. in. Find: a. Cross-sectional area b. Tensile strength a. Area Area Area = = = 0.7854 0.7854 X D X f X 0.1104 sq. 2 in. 1 Ans. wire. Strength of Materials Tensile strength Tensile strength Tensile strength 6. 1 57 = A X U.T.S. = 0.1104 X 13,000 - 1,435.2 Ib. Ans. Examples: 1. Find the 2. How 3. Find the strength 4. What tensile strength of a ^j-in. dural wire. strong is a \|~ by 2^-in. cast-iron bar in tension? of a i%-in. brass wire. load will cause failure of a f-in. square dural rod in tension (sec Fig. 209) Fig. 209. ? Tie rod, square cross section. Find the strength in tension of a dural fitting at a point where its cross section is -^ by ^ in. 6. Two copper wires are holding a sign. Find the great5. est possible weight of the sign in diameter. 7. A load is bolts in tension. 8. A if the wires are each ^ in. being supported by four ^-in. nickel-steel What is the strength of this arrangement? mechanic tried to use 6 aluminum ^2-in. rivets to support a weight of 200 Ib. in tension. Will the rivets hold? 9. Which has the greater tensile strength: (a) 5 H.C. steel ^r-in. wires, or (6) 26 dural wires each eV in. in diameter? 10. H" What is the greatest weight that a dural strap in tension? What would be the by 3^-in. can support effect of drilling Job 2: a ^-in, hole in the center of the strap? Compression There are some ductile materials like lead, silver, copper, steel, etc., which do not break, no matter how much pressure is put on them. If the compressive force is great enough, the material will become deformed (see aluminum, Fig. 211). 158 Mathematics for the Aviation Trades On the other hand, if concrete or cast iron or woods of various kinds are put in compression, they will shatter C o repression Fi g . Fig. lead into 210 211. Due file material pieces if too much load is applied. Think of what happen to a stick of chalk in a case like that shown in many might Fig. 212. 10 Ib. iron Brittle material Fi 3 . 212. Definitions: Ultimate compressive strength (for brittle materials) is the number of pounds 1 sq. in. of the material will support in compression before it breaks. Ultimate compressive strength (for ductile materials) is the number of pounds 1 sq. in. of the material will support in compression before it becomes deformed. For ductile materials the compressive strength to the tensile strength. is equal 160 3. Mathematics Four blocks of concrete, each 2 up a are used to hold they 4. for the Aviation Trades structure. by 2 ft. in cross section, Under what load would fail? What is the strength of a f-ft. round column of concrete ? (OL) m r" +--5 --H if. (ct) Fig. 21 6. What is 3. All blocks are of spruce. the strength of a 2-in. H.C. steel rod in compression? 6. Find the strength in compression of each of the blocks of spruce Job 3: Two shown in Fig. 213. Shear plates, as shown in Fig. 215, have a tendency to cut a rivet just as a pair of scissors cuts a thread. 161 Strength of Materials SHEAR Fig. 214. The strength of a rivet, or any other material, in shear, that is, its resistance to being cut, depends upon its crosssectional area and its ultimate shear strength. Formula: Shear strength where A = = U.S.S. =AX U.S.S. cross-sectional area in shear. ultimate shear strength. (M) Fi g . TABLE 10. 215. ULTIMATE SHEAR STRENGTHS (In Ib. per sq. in.) The strength in shear of aluminum and aluminum alloy Chap. XI. Do these examples without any assistance from an illustrative example. rivets is given in Examples: 1. Find the strength 2330) fk in. in in shear of diameter. a nickel steel pin (S.A.E. 162 2. Mathematics A for the Aviation Trades chrome-molybdenum pin (S.A.E. X-4130) What is its strength in shear? What is the strength in shear of a -fV-in. is -f in. in diameter. 3. A brass rivet? spruce beam will withstand what maximum shearing load? 5. What is the strength in shear of three ^-in. S.A.E. 1015 rivets? 4. Job 4: 2\|- by 1^-in. Bearing Bearing stress is a kind of compressing or crushing force which is met most commonly in riveted joints. It usually shows up by stretching the rivet hole and crushing the surrounding plate as shown in Fig. 216. o Failure in bearing Original plate Fig. The bearing 216. strength of a material depends upon 3 factors: (a) the kind of material; (b) the bearing area; the edge distance of the plate. (c) The material itself, whether dural or steel or brass, will determine the ultimate bearing strength (U.B.S.), which is approximately equal to f times the ultimate tensile strength. TABLE 11. ULTIMATE BEARING STRENGTH (In Ib. per sq. in.) Material U.B.S. Aluminum 18,000 Dural-tempered Cast iron Low-carbon High-carbon Nickel steel steel steel 75,000 100,000 75,000 220,000 200,000 164 2. Mathematics for the Aviation Trades Find the bearing strength of a nickel-steel lug ^ is drilled to carry a A-in. pin. in. thick which 3. What 4. (a) the strength in bearing of a Tfr-in. dural plate with two A-in. rivet holes? Does the bearing strength depend upon the relative position of the rivet holes ? is What is the strength in bearing of the fitting in Fig. 218? Drilled^hole Fig. 218. (6) How many times is the edge distance greater than the diameter of the hole? Measure edge distance from the center of the hole. %Duralplate Hole drilled Wdiameter Fig. 5. 219. Find the bearing strength from the dimensions given in Fig. 219. Job 5: Required Cross-sectional Area has probably been noticed that the formulas for tension, compression, shear, and bearing are practically the same. = area X U.T.S. Strength in tension It Strength in compression Strength in shear Strength in bearing = = = area area area X X X U.C.S. TJ.S.S. TJ.B.S. Consequently, instead of dealing with four different formulas, it is much simpler to remember the following: Formula: Strength = cross-sectional area X ultimate strength Strength of Materials 1 65 In this general formula, it can be seen that the strength whether in tension, compression, shear, or bearing, depends upon the cross-sectional area opposing the stress and the ultimate strength of the kind of material. Heretofore the strength has been found when the dimensions of the material were given. For example, the tensile strength of a wire was found when its diameter was given. Suppose, however, that it is necessary to find the size (diameter) of a wire so that it be of a certain required strength, that is, able to hold a certain amount of weight. How was this formula obtained ? of a material, ~ ,. c Formula: Cross-sectional area strength required i , -1 -r- ~, r-. ultimate strength It will what be necessary to decide from reading the example stress is being considered and to look up the right any work with the numbers involved. table before doing EXAMPLE ILLUSTRATIVE What cross-sectional area should a dural wire hold 800 Ib. in Given: Required strength Material = dural U.T.S. Find: have in order to tension? = 55,000 = Ib. 800 Ib. per sq. in. Cross-sectional area 4 Area = strength required -^ -r u ultimate strength . 80 5^000 Area = 0.01454 Check: t.s. = A X U.T.S. = sq. in. 0.01454 Arts. X 55,000 = 799.70 Ib. Questions: 1. 2. Why How doesn't the answer check exactly ? would you find the diameter of the dural wire? suggested that at this point the student review the method of finding the diameter of a circle whose area is It is given. 166 Mathematics for the Aviation Trades Examples: Find the cross-sectional area of a low-carbon steel wire whose required strength is 3,500 Ib. in tension. Check 1. the answer. 2. What is the cross-sectional area of a square oak Fig. 220. beam Tie rod, circular cross section. which must hold 12,650 Ib. in compression parallel to the grain ? 3. What is the length of the side of the beam in Example 2? Check the answer. 4. A rectangular block of spruce used parallel to the grain must have a required strength in compression of 38,235 Ib. If its width is 2 in., what is the cross-sectional length ? A copper rivet is required to hold 450 Ib. in shear. What is the diameter of the rivet? Check the answer. 6. Four round high-carbon steel tie rods are required to hold a total weight of 25,000 Ib. What must be the diameter of each tie rod, if they are all alike? (See Fig. 220.) 5. Job 1. 6: Review Test Find the steel tie tensile strength rod measuring -^ in. 2 Holes drilled '/^radius Fig. of on a 221. a square high-carbon side. 167 Strength of Materials 2. Find the strength beam 3. if What is What 5. What is compression of a 2 by 1-in. oak is 1025) whose diameter 4. in applied parallel to the grain. the strength in shear of a steel rivet (S.A.E. the load is 3^ in.? the bearing strength of the fitting in Fig. 221 ? should be the thickness of a f-in. dural strip in order to hold 5,000 Ib. in tension (see Fig. 222) ? 6. If the strip in Example 5 were 2 ft. 6 in. long, much would it weigh? how Gupter XI FITTINGS, TUBING, The purpose of this chapter is AND RIVETS to apply the information strength of materials to learned about calculating the aircraft parts such as fittings, tubing, and rivets. common Job 1: SafeWorking Stress Is it considered safe to load a material until it is just about ready to break? For example, if a TS-ITL. low-carbon steel wire were used to hold up a load of 140 lb., would it be safe to stand beneath it as shown in Fig. 223? Applying the formula tensile strength = A X TJ.T.S. will show that this wire will hold nearly 145 lb. Yet it would not be safe to stand under it because This particular wire might not be so strong as it should be. 1. 2. The slightest movement of the weight or of the surrounding structure might break the wire. 3. Fis, 223. Is this safe? Our wrong, in calculations might be which case the weight might snap the wire at once. For the sake of safety, therefore, it would be wiser to use a table of safe working strengths instead of a table of ultimate strengths in calculating the load a structure can withstand. Using the table of ultimate strengths will 168 tell Fittings, Tubing, and 169 Rivets how much loading a structure can stand breaks. Using Table 12 will tell the maximum load- approximately before it ing that can be piled safely on a structure. TABLE 12. SAFE WORKING STRENGTHS (In Ib. per sq. in.) Examples: 1. What is the safe working strength of a i%-in. dural wire in tension ? 2. What diameter H.C. steel wire can be safely used to hold a weight of 5,500 Ib.? 3. A nickel-steel pin stress of 3,500 Ib. is required to withstand a shearing pin should be selected? What diameter What should be the thickness of a L.C. steel fitting necessary to withstand a bearing stress of 10,800 Ib., if the hole diameter is 0.125 in.? 4. Job 2: Aircraft Fittings a plane is very serious, and not an uncommon occurrence. This is sometimes due to a lack of understanding of the stresses in materials and how The failure of their strength is any fitting in affected by drilling holes, bending opera- tions, etc. Figure 224 shows an internal drag- wire the holes are drilled. fitting, just before Questions: 1. What at line BB are the cross-sectional area 1 , using Table 12? and tensile strength 170 2. Mathematics Two l-in- holes are drilled. sectional shape 3. for the Aviation Trades What is and area at line What are now the cross- A A'? the tensile strength at SAE I025 L.C. STEEL D < Fis. 224. Examples: 1. Using Table in Fig. 225, material is at section BB', (a) iV 12, find the tensile strength of the fitting (6) at section AA'. The ' 1 11 - low-carbon steel. Suppose that a ^-in. hole were drilled by a careless mechanic in the fitting in Fig. 225. What is the strength of this fitting in Fig. 226 now? 2. Is this statement true: "The strength of a fitting is lowered by drilling holes in it"? Find the strength of the fittings in Fig. 227, using Table 12. Fittings, Tubing/ and Rivets 171 3 4iH.C. STEEL Example 4 Example 3 3 H / S.A.E.I025 64 Example 6 3 /JS.A.E.I095 Example 8 Fig. Job 3: Aircraft 227 Tubing The cross-sectional shapes of the 4 types of tubing most commonly used in aircraft work are shown in Fig. 228. Tubing is made either by the welding of flat stock or by cold-drawing. Dural, low-carbon chrome-molybdenum, and Round steel, S.A.E. X-4130 or stainless steel are Streamlined Fig. Square 228. among the aircraft tubing. Four types of materials used. Almost any Rectangular size, shape, or thickness can be purchased upon special order, but commercially the outside diameter varies from T to 3 in. and the wall thickness varies from 0.022 to 0.375 in. 172 Mathematics for the Aviation Trades Round Tubing. What is the connection between the outside diameter (D), the inside diameter (d} > and the wall thickness ()? A. Are the statements in Fig. Fig. 229 true? 229. Complete the following table : B. The Cross-sectional Area of Tubing. Figure 230 shows that the cross-sectional area of any tube may be obtained by taking the area A of a solid bar and subtracting the area a of a removed center portion. D Minus S Minus [ a ) d Equals a Fig. Equals 230. Fittings/ Tubing, and 173 Rivets Formula: Cross-sectional area = A a For round tubes: A = 0.7854D a = 0.7854d 2 = S2 For square tubes: A a = s2 It will therefore be necessary to work out the areas of both A and a before the area of the cross section of a tube can be found. 2 . , . ILLUSTRATIVE is EXAMPLE Find the cross-sectional area of a tube whose outside diameter in. and whose inside diameter is 1^ in. Given: D = 2 in. % d Find: = l in. A (1) () a (3) Area (1) A = A = yl (2) a of tube 0.7854 0.7854 XD X 2X 2 2 = is 3.1416 sq. in. Ans. found in a similar manner = 1.7667 sq. in. Ans. == A a (3) Cross-sectional area = 3.1416 Cross-sectional area a Cross-sectional area Complete = - 1.3749 sq. 1.7667 in. Ans. this table: The struts of a biplane are kept in compression, between the spars of the upper and lower wings, by means of the 1 74 Mathematics for the Aviation Trades tension in the bracing wires and tie rods. almost all struts were of solid A few years ago wooden form, but they are now being replaced by metal tubes. Answer the following questions because they will help to make clear the change from wood to metal parts in aircraft : 1. What is the compressive strength of a round spruce whose diameter is 2^ in. ? 2. What would be the strength same size and shape? strut 3. so Why much a. If of a dural strut of the are solid metal struts not used, since they are stronger than wooden ones ? the spruce strut were 3 ft. long, what would it weigh ? b. What would 4. Would a the dural strut weigh ? ^-in. H.C. steel round rod be as strong in compression as the spruce strut whose diameter is 2j in.? 5. Why then are steel rods not used for struts ? Rods should never be used in compression because they bend under a very small load. Tubing has great com- will compared to weight. Its compressive strength can be calculated just like the strength of any other material. It should always be remembered, however, pressive strength its that tubing in compression will long before its full because it will either developed, compressive strength bend or buckle. The length of a tube, compared to its fail is diameter, is extremely important in determining the compressive load that the tube can withstand. The longer the tube the more easily it will fail. This fact should be kept in mind when doing the following examples. Examples: Use Table 12 in the calculations. 1. Find the strength in compression of a S.A.E. 1015 round tube, whose outside diameter is f in. and whose inside diameter is 0.622 in. Fittings, Tubing, anc/ Rivets 1 75 2. Find the strength of a square tube, S.A.E. 1025, whose outside measurement is 1^ in. and whose wall thickness is What O.OH3 in. the strength in tension of a 16 gage round H.C. steel tube whose inside diameter is 0.0930 in.? 3. is 4. A nickel-steel tube, whose wall thickness and whose outside diameter is 1^ in., is placed sion. What load could it carry before breaking is 0.028 in compresdid not in. if it bend or buckle? Job 4: Aircraft Rivets A. Types of Rivets. No study of aircraft materials would be complete without some attention to rivets and riveted joints. Since it is important that a mechanic be able to recognize each type of rivet, study Fig. 231 carefully, and notice that Most dimensions of a rivet, such as width of the head and the radius of the head, depend upon the 1. of the diameter of the 2. head The rivet, indicated length of the rivet (except in is by A in Fig. 231. measured from under the and is naturally countersunk rivets) independent of the diameter. Examples: Find all the dimensions for a button head aluminum whose diameter is ^ in. (see Fig. 231). 2. A countersunk head dural rivet has a diameter of f in. Find the dimensions of the head. 3. Make a drawing, accurate to the nearest 32nd in., of a round head aluminum rivet whose diameter is f in. and whose length is 2 in. 4. Make a drawing of a countersunk rivet whose diameter is in. and whose length is 3 in. 1. rivet -3- The Strength of Rivets in Shear. Many different kinds of aluminum alloys have been classified, and the B. 176 Mathematics for the Aviation Trades l*1. -j R- c A 5 ^-76^ * Fig. In sizes J in. 231. and Common larger. f For sizes up types of aluminum-alloy to and including j^ in. diameter. (From "The Riveting of rivets. Aluminum" by The Aluminum Co. of America.) Fittings/ Tubing, and 177 Rivets strength of each determined by direct test. The method of driving rivets also has an important effect upon strength as Table 13 shows. TABLE 13. STRESSES FOR DRIVEN KIVETS Examples: 1. Find the strength in shear of a or-in. button head 17S-T rivet, driven cold, immediately after quenching. 2. What is the strength in shear of a ^-in. round head 24S-T rivet driven cold immediately after quenching? 3. Find the strength in shear of a f-in. flat head 2S rivet driven cold. 4. Two 53S-W combined strength rivets are in shear, driven cold. if What is their the diameter of each rivet is -YQ in. ? 5. Draw up a table of the shear strength of 2S rivets, driven cold, of these diameters: f in., in., f- in., ^ in., f in., n. C. Riveted Joints. There are two main classifications of riveted joints: lap and butt joints, as illustrated in Fig. 232. In a lap joint, the strength of the structure in shear is equal to the combined strength of all the rivets. In a butt joint, on the other hand, the shear strength of the structure is equal to the strength of the rivets on one side of the joint only. Why? Fittings, Tubing, and 179 Rivets 3. What would be the strength in shear of a lap joint with one row of ten ITS -in. 2S rivets driven cold, as received ? 4. Find the strength of the lap joint shown in Fig. 234. 5. Find the strength of the butt joint shown in Fig. 235. Job 5: Review lest In a properly designed structure, no one item is disproportionately stronger or weaker than any other. Why? 1. VM Fig. 236. Lap joint, dural plates, immediately The (d) riveted joint shown 3/16 after in. diameter 17 S-T in Fig. 236 the ultimate strength in tension; T rivets, driven in tension. Find quenching. is (b) the strength of L All maferia/s: High carbon $feel (b) (a) Fig. 237. (a) Tie rod terminal; (b) clevis pin; (c) (c) fitting. the rivets in shear; (c) the strength of the joint in bearing. If this joint were subjected to a breaking load, where would it break first? What changes might be suggested? 180 Mathematics for the Aviation Trades Examine the structure in Fig. 237 very carefully. Find the strength of (a) the tie rod terminal in tension; (6) the tie rod terminal in bearing; (c) the clevis pin in shear; (d) the fitting in tension; (e) the fitting in bearing. If the 2. rod terminal were joined to the fitting by means of the and subjected to a breaking load in tension, where would failure occur first ? What improvements might be suggested? NOTE: It will be necessary to find the ultimate strength in each of the parts of the above example. tie clevis pin Chapter XII ALLOWANCE BEND A large number of aircraft factories are beginning to consider a knowledge of bend allowance as a prerequisite to the hiring of certain types of mechanics. Aircraft manufacturers in their some cases have issued special instructions to employees on this subject. Angle of bend ^- V __ * Fig. Many to instructions drawings. The amount and is metal be bent from the fittings require that according piece, of J 238. given bending is in flat blueprints measured or in degrees called the angle of bend (see Fig. 238). R = Radius r* 4J I Good bend Bad bend Fig. 239. When I a piece of metal is bent, it is important to round he vertex of the angle of bend or the metal may break. A form is, therefore, used to assist the mechanic in making a good bend. The radius of this form as shown in Fig. 239 called the radius of bend. 181 is 182 A Mathematics for the Aviation Trades bend means a gradual curve; a very means a sharp bend. Experience has shown that the radius of bend depends on the thickness of the large radius of small radius metal. In the case of steel, for example, for cold bending, the radius of bend should not be smaller than the thickness of the metal. Job 1 : The Bend Allowance Formula This job is the basis of all the work in this chapter. it is understood before the next job is undertaken. Be sure Definition: Bend allowance (B.A.) the length of the curved part of practically equal to the length of an arc is the fitting. It is of a circle as shown in Fig. 40. Bend allowance Fis. 240. The amount of material needed for the bend depends the radius of bend (jR); the thickness of the metal upon (T} the angle of bend in degrees (N). Formula: B.A. - (0.0 1 743 X ILLUSTRATIVE R + 0.0078 X T) X N EXAMPLE Find the bend allowance for a f-in. steel fitting to be bent 90 over a ^-in. radius, as shown in Fig. 241. 90* Fi S . 241. Allowance fienc/ Given: R = 1 8 3 in. T = \| in. N= 90 B.A. Find: B.A. B.A. B.A. B.A. B.A. To + 0.0078 X T) X N + 0.0078 X 1) X 90 (0.01743 X X i (0.00872 + 0.00098) (0.01743 = = = = (0.00970) 0.8730 fl X X 90 90 in. the nearest 64th, in. Ans. Method: a. Multiply, as indicated the by formula, within the parentheses. b. Add within the parentheses. c. Multiply the sum by the number outside the parentheses. Examples: 1. Find the bend allowance for a Te-in. steel be bent 90 over a form whose radius is % in. 2. What is the bend allowance needed for fitting to be bent over a a 45 form whose radius is f fitting to -J-in. in. to dural make angle of bend ? (b) Fi 3 . 3. bend 4. A -g-i n 242. steel fitting is to be bent 00. The radius is in. What is the bend allowance? Find the bend allowance for each of the fittings of shown in Fig. 242. Complete the following table, keeping in mind that the thickness of the metal, R is the radius of bend, and that all dimensions are in inches. 5. T is 184 Mathematics for the Aviation Trades BEND ALLOWANCE CHART (90 angle of bend) R t 0.120 005 0.032 Job 2: The Over-all Length of the Flat Pattern Before the fitting can be laid out on flat stock from a drawing or blueprint such as shown in Fig. 243 (a), it is N FLAT PATTERN "BENT-UP"VIEW () (b) Fis- 243. important to know the over-all or developed length of the pattern, which can be calculated from the bent-up drawing. If the straight portions of the fitting are called A and J5, the following formula can be used: flat Formula: Over-all length ILLUSTRATIVE = A+ B + B.A. EXAMPLE Find the over-all length of the flat pattern in Fig. 244. Notice that the bend allowance has already been calculated. 64 Fi 9 . 244. 1 86 Mathematics Job 3iWhen for the Aviation Trades Are Given Inside Dimensions easy enough to find the over-all length when the exact length of the straight portions of the fitting are given. It is these However, i ^3 Fig. each 248. will show that the problem individually, Formula: B, which fitting, than A = must straight portion A is d to apply a formula R length of one straight portion. inner dimension. radius of bend. is the length of the other straight portion of the in a similar manner. can be found ILLUSTRATIVE Find the over-all length shown in Fig. 249. EXAMPLE of the flat pattern for the fitting f Bend 90"J r _i rfe, Fig. Given: R = T - N= B.A. Find: portions equal to the inner dimension d, minus the radius of bend jR. This can be put in terms of a formula, but it will be easier to solve mechanically. where A = d = R = straight usually be found from other dimensions given in the drawing or blueprint. In this case, an examination of Fig. 248 = ? in. * in. 90 \|i in. Over-all length A - 2* B = If - \| i = - 249. Bend Allowance Over-all length Over-all length Over-all length 187 = A + B + B.A. = t + If + ft = 4^ in. Ans. Method: a. First calculate the length of the straight portions, A and B, from the drawing. b Then use the formula: over-all length = A + B + A.B. In the foregoing illustrative example, the bend allowance it have been calculated, if it had not been was given. Could given? How? Examples: 1. Find the over-all length shown tings to find the in Fig. 250. of the flat pattern of the Notice that it will first fit- be necessary bend allowance. Bend 90 A* J" (a) Fi 3 . 2. 250. Find the bend allowance and over-all length of the Observe that in patterns of the fittings in Fig. 251. one outside dimension is given. (a) flat -is I'"" 64 fa) (b) 251. 3. Find the bend allowance and fitting shown in Fig. 252. Draw accurate to the nearest 64th. over-all length of the a full-scale flat pattern 188 Job Mathematics 4: When for the Aviation Trades Outside Dimensions Are Given In this case, not only the radius but also the thickness metal must be subtracted from the outside dimension of the in order to find the length of the straight portion. Formula: A = D A = D = R = T radius of bend. T = where R thickness of the metal. length of one straight portion. outer dimension. B can be found in a similar manner. Here again no formulas should be memorized. A careful analysis of Fig. 253 will show how the straight portion L -A jr> Fis. of the fitting 253. can be found from the dimensions given on the blueprint. Examples: 1. Find the length shown 2. 3. of the straight parts of the fitting in Fig. 254. Find the bend allowance of the fitting in Fig. 254. Find the over-all length of the flat pattern of the fitting in Fig. 254. Bend Allowance 189 0.049- Fig.254. What Fi g . 3/64-in.L.C.rteel bent 255. 90. the over-all length of the flat pattern of the fitting shown in Fig. 255 ? The angle of bend is 90. 4. is r --/A-~, U-~// Fig. 6. Make fitting 6. Job 257. What 5: 0.035 in. thickness, 2 bends of 90 a full-scale drawing of the shown Fig. 256. -> is flat each. pattern of the in Fig. 256. 1 /8-in. H.C. steel bent 90, 1 /4 in. radius of bend. thd over-all length of the fitting in Fig. 257? Review Test Figure 258 shows the diagram of a 0.125-in. low-carbon steel fitting. Find the bend allowance for each of the three bends, if the radius of bend is ^ in. (6) Find the over-all dimensions of this fitting. 1. (a) 190 Mathematics for the Aviation Trades Bencl45 fle " 2i ..I -r5 QW AH dimensions are in inches Fig. 2. Find the 258. tensile strength of the fitting in Fig. the diameter of all holes is in., (a) each end; 258 if (6) at the having two holes. Use the table of safe working strengths page 169. 3. Make a full-scale diagram of the fitting (Fig. 258), side including the bend allowance. 4. Calculate the total bend allowance and over-all length of the flat pattern for the fitting in Fig. 259. All bends are 90 Chapter XIII HORSEPOWER What is the main purpose of the aircraft engine? It provides the forward thrust to overcome the resistance of the airplane. What part of the engine provides the thrust? The rotation of the propeller provides the thrust (see Fig. 260). Fig. But what makes The 260. the propeller rotate? revolution of the crankshaft (Fig. 261) turns the propeller. What makes the shaft rotate? The force exerted by the connecting rod (Fig. 262) turns the crankshaft. What forces the rod to drive The piston drives the rod. the heavy shaft around? Trace the entire process from piston to propeller. can easily be seen that a great deal of work is required to keep the propeller rotating. This energy comes from the It burning of gasoline, or any other 193 fuel, in the cylinder. 195 Horsepower In a very powerful engine, a great deal of fuel will be used and a large amount of work developed. We say such an engine develops a great deal of horsepower. In order to understand horsepower, we must first learn the important subtopics upon which this subject depends. Job 1 : Piston Area The greater the area of the piston, the more horsepower the engine will be able to deliver. It will be necessary to find the area of the piston before the horsepower of the engine can be calculated. Fig. The 263. Piston. top* of the piston, called the head, is known to be a To find its area, the formula for the circle (see Fig. 263). area of a circle is needed. Formula: A= 0.7854 ILLUSTRATIVE X D2 EXAMPLE Find the area of a piston whose diameter Given: Diameter = 3 in. Find Area : is 3 in. 196 Mathematics A = A = A = for the Aviation Trades 0.7854 0.7854 XD X3 X 7.0686 sq. 2 3 Ans. in. Specifications of aircraft engines do not give the diameter but do give the diameter of the cylinder, or of the piston, the bore. Definition: equal to the diameter of the cylinder, but may be considered the effective diameter of the piston. correctly Bore is Examples: 1. 6 in., Find the area 7 2-9. 10. of the pistons whose diameters are in., 3.5 in., 1.25 in. Complete the following: The Jacobs has 7 cylinders. What is its total piston area? 11. What The Kinner C-7 has is its 7 cylinders total piston area? and a bore of 5f in. 197 Horsepower 12. A 6 cylinder Menasco engine has a bore of 4.75 in. Find the total piston area. The head of the piston may be flat, concave, or domed, as shown in Fig. 264, depending on how it was built by Effective piston Concave Flat S. 264. Dome Three types of piston heads. the designer and manufacturer. The effective piston area in all cases, however, can be found by the? method used in this job. Job 2: Displacement of the Piston When down running, the piston moves up and cylinder. It never touches the top of the cylinder the engine in its is Top center Bottom center Displacement Fig. 265. on the upstroke, and never comes too near the bottom of the cylinder on the downstroke (see Fig. 265). 198 Mathematics for the Aviation Trades ' Definitions: Top . center is the highest point the piston reaches on its upstroke. Bottom center is the lowest point the head of the piston reaches on the downstroke. Stroke is center. It is the distance between top center and bottom measured in inches or in feet. the volume swept through by the piston in moving from bottom center to top center. It is measured in cubic inches. It will depend upon the area of the moving Displacement piston and is upon the distance moves, that it area Formula: Displacement X is, its stroke. stroke EXAMPLE ILLUSTRATIVE Find the displacement of a piston whose diameter is 6 in. and whose stroke is 5% in. Express the answer to the nearest tenth. Given: Diameter = 6 in. = Stroke 5^ Displacement Find: = A = A = A = Disp. Disp. Disp. Note that it is 5.5 in. 0.7854 0.7854 X X Z> 2 6 28.2744 sq. X = A X S = 29.2744 X 5.5 = 155.5 cu. in. Ans. first necessary to find the area of the piston. Examples: 1-3. 6 in. Complete the following table: Horsepower 4. 1 The Aeronca E-113A has a bore stroke of 4 in. It What has 2 cylinders. of 4.25 in. is its 99 and a total piston displacement? 5. of The Aeronca E-107, which has 4-g- and a stroke in. 2 cylinders, has a bore is its total cubic What of 4 in. displacement? 6. A 6.12 in. Job 3: 9 cylinder radial Wright Cyclone has a bore of of 6.87 in. Find the total displacement. and a stroke Number of Power Strokes In the four-cycle engine the order of strokes is intake, compression, power, and exhaust. Each cylinder has one power stroke for two revolutions of the shaft. How many power strokes would there be in 4 revolutions? in 10 revolutions? in 2,000 r.p.m.? Every engine has an attachment on its crankshaft to which a tachometer, such as shown in Fig. 266, can be fas- Fig. tened. 266. Tachometer. (Courtesy of Aviation.) The tachometer has of revolutions the shaft Formula: where N= * R.p.m. = is N a dial that registers the making ' - number in 1 minute. X cylinders number of power strokes per minute. revolutions per minute of the crankshaft. 200 Mathematics for the Aviation Trades ILLUSTRATIVE EXAMPLE A 5 cylinder engine is making 1,800 r.p.m. strokes does it make in 1 miri.? in 1 sec.? How many power Given 5 cylinders : 1,800 r.p.m. N N = r.p.m. N Find: = 4,500 power strokes per minute , r There are 60 Number N = of sec. in 1 power 4,500 . = , X ,. , cylinders min. strokes per second _ Ana. . : . 75 power strokes per second A Ans. Examples: 1-7. 8. when Complete the following table in your own notebook: How many it r.p.m. does a 5 cylinder engine delivers 5,500 power strokes per minute? make 201 Horsepower A 9 cylinder Cyclone delivers 9,000 power strokes per the tachometer reading? 10. A 5 cylinder Lambert is tested at various r.p.m. as listed. Complete the following table and graph the results. 9. minute. Job 4: What is Types of Horsepower The fundamental purpose to turn the propeller. This of the aircraft engine work done by the engine is is expressed in terms of horsepower. Definition: One horsepower of work equal to 33,000 is raised one foot in one minute. The horsepower necessary developed inside the cylinders bustion of the peller. Part of fuel. But not it is lost in oil to by the heat of the com- ever reaches the proovercoming the friction of the all of it it is used to operate etc. pumps, There are three 268). being explain Fig. 267? turn the propeller is shaft that turns the propeller; part of the Ib. Can you different types of horsepower (see Fig. 202 Mathematics for the Aviation Trades Definitions: Indicated developed horsepower is (i.hp.) the total horsepower in the cylinders. is that part of the indicated used in overcoming friction at the bearings, driving fuel pumps, operating instruments, etc. Friction horsepower (f.hp.) horsepower that 267. 1 ft.-lb. hp. = is 268. 33,000 Brake horsepower (b.hp.) drive the propeller. Formulas: Three types of horsepower. per min. Indicated the horsepower available to is horsepower I.hp. = brake -f- friction horsepower = ILLUSTRATIVE horsepower b.hp. -[- f.hp. EXAMPLE Find the brake horsepower of an engine when the indicated horsepower is 45 and the friction horsepower is 3. Given: I.hp. = 45 F.hp. Find: = 3 B.hp. I.hp. 45 B.hp. = = = b.hp. b.hp. 42 + f.hp. +3 Ans. Examples: 1. The indicated horsepower of an engine 43 hp. is lost as horsepower ? friction horsepower, what is is 750. If the brake 203 Horsepower 2-7. Complete the following table: Figure out the percentage of the total horsepower that is used as brake horsepower in Example 7. This percentage is called the mechanical efficiency of the 8. engine. 9. What is the mechanical efficiency of an engine whose indicated horsepower is is 95.5 and whose brake horsepower 65? 10. An engine developes efficiency Job 5: The sq. in. if 25 hp. Mean air is 155 b.hp. What is its mechanical lost in friction? Effective Pressure pressure all about us is approximately 15 Ib. per is also true for the inside of the cylinders before This started; but once the shaft begins to turn, the pressure inside becomes altogether different. Read the following description of the 4 strokes of a 4-cycle engine very the engine is carefully and study Fig. 269. 1. Intake: The piston, moving downward, acts like a pump and pulls the inflammable mixture from the carburetor, through the manifolds and open intake valve into the cylinder. closes. When the cylinder is full, the intake valve 204 Mathematics During the intake for the Aviation Trades moves down, making stroke the piston the pressure inside less than 15 Ib. per sq. in. This pressure is not constant at any time but rises as the mixture fills the chamber. Compression: With both valves closed and with a cylinder full of the mixture, the piston travels upward compressing the gas into the small clearance space above 2. the piston. The pressure mixture from about 15 per sq. is raised Ib. this squeezing of the by per sq. in. to 100 or 125 Ib. in. (1) (2) (4) Intake stroke Compression stroke Exhaust stroke Fig. 269. 3. Power: The spark plug supplies the light that starts the mixture burning. Between the compression and power strokes, when the mixture is compressed into the clearance The pressure rises to 400 Ib. per gases, expanding against the walls of the enclosed chamber, push the only movable part, the piston, space, ignition occurs. sq. in. The hot downward. This movement by the connecting 4. Exhaust: is transferred to the crankshaft rod. The last stroke in the cycle is the exhaust gases have now spent their energy in pushing the piston downward and it is necessary to clear the stroke. The cylinder in order to make room for a new charge. The ex- 205 Horsepower haust valve opens and the piston, moving upward, forces the burned gases out through the exhaust port and exhaust manifold. During the exhaust stroke the exhaust valve remains open. Since the pressure inside the cylinder is greater than atmospheric pressure, the mixture expands into the air. It Fig. 370. further helped by the stroke of the piston. inside the cylinder naturally keeps falling off. is The pressure chart in Fig. 270 shows how the pressure changes through the intake, compression, power, and exhaust strokes. The horsepower of the engine depends upon the The average of all these changing pressures. Definitions: Mean effective pressure is the average of the changing be abbreviated pressures for all 4 strokes. It will henceforth M.E.P. Indicated obtained by mean pressure is the actual average using an indicator card somewhat similar to the diagram. This Brake mean effective is abbreviated I.M.E.P. that percentage of the indicated mean effective pressure that is not lost in friction but goes toward useful work in turning the propeller. This is effective pressure abbreviated B.M.E.P. is 206 Job Mathematics 6: How for the Aviation Trades to Calculate Brake Horsepower We have already learned that the brake horsepower depends upon 4 factors: 1. The B.M.E.P. 2. 3. 4. The length of the stroke. The area of the piston. The number of power strokes Remember per minute. these abbreviations: B.M.E.P. Formula: B.hp. X L XAX N 33,000 iLLUSTRATIVE EXAMPLE Given B.M.E.P. : = Stroke = Area = N = 120 0.5 Ib. per sq. in. ft. 50 sq. in. 3,600 per min. Find: Brake horsepower B.hp. = B.M.E.P. X L X A X B.hp. = B.hp. = 327 N 33,000 120 X 0.5 X 50 X 3,600 33,000 A ns. be necessary to calculate the area of the piston of power strokes per minute in most of the problems in brake horsepower. Remember that the stroke must be expressed in feet, before it is used in the formula. It will and the number 207 Horsepower Examples: Find the brake horsepower of an engine whose stroke and whose piston area is 7 sq. in. The number of power strokes is 4,000 per min. and the B.M.E.P. is 120 Ib. 1. is 3 ft. per sq. 2. Find in. The area of a piston is 8 sq. in. and its stroke 4 in. is brake horsepower if the B.M.E.P. is 100 Ib. per in. This is a 3 cylinder engine going at 2,000 r.p.m. sq. Hint: Do not forget to change the stroke from inches to its feet. 3. The diameter of a piston is 2 in., its stroke is 2 in., and it has 9 cylinders. When it is going at 1,800 r.p.m., the B.M.E.P. is 120 Ib. per sq. in. Find the brake horsepower. 4-8. Calculate the brake horsepower of each of these engines: 120 . 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 R.p.m. Fig. 271. Graph of B.M.E.P. for Jacobs aircraft engine. shows how the B.M.E.P. keeps changing with the r.p.m. Complete the table of data in your own notebook from the graph. 9. The graph in Fig. 271 208 Mathematics 10. for the Aviation Trades Find the brake horsepower of the Jacobs L-5 at each r.p.m. in the foregoing table, if the bore is 5.5 in. stroke is 5.5 in. This engine has 7 cylinders. Job 7: and the The Prony Brake In most aircraft engine factories, brake horsepower is calculated by means of the formula just studied. There are, in addition, other methods Fig. The Prony brake is 272. of obtaining it. Prony brake. built in many different ways. The to be determined is flywheel of the engine whose power clamped by means of the adjustable screws between friction blocks. Since the flywheel tends to pull the brake in the is same direction as pushes the it would normally move, arm downward. The force F it naturally with which it 209 Horsepower pushes downward is Formula: DL B.hp. = , F may 2* X F X scale in pounds. D X r.p.m. is TT - measured on the the reading on the scale and is measured in pounds. This does not include the weight of the arm. D is the distance in feet from the center of the flywheel to the scale. be used as 3.14. ILLUSTRATIVE EXAMPLE The scale of a brake dynamometer reads 25 Ib. when the shaft an engine going 2,000 r.p.m. is 2 ft. from the scale. What is the brake horsepower? Given: F = 25 Ib. of D = 2 ft. 2,000 r.p.m. Find: B.hp. B.hp. = B.hp. = B.hp. = 27r 2 XFXD X X 33,000 3.14 X 25 19.0 hp. r.p.m. X 2 X 2,000 33,000 Ans. Examples: 1. 12 The scale of a Prony brake 2 ft. from the shaft when the engine is going at 1,400 r.p.m. What Ib. reads is the brake horsepower 2. A Prony brake has its scale 3 ft. 6 in. from the shaft of an engine going at 700 r.p.m. What horsepower is being developed when the scale reads 35 Ib. ? of the engine? 3. The away. when the shaft is 1 ft. 3 in. the brake horsepower when the tachometer scale reads 58 Ib. What is reads 1,250 r.p.m.? important to test engines at various r.p.m. Find the brake horsepower of an engine at the following tachom4. It is eter readings if the scale is 3 ft. from the shaft: Giapter FUEL AND OIL XIV CONSUMPTION Mechanics and pilots are extremely interested in how gasoline and oil their engine will use, because aviation gasoline costs about 30ff a gallon, and an engine that wastes gasoline soon becomes too expensive to operate. That is why the manufacturers of aircraft list the fuel and oil much consumption, in the specifications that accompany each engine. Fuel consumption is sometimes given in gallons per hour, or in miles per gallon as in an automobile. But both these methods are very inaccurate and seldom used for aircraft engines. The quantity of fuel and increasing as the throttle is opened increases. Also, the longer the engine and oil are used. consumed keeps and the horsepower is run, the more fuel oil We can, therefore, say that the fuel and oil consumption depends upon the horsepower of the engine and the hours of operation. Job 1 : Horsepower-hours Definition: The horsepower-hours show both the horsepower and running time of the engine in one number. Formula: Horsepower-hours where horsepower hour = = = horsepower X hours horsepower of the engine. length of time of operation in hours. 212 Fuel and Oil Consumption EXAMPLE ILLUSTRATIVE A 65-hp. engine runs for hr. 213 What is the number of horse- power-hours Given: 65 hp. ? Find: Hp.-hr. Hp.-hr. Hp.-hr. Hp.-hr. = = = hp. X hr. 65 X 2 Am. 130 Examples: 1. A 130-hp. engine is run for 1 hr. What is the number Compare your answer with the answer to the illustrative example above. 2. A 90-hp. Lambert is run for 3 hr. 30 min. What is the of horsepower-hours number of ? horsepower-hours? 3-9. Find the horsepower-hours for the following engines Job A 2: Specific typical : Fuel Consumption method of listing fuel consumption is in the of fuel consumed per horsepower-hour, amount consumed by each horsepower for 1 hr. For instance a LeBlond engine uses about ^ Ib. of gasoline number that is, of pounds the to produce 1 hp. for fuel consumption 1 hr. of the We therefore say that the specific LeBlond ^ Ib. per hp.-hr. This can few of the different forms is be abbreviated in many ways. A used by various manufacturers follow: 214 Mat/iemat/cs for the Aviation Trac/es BHP lb. lb. .50 lb. per HP. hour .50 lb. per HP.-hr. per hour /BHP /hour /BHP-hour lb. per For the sake o.70i i i 0.50 lb. /hp. hr. form lb. per hp.-hr. used for the work of simplicity the i i i i i jo.65 1300 1500 1700 1900 2100 Revolutions per minute 275. be in this chapter. The Fig. will Specific fuel consumption of the Menasco B-4. specific con- fuel sumption changes with the r.p.m. The graph in Fig. 275 shows that there is a different specific fuel consumption at each throttle As the number of revolutions per minute of the crankshaft increases, the specific fuel consumption changes. setting. Complete the following table of data from the graph : Questions: At what r.p.m. engine shown in Fig. 1. concerned ? is it most economical to operate the consumption is 276, as far as gasoline Fuel and Oil Consumption 2. The engine specific fuel 21 5 rated 95 hp. at 2,000 r.p.m. is What is consumption given in specifications for the this horsepower? 40 276. Fig. Fuel sage. eo '; (Courtesy of Pioneer Instrument Aviation Corp.) Division of Bendix ' Job 3: Gallons and Cost The number of pounds of gasoline an engine will consume can be easily calculated, if we know (a) the specific consumption; (6) the horsepower of the engine; (c) the running time. Formula: Total consumption specific consumption X horsepowerhours ILLUSTRATIVE A Lycoming EXAMPLE 240-hp. engine runs for 3 hr. Its specific fuel con- How many pounds sumption it consume? Given: 240 hp. for 3 hr. at 0.55 Ib. per hp.-hr. Find: Total consumption is 0.55 Ib. per hp.-hr. Total Total Total = = = specific 0.55 396 X Ib. consumption X of gasoline will hp.-hr. 7-20 Ans. Hint: First find the horsepower-hours. Examples: 1-9. Find the total fuel of the following engines: consumption in pounds of each 216 Mathematics for the Aviation Trades Find the total consumption wind in Examples 8 and 9. 10. in pounds for the Whirl- the weights that airplanes must carry in fuel only seem amazing, consider the following: If The Bellanca Transport carries 1,800 Ib. of fuel The Bellanca Monoplane carries 3,600 Ib. of fuel The Douglas DC-2 carries 3,060 Ib. of fuel Look up the fuel capacity of 5 other airplanes and compare the weight of fuel to the total weight of the airplane. You now know how to find the number of pounds of gasoline the engine will need to operate for a certain number many But gallons will One about of hours. gasoline be needed? is bought by the How much gallon. How will it cost? gallon of aviation gasoline weighs 6 Ib. and costs 30ff. ILLUSTRATIVE EXAMPLE A mechanic needs 464 Ib. of gasoline. How many gallons should How much would this cost? he buy at 30^f a gallon? Given: 464 Ib. Find: (a) Gallons (6) Cost (a) A\|A = 77.3 (b) 77.3 X .30 gal. = $3.19 Ana. Fue/ and Oil Consumption 21 7 Method: To get the number of gallons, divide the number of pounds by 6. Examples: A mechanic needs 350 Ib. of gasoline. gallons does he need? If the price is 28^ per 1. is How many gallon, what the cost? The 2. price of gasoline chanic needs 42 Ib. A pilot stops 3. different times is per gallon, and a meshould he pay? 20jS How much at three airports and buys gasoline three : La Guardia Airport, 40 Ib. at 30^ per gallon. Newark Airport, 50 Ib. at 28ji per gallon. Floyd Bennett Field, 48 Find the total cost for gasoline Ib. at on 29^ per gallon. this trip. Do this problem without further explanation: Bellanca Transport has a Cyclone 650-hp. engine whose specific fuel consumption is 0.55 Ib. per hp.-hr. On a 4. A trip to Chicago, the engine runs for 7 hr. Find the number of gallons of gasoline needed and the cost of this gasoline at 25 per gallon. Note: The assumption here is that the engine operates at a constant fuel consumption for the entire trip. Is this entirely true? Shop Problem: What is efficiently Job meant by octane rating? Can all engines operate using fuel of the same octane rating? 4: Specific Oil Consumption The work in specific oil consumption is very much like the work in fuel consumption, the only point in the fact that much of difference smaller quantities of oil being are used. 218 Mathematics The average for the Aviation Trades specific fuel 0.49 Ib. per is consumption hp.-hr. One gallon The average of gasoline weighs 6 specific oil Ib. consumption is 0.035 Ib. per hp.-hr. One gallon of oil weighs 7.5 Ib. Examples: Do 1. the following examples by yourself: 575-hp. engine has a specific The Hornet and runs consume? for 3 hr. tion of 0.035 Ib. per hp.-hr. pounds of oil does it oil consump- How many A 425-hp. Wasp runs for 2-g- hr. If its specific oil consumption is 0.035 Ib. per hp.-hr., find the number of 2. pounds of oil it uses. 3-7. Find the weight and the number of gallons of used by each of the following engines oil : 8. ALeBlond 70-hp. of 0.015 Ib. per hp.-hr. used in 2 hr. 30 min. ? Job 5: How engine has a specific How many oil quarts of consumption oil would be Long Can an Airplane Stay Up? The calculation of the exact time that an airplane can fly nonstop is not a simple matter. It involves consideration of the decreasing gross weight of the airplane due to the consumption of gas6line during flight, changes in horsepower at various times, and many other factors. However, Fuel and Oil Consumption 219 the method shown here will give a fair approximation of the answer. nothing goes wrong with the engine, the airplane will stay aloft as long as there is gasoline left to operate the engine. That depends upon (a) the number of gallons of If gasoline in the fuel tanks, used per hour. The capacity and (6) amount the of the fuel tanks in gallons is of gasoline always given in aircraft specifications. An instrument such as that appearing in Fig. 276 shows the number of gallons of fuel in the tanks at all times. is.. ,. Formula: Cruising time gallons in fuel tanks = jp- ILLUSTRATIVE An is airplane per hr. it stay up, if consumed EXAMPLE powered with a Kinner How long can . j gallons per hour K5 which uses 8 gal. there are 50 gal. of fuel in the tanks ? ~ gallons in fuel tanks -? n gallons per hour consumed Cruising time = Cruising time = %- = 6^ . . . ; hr. ^4rw. Examples: An Aeronca has an engine which consumes gasoline at the rate of 3 gal. per hr. How long can the Aeronca stay 1. up, 2. if it At started with 8 gal. of fuel? cruising speed an airplane using a LeBlond engine per hr. How long can this airplane fly at has 12^ gal. of fuel in its tanks? 3. The Bellanca Airbus uses a 575-hp. engine whose fuel consumption is 0.48 Ib. per hp.-hr. How long can this airplane stay up if its fuel tanks hold 200 gal.? consumes 4f this speed, 4. gal. if it The Cargo Aircruiser uses a 650-hp. engine whose per hp.-hr. The capacity of the tank consumption is 150 gal. How long could it stay up? 5. A Kinner airplane powered with a Kinner engine has 50 gal. of fuel. When the engine operates at 75 hp., the is 0.50 Ib. 220 Mathematics specific it is consumption for the Aviation Trades 0.42 Ib. per hp.-hr. How long could fly? An airplane has a LeBlond 110-hp. engine whose specific consumption is 0.48 Ib. per hp.-hr. If only 10 gal. of gas are left, how long can it run ? 6. A large transport airplane is lost. It has 2 engines of 715 hp. each, and the fuel tanks have only 5 gal. altogether. If the lowest possible specific fuel consumption is 7. 0.48 Ib. per hp.-hr. for each engine, airplane stay aloft? how long can the Job 6: Review Test 1. A Vultee consumption Fig. 277. consumption is is powered by an engine whose specific fuel and whose specific oil 0.60 Ib. per hp.-hr. Vultee is military 0.025 Ib. monoplane. (Courtesy of Aviation.) per hp.-hr. when operating at 735 hp. (see Fig. 277). a. How many would be used in 2 hr. would be consumed in 1 hr. gallons of gasoline 15 min.? b. How many quarts of oil 20 min. ? c. The 206 gal. How long can the tanks are empty, if it fuel tanks of the Vultee hold the airplane stay up before all operates continuously at 735 hp. ? d. The oil tanks of the Vultee have a capacity of 15 gal. How long would th engine operate before the oil tanks were empty? 221 fuel and Oil Consumption 2. The Wright GR-2600-A5A whose bore radial engine 6fV is in. 168 When Ib. operating at per sq. Fig. specific fuel is 278. in. is a 14 cylinder staggered and whose stroke is 2,300 r.p.m. its B.M.E.P. 6^ in. At rated horsepower, Performance curves: Ranger consumption is 6 aircraft this engine's engine. 0.80 Ib. per hp.-hr. Find how in 4 hr. gallons of gasoline will be consumed of the engine. Hint: First find the horsepower curve for the Ranger 6 cylinder, 3. The many performance in-line engine, specifications. this graph: was taken from company Complete the following table of data from shown in Fig. 278, 222 Mathematics for t/ie Aviation Trades Complete the following tables and represent the results a line graph for each set of data. by 4. FUEL CONSUMPTION OF THE RANGER 6 Aircraft engine performance curves generally show two types of horsepower: Full throttle horsepower. This is the power that the engine can develop at any r.p.m. Using the formula for b.hp. (Chap. XIII) will generally give this curve. 1. Propeller load horsepower. This will show the horsepower required to turn the propeller at any speed. 2. C/iapterXV COMPRESSION RATIO AND VALVE TIMING In an actual engine cylinder, the piston at top center does not touch the top of the cylinder. The space left near the top of the cylinder after the piston has reached top center may have any of a wide variety of shapes depending >) Fig. ~(c) (b) 280. upon the engine design. Some are shown in Fig. 280. Job 1 : (6) (d) Types of combustion chambers from "The Airplane and Chatfield, Taylor, and Ober. of the Its Engine' by more common shapes Cylinder Volume The number of cylinders in aircraft engines ranges 2 for the Aeronca the from to 14 cylinders for certain way up Wright or Pratt and Whitney engines. For all practical purposes, all cylinders of a multicylinder engine may be considered identical. It was therefore conall 224 Compression Ratio and Valve Timing 225 sidered best to base the definitions and formulas in this job upon a consideration of one cylinder only, as shown in Fig. 281. However, these same definitions and formulas will also hold true for the entire engine. B.C. Fig. 281 .Cylinder from Pratt and Whitney Wasp. (Courtesy of Aviation.) Definitions: 1. Clearance volume is the volume of the space left above the piston when it is at top center. Note: This is sometimes called the volume of the com- bustion chamber. the volume that the piston moves through from bottom center to top center. 3. Total volume of 1 cylinder is equal to the displacement plus the clearance volume. 2. is Displacement Formula :V, where V c ^ Disp. +V c means clearance volume. Disp. means displacement for one cylinder. V t means total volume of ILLUSTRATIVE The displacement volume is 10 cu. in. of a cylinder one cylinder. EXAMPLE is 70 cu. Find the total volume in. and the clearance of 1 cylinder. 226 Mathematics = V = for the Aviation Trades Given: Disp. 70 cu. in. c 10 cu. in. V Find: t V = V = Vt = t t + V Disp. 10 70 c + 80 cu. Ans. in. Examples: The displacement of one cylinder 98 cu. in. The volume above the piston 1. 24.5 cu. in. Each 2. ment is is is cylinder of a Whirlwind engine has a displaceand a clearance volume of 18 cu. in. of 108 cu. in. What 3. What a Kinner at top center the total volume of one cylinder? of the volume of one cylinder? is The Whirlwind engine What is in Example 2 has the total displacement? What is 9 cylinders. the total volume of all cylinders? 4. The total craft engine is volume 105 cu. of in. each cylinder of an Axelson airFind the clearance volume if the displacement for one cylinder Job 2: is 85.5 cu. in. Compression Ratio The words compression ratio are now in trade literature, instruction manuals, being used so much and ordinary auto- Totat cyUnder volume Bottom center Fig. 282. The ratio of these two volumes is called the "compression ratio." mobile advertisements, that every mechanic ought to what they mean. know Compression Ratio and Valve Timing 227 been pointed out that the piston at top center does not touch the top of the cylinder. There is always a compression space left, the volume of which is called the clearance volume (see Fig. 282). It has Definition: of Compression ratio is the ratio between the total volume one cylinder and its clearance volume. w Formula: C.R. =~ Ve = = Vt V = where C.R. c ratio. compression volume of one cylinder. clearance volume of one cylinder. total Here are some actual compression ratios for various aircraft engines: TABLE 14 Compression Ratio 6:1* 5.4:1 Engine Jacobs L-5 Aeronca E-113-C. Pratt and Whitney Ranger 6. . Wasp . Jr 6:1 6.5:1 . Guiberson Diesel * . 15: 1 Pronounced "6 to 1." Notice that the compression ratio of the diesel engine higher than that of the others. Why? is much ILLUSTRATIVE Find the compression ratio of the the total volume of one cylinder volume is Given: is Aeronca E-113-C in which and the clearance 69.65 cu. in. 12.9 cu. in. V = V = t 69.65 cu. c Find: EXAMPLE 12.9 cu. in. in. C.R. C.R. - c C>R ~ - C.R. = 69.65 12^ 5.4 Ans. 228 Mathematics for the Aviation Trades Examples: 1. The volume total Allison V-1710-C6 pression chamber is of of one cylinder 171.0 cu. in. one cylinder of the water-cooled The volume is 28.5 cu. in. of the com- WhaHs the compression ratio? 2. The Jacobs L-4M radial engine has a clearance volume for one cylinder equal to 24.7 cu. in. Find the compression ratio if the total volume of one cylinder is 132.8 cu. in. Find the compression ratio of the 4 cylinder Menasco Pirate, if the total volume of all 4 cylinders is 443.6 cu. in. and the total volume of all 4 combustion chambers is 3. 80.6 cu. in. Job 3: How to The shape Find the Clearance Volume compression chamber above the piston at top center will depend upon the type of engine, the number of valves, spark plugs, etc. Yet there is a simple method of calculating its volume, if we know the compression ratio of the and the displacement. Do specifications give these facts ? Notice that the displacement for one cylinder must be calculated, since only total displacement is given in specifications. .. r Formula: Vc , = displacement -7^-5 N^.K. I where Vc C.R. = = clearance volume for one cylinder. compression ratio. ILLUSTRATIVE The displacement ratio is 6:1. Find Given: Disp. C.R. = = it EXAMPLE for one cylinder is clearance volume. 25 cu. 6:1 in. 25 cu. in.; its compression Compression Ratio and Valve Timing V Find: 229 c V * c V = V = Disp. C.R. 1 25 c c - f^ 6 - 1 5 cu. in. Ans. Check the answer. Examples: The displacement one cylinder of a LeBlond engine is 54 cu. in.; its compression ratio is 5.5 to 1. Find the clearance volume of one cylinder. 2. The compression ratio of the Franklin is 5.5:1, and the displacement for one cylinder is 37.5 cu. in. Find the 1. volume of of the for compression chamber and the total volume one cylinder. Check the answers. The displacement for all 4 cylinders of a Lycoming 144.5 cu. in. Find the clearance volume for one cylinder, 3. is the compression ratio is 5.65 to 1. 4. The Allison V water-cooled engine has a bore and stroke of 5^ by 6 in. Find the total volume of all 12 cylinders if the compression ratio is 6.00:1. if Job 4: Valve Timing Diagrams The exact time open and at which the intake close has been carefully set and exhaust valves by the designer, so as to obtain the best possible operation of the engine. After the engifte has been running for some time, however, the valve timing will often be found to need adjustment. Failure to make such corrections will result in a serious loss of power and in eventual damage to the engine. Valve timing, therefore, is an essential part of the specifications of an engine, whether aircraft, automobile, marine, or any other kind. All valve timing checks and adjustments that the mechanic makes from time to time depend upon this information. 230 Mathematics for the Aviation Trades A. Intake. Many students are under the impression that the intake valve always opens just as the piston begins to move downward on the intake stroke. Although this may at first glance seem natural, very seldom correct for it is aircraft engines. Intake valve opens ^^ v Intake valve opens 22B.T.C. before top center, Arrowshows^ direction of rotation of the Intake valve crankshaft doses 62'A.B.C} Bottom center Bottom center Fi 9 . 283. Fig. 283b. Valve timing data is given in degrees. For instance, the intake valve of the Khmer K-5 opens 22 before top center. This can be diagrammed as shown in Fig. 283 (a). Notice that the direction of rotation of the crankshaft is given by the arrow Intake In opens most aircraft the engines, intake valve does not close as soon as 22B.T.C. the piston reaches the bottom of downward stroke, but remains for a considerable length of its open time thereafter. The intake valve of the engine shown in Fig. 283(6) closes 82 after bottom The diagram center. This information can be put on the same diagram. shows the valve timing diagram in Fig. 284 for the intake stroke. These abbreviations are used: 232 MatAemat/cs for the Aviation Trades Examples: 1-5. Draw the timing diagram for the following engines : Figure 286 shows how the Instruction Book of the Axelson Engine Company gives the timing diagram for TC 1 poinnAdwcedW&C. Inlet opens V overlap ~%J&:Et<'<6.T.C Exhaust opens 60 B.B.C. x Intake valve remains open 246 Exhaust valve remains open 246' 286. Valves honfe 6B.TC:"/ In let closes 60A.B.C. Fig. K Bofhm cenfer Valve-timing diagram: Axelson aircraft engine. one of their engines. Can you obtain the data used in making this chart? Notice that the number of degrees that the valves remain open is neatly printed on the diagram, as well as the firing points and valve overlap. How Long Does Each Valve Remain Open? When the piston is at top center, the throw on the shaft is Job 5: pointing directly up toward the cylinder, as in Fig. 287. Compression Ratio and Valve Timing 233 When the piston is at bottom center, the throw is at its farthest point away from the cylinder. The shaft has turned through an angle of 180 just for the downward movement of the piston from top center to bottom center. Bofhi cento Fig. 287. The intake valve of the Kinner K-5 opens 22 B.T.C., and closes 82 A.B.C. The intake valve of the Kinner, therefore, remains open 22 + 180 + 82 or a total of 284. The exhaust valve of the Kinner opens 68 B.B.C., and closes 1C. idO B.C. B.C. Intake Exhaust Fig. 36 A.T.C. It total of 284 is, therefore, 288. open 68 + 180 + 36 or a (see Fig. 288). Examples: Draw the valve timing diagrams for the following engines and find the number of degrees that each valve 1-3. remains open: 234 Job Mathematics 6: for the Aviation Trades Valve Overlap From the specifications given in previous jobs, it may in most aircraft engines the intake valve opens before the exhaust valve closes. Of course, this have been noticed that means that some be wasted. However, the rush of gasoline from the intake manifold serves to drive out all previous exhaust vapor and ^-V&f/ve over-fa. leave the mixture in the cylinder clean for the next stroke. This fuel will 'Exhaust valve closes is very important in a highcompression engine, since an improper mixture might cause detonation or engine knock. Definition: Valve overlap l9 ' ' is the length of time that both valves remain open at the same time. It is measured in degrees. In finding the valve overlap, it will only be necessary to consider when the exhaust valve closes and the intake valve opens as shown in Fig. 289. ILLUSTRATIVE What EXAMPLE is the valve overlap for the Kinner K-5 ? Given: Exhaust valve closes 36 A.T.C. Intake valve opens 22 B.T.C. Compression Ratio and Valve Timing 237 Examples: 1. Find the area 2. What 3. 4. 5. 6. of 1 piston. Find the total piston area, the total displacement for all cylinders? Calculate the brake horsepower at 2,100 r.p.m. Complete these tables from the performance curves: is Find the clearance volume for 1 cylinder. Find the total volume of 1 cylinder. 7. Draw 8. How many 9. 10. What the valve timing diagram. degrees does each valve remain open ? is the valve overlap in degrees ? How many consume operating gallons of gasoline would this engine at 2,100 r.p.m. for 1 hr. 35 min.? Party REVIEW S39 Chapter XVI ONE HUNDRED SELECTED REVIEW EXAMPLES Can you read 1. the rule? Measure the distances in Fig. 292: H/> A -+B E+ F Fig. (a) AB AD (c) EF (d) GE OF (e) i DK +f+f (/) Add: 2. () i (b) 292. +1+i+ iV (b) (c) 3. Which fraction in each group is the larger and how much ? () iorff (c) -fa or i 4. Find the (6) (rf) fVor^ ^ or i over-all dimensions of the piece in Fig. 293. Fi S . 293. 241 One Hundred 10. weight Find the weight is 1.043 11. If 1-in. ft. Selected Review Examples Ib. of length, find the 35 ft. of round steel rod, if the of length. stainless steel bar weighs 2.934 Ib. per ft. per round of 243 weight of 7 bars, each 18 ft. long. 12. Divide: (a) 2i by 4 43.625 by 9 The 12f by (c) Obtain answers to the nearest hundredth, 13. Divide. (a) 4^ by f (6) (6) 2.03726 by 3.14 metal in Fig. 297 (c) 0.625 by 0.032 have the centers of 7 holes equally spaced. Find the distance between centers to the nearest 64th of an inch. 14. strip of is to (J) 20'^ Fig. 16. " How many 297. round pieces \|- punched" from a strip of steel 36 between punchings (see Fig. 298) ? in. diameter can be long, allowing iV in. Stock: '/Q thick, /"wide ^f Fig. 16. a. in. in What 298. is The steel strip is 36 in. long. the weight of the unpunched strip in Fig. 298? b. What is the total weight of c. What is the weight of the punched strip ? all the round punch- ings? 17. Find the area of each figure in Fig. 299. 244 Mathematics for the Aviation Trades fa) (b) Fig. 18. 19. Find the perimeter of each figure Find the area and circumference diameter 20. 299. in Fig. 299. of a circle whose is 4ijr in. Find the area in square inches of each figure in Fig. 300. (6) (a.) Fi 3 . 21. Find the area 300. of the irregular flat surface shown in Fig. 801. : -J./J0" Fig. -H V-0.500" 301. 22. Calculate the area of the cutout portions of Fig. 302. (b Fig. 302. One Hundred 23. Selected Review Examples Express answers to the nearest 10th: VlS.374 (6) is 245 (c) V0.9378 24. What is the length of the side of a square whose area 396.255 sq. in.? 25. Find the diameter of a piston whose face area is 30.25 sq. in. 26. Find the radius 27. A length of 28. steel rectangular 275 ft. What whose area is 3.1416 sq. ft. whose area is 576 sq. yd. has a of circle field width ? is its For mass production of aircraft, a modern brick and structure was recently suggested comprising the following sections: Section Dimension, Ft. 600 by 1,400 Manufacturing 100 by 50 by Truck garage Boiler house Flight hangar Calculate the 150 400 100 75 by 200 by Oil house a. 900 120 by 100 by Engineering Office amount 150 200 of space in square feet assigned to each section. Find the total amount of floor space. 29. Find the volume in cubic inches, of each 6. solid in Fig. 303. Fig. 30. How many tank, 12 feet? if gallons of the diameter of its 303. oil base can be stored in a circular is 25 ft. and its height is 246 31. is Mathematics for t/)e Aviation Trades A circular boiler, 8 ft. long and 4 ft. 6 in. in diameter, completely filled with gasoline. What is the weight of the gasoline ? 32. What 34. Find the weight the weight of 50 oak beams each 2 by 4 in. by 12 ft. long? 33. Calculate the weight of 5,000 of the steel items in Fig. 304. . 304. is copper dimensions shown plates in Fig. 305. cut of one dozen according the to 12 Pieces " '/ thick 4 V- 15" 4 22Fi g . 4<-7-> 305. 35. Calculate the weight of 144 steel pins as shown in Fig. 306. 36. 2 How many in. thick, 37. How many board feet build the platform 38. board feet are there in a piece of lumber 9 in. wide, and 12 ft. of long? lumber would be needed to shown What would in Fig. 307 ? be the cost of this bill of material ? One Hundred Selected Review Examples Find the number 39. of board of 15 spruce planks each weight feet, f by 12 247 the cost, and the by 10 ft., if the in. price is $.18 per board foot. 40. Calculate the number of board feet needed to con- box shown struct the open in Fig. 308, if 1-in. white pine is used throughout. 41. (a) What is the weight of the box (Fig. 308) ? (6) What would be the weight of a similar steel box? 42. What weight of concrete would the box (Fig, 308) contain 43. Nov. when filled? Concrete weighs 150 Ib. per cu. ft. The graph shown in Fig. 309 appeared in the 1940, issue of the Civil Aeronautics Journal. 15, Notice how much information is given in this small space. UNITED STATES Aiu TRANSPORTATION REVENUE MILES FLOWN 12.0 10.5 1940 9.0 o -7.5 7 ~7 6.0 // 4.5 Join. Feb. Fig. Mar. Apr. 309. May June July Aug. Sept, Oct. Nov. Dec. (Courtesy of Civil Aeronautics Journal.) 248 a. Mathematics What is the worst for the Aviation Trades month of every year graph as far as "revenue miles flown" shown is in the concerned? Why? 6. How many revenue miles were flown in March, 1938? In March, 1939? In March, 1940? 44. Complete a table of data showing the number of revenue miles flown in 1939 (see Fig. 309). 46. The following table shows how four major airlines compare with respect to the number of paid passengers carried during September, 1940. Operator American Passengers Airlines 93,876 Eastern Airlines 33,878 T.W.A 35,701 United Air Lines 48,836 Draw 46. a bar graph of this information. Find the over-all length of the fitting shown in Fig. 310. Section A-A Fi 3 . 310. 1/8-in. cold-rolled, S.A.E. 1025, 2 holes drilled 47. What What in. diameter. Make 60. 3/16 a full-scale drawing of the fitting in Fig. 310. 48. Find the top surface area of the fitting in Fig. 310. 49. What is the volume of one fitting? 51. is is the weight of 1,000 such items? the tensile strength at section AA (Fig. 310)? What AA the bearing strength at (Fig. 310)? a table of data in inches to the nearest Complete 64th for a 30-in. chord of airfoil section N.A.C.A. 22 from 62. is 53. the data shown in Fig. 311, One Hundred Fig. 311. Se/ectec/ Review Examples Airfoil section: 249 N.A.C.A. 22. N.A.C.A. 22 54. Draw the nosepiece (0-15 per cent) from the data obtained in Example 53, and construct a solid wood nosepiece from the drawing. 56. What is the thickness in inches complete airfoil for Draw a 30-in. chord at each station of the ? (75-100 per cent) for a 5-ft chord length of the N.A.C.A. 22. 57. Make a table of data to fit the airfoil shown in 56. the tail section Fig. 312, accurate to the nearest 64th. 250 Mathematics Fig. for the Aviation Trades 312. Airfoil section. 58. Design an original airfoil section on graph paper and complete a table of data to go with it, 59. What is the difference between the airfoil section in Example 58 and those found in N.A.C.A. references? Complete a table of data, accurate to the nearest tenth of an inch, for a20-in. chord of airfoil section N.A.C.A. 4412 (see Fig. 313). 60. AIRFOIL SECTION: N.A.C.A. 4412 Data in per cent of chord 20 40 20 80 60 Per cent of chord Fig. 313. Airfoil section: N.A.C.A. 4412 is used on the Luscombe Model 50 two-place monoplane. 61. Draw airfoil section a nosepiece (0-15 per cent) for a N.A.C.A. 4412 (see Fig. 313). 4-ft. chord of One Hundred 62. What is Selected Review Examples 251 the thickness at each station of the nose- piece drawn in Example 61 ? Check the answers by actual measurement or by calculation from the data. 63. Find the useful load Fig. 31 4. of the airplane (Fig. 314). Lockheed Lodestar twin-engine transport. (Courtesy of Aviation.) LOCKHEED LODESTAR Weight, empty Gross weight 12,045 Ib. 17,500 Ib. Engines 2 Pratt and Whitney, 1200 hp. each Wing area Wing span 551 sq. 65 ft. 6 64. What is ft. in. the wing loading? What is the power loading? 66. 66. 67. What is the mean chord of the wing? Find the aspect ratio of the wing. Estimate the dihedral angle of the wing from Fig. 314. 68. Estimate the angle 69. What per cent of sweepback? of the gross weight is the useful load ? 70. This airplane (Fig. 314) carries 644 gal. of gasoline, and at cruising speed each engine consumes 27.5 gal. per hr* Approximately how long can it stay aloft? 71. What is the formula you would use to a. Area of a piston? 6. c. d. Displacement? of power strokes per minute? Brake horsepower of an engine? Number find: 254 Mathematics The for the for the Aviation Trades and performance curves (Fig. 316) are Lycoming geared 75-hp. engine shown in Fig. 317. specification Number of cylinders 4 Bore 3.625 Stroke 8.50 Engine r.p.m 8,200 at rated horsepower in. in. B.M.E.P 1$8 Compression ratio Weight, dry Specific fuel consumption 0.5: 1 Specific oil lb. per sq. in. 181 lb 0.50 Ib./b.hp./hr. 0.010 Ib./b.hp./hr. consumption Valve Timing Information Intake valve opens 20 B.T.C.; closes 65 A. B.C. Exhaust valve opens 65 B.B.C.; closes 20 A.T.C. 74. 75. What What is is the total piston area? the total displacement? 76. Calculate the rated horsepower of this engine. Is it exactly 75 hp.? 77. 78. if the Why? What is the weight per horsepower of the Lycoming ? How many gallons of gasoline would be consumed Lycoming operated 79. How many for 2 hr. 15 min. at 75 hp.? quarts of oil would be consumed during this interval? Complete the following table performance curves: 80. 81. depend On what ? of data from the three factors does the bend allowance One Hundred 82. Calculate the Selected Review Examples bend allowance for the fitting 255 shown in Fig. 318. + +0.032" /&' 318. Fis. 83. Find the of Ansle bend 90. over-all or developed length of the fitting in Fig. 318. 84. Complete the following table: BEND ALLOWANCE CHART: (All 90 ANGLE OF BEND dimensions are in inches) 0.049 0.035 0.028 Use the above table to help solve the examples that follow. 85. Find the developed length of the fitting shown in Fig. 319. s- 319. An 9 le of bend 90. 256 Mat/iemat/cs for the Aviation Trades 86. Calculate the developed length of the part shown in Fig. 320. 0.028- ,/L Fig. 320. 87. bends, each 90. What is diameter 88. Two the strength in tension of a dural rod whose 0.125 in.? is Find the strength in compression parallel to the an oak beam whose cross section is 2-g- by 3f in. What would be the weight of the beam in Example were 7 ft. long? grain of 89. 88 if it 90. Calculate the strength in shear of a ^V-in. copper rivet. 91. What in Fig. 321 is the strength in shear of the lap joint shown ? o Fig. What 321. Two 1/16-in. S.A.E. X-4130 rivets. the strength in bearing of a 0.238-in. dural plate with a ^-in. rivet hole? 93. Find the strength in tension and bearing of the 92. cast-iron lug is shown in Fig. 322. Fig. 94. hold a 322. What is the diameter of a L.C. maximum load of 1,500 lb.? steel wire that can One Hundred 95. Selected Review Examples A dural tube has an outside diameter of l 257 in. and a wall thickness of 0.083 in. What is the inside diameter? What is the cross-sectional area? 96. What would 100 ft. of the tubing a. b. in Example 95 weigh ? 97. If be the no bending or buckling took place, what would compressive strength that a 22 gage maximum (0.028 in.) S.A.E. 1015 tube could develop, if its inside diameter were 0.930 in.? 98. Find the strength in tension of the riveted strap shown in Fig. 323. -1X9 Fig. 99. 100. in Fig. 323. Lap joint, dural straps. Two 1 /8 in., 2S rivets, driven cold. What is the strength in shear of the joint in Fig. 323 ? What is the strength in shear of the butt joint shown 324? Fig. 324. All rivets 3/64 in. 17 S-T, driven hot.
Find a South Houston help students develop the ability to see computational problems from a mathematical perspective. Discrete math is normally divided into six areas: sets, functions, and relations; basic logic; proof techniques; counting basics; graphs and trees; and discrete probability. I show students how these topics are interwoven with computer science applications.
Summary: KEY MESSAGE:Gary Rockswold and Terry Kriegerfocus on teaching algebra in context, giving students realistic and convincing answers to the perennial question, ''When will I ever use this?'' The authors' consistent use of real data, graphs, and tables throughout the examples and exercise sets gives meaning to the numbers and equations as students encounter them. This new edition further enhances Rockswold and Krieger's focus on math in the real world with a new features and updated ap...show moreplications to engage today's students. KEY TOPICS: Introduction to Algebra, Linear Equations and inequalities, Graphing Equations, Systems of Linear Equations in Two Variables, Polynomials and Exponents, Factoring Polynomials and Solving Equations, Introduction to Rational Expressions, Linear Functions and Models, Matrices and Systems of Linear Equations, Radical Expressions and Functions, Quadratic Functions and Equations, Exponential and Logarithmic Functions, Conic Sections, Sequences and Series MARKET: For all readers interested in algebra
Created for the Connected Curriculum Project, the purpose of this module is to learn the basics of Mathematica for use in a linear algebra course. This is one resource within a larger set of learning modules hosted by... Created by Lang Moore, David Smith and Jim Tomberg for the Connected Curriculum Project, the purpose of this module is to earn the basics of Maple for use in a linear algebra course. This is a portion of a larger set of... Created by William Barker and David Smith for the Connected Curriculum Project, the purposes of this module are to develop a mathematical model for decay of radioactive substances, and to develop a technique for... Created by Lawrence Moore and David Smith for the Connected Curriculum Project, the purpose of this module is to carry out an exploration of functions defined by data; to learn about data entry and plotting operations. ... Created by David Smith for the Connected Curriculum Project, the purpose of this module is to apply linear algebra concepts to study the properties of sequences defined by difference equations. This is one within a...
Find a Wakefield, MA PrecalculusProperly, finite mathematics is equivalent to discrete mathematics, and formal definitions of what that means can be had at the asking. The typical class is trying to deal with the applications to real-world problems where the results of using mathematics are emphasized over the reasons why they...
Pirnot believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking behind the subject matter, so that students are able to truly understand the material and apply it to their lives. This textbook maintains a conversational tone throughout and focuses on motivating students and the mathematics through current applications. Ultimately, students who use this book will become more educated consumers of the vast amount of technical and mathematical information that they encounter daily, transforming them into mathematically aware citizens.
College Algebra Providing students who need a solid understanding of algebra with an excellent start, this textbook encourages student understanding of algebra ...Show synopsisProviding students who need a solid understanding of algebra with an excellent start, this textbook encourages student understanding of algebra through the use of modelling techniques and real-data applications.Hide synopsis961185961185Fine. Hardcover. Instructor Edition: Same as student edition...Fine. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Almost new condition. SKU: 9781133961185Hardcover. Instructor Edition: Same as student edition with...Hardcover. Instructor Edition: Same as student edition with additional notes or answers. New Condition. SKU: 9781133961185-133963028Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 9781133963028111856108498
Elementary Statistics: A Step by Step Approach This text is aimed at students who do not have a mathematical background. It therefore uses a non-theoretical approach, and concepts are explained ...Show synopsisThis text is aimed at students who do not have a mathematical background. It therefore uses a non-theoretical approach, and concepts are explained intuitively, without the use of formal proofs; they are instead supported by example. The statistical applications are drawn from various disciplines, including natural, social and computer science and business. There are margin articles with interesting trivia related to statistics.Hide synopsis Description:Good. Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 978007353498534985. Description:Good. Formulas card Item may show signs of shelf wear. Pages...Good. Formulas card Item may show signs of shelf wear. Pages may include limited notes and highlighting. Includes supplemental or companion materials if applicable. Access codes may or may not work. Connecting readers since 1972. Customer service is our top priority 0073534986 -used book-free tracking number with every...Good. 0073534986 -used book-free tracking number with every order. book may have some writing or highlighting, or used book stickers on front or back Description:Fine. Never used, pages are clean and crisp! Cover has minor...Fine. Never used, pages are clean and crisp! Cover has minor handling wear or this would have been listed as NEW. LIKE NEW-NEVER READ! Book is an overstock and may show minor wear or have marker line (remainder mark) on edge
Why Graph Theory ?  Graphs used to model pair wise relations between objects  Generally a network can be represented by a graph  Many practical problems can be easily represented in terms of graph theory Graph Theory - History The origin of graph theory can be traced back to Euler's work on the Konigsberg bridges problem (1735), which led to the concept of an Eulerian graph. The study of cycles on polyhedra by the Thomas P. Kirkman (1806 - 95) and William R. Hamilton (1805-65) led to the concept of a Hamiltonian graph. Vertex & Edge  Vertex /Node    Basic Element Drawn as a node or a dot. Vertex set of G is usually denoted by V(G), or V or VG  Edge /Arcs    A set of two elements Drawn as a line connecting two vertices, called end vertices, or endpoints. The edge set of G is usually denoted by E(G), or E or EG  Neighborhood  For any node v, the set of nodes it is connected to via an edge is called its neighborhood and is represented as N(v) Classification of Graph Terms   Global terms refer to a whole graph Local terms refer to a single node in a graph Connected and Isolated vertex  Two vertices are connected if there is a path between them  Isolated vertex – not connected 1 isolated vertex 2 3 4 5 6 Adjacent nodes  Adjacent nodes -Two nodes are adjacent if they are connected via an edge.  If edge e={u,v} ∈ E(G), we say that u and v are adjacent or neigbors  An edge where the two end vertices are the same is called a loop, or a self-loop Degree (Un Directed Graphs)  Number of edges incident on a node The degree of 5 is 3 Walk  trail: no edge can be repeat a-b-c-d-e-b-d  walk: a path in which edges/nodes can be repeated. a-b-d-a-b-c  A walk is closed is a=c Paths  Path: is a sequence P of nodes v1, v2, …, vk-1, vk  No vertex can be repeated  A closed path is called a cycle  The length of a path or cycle is the number of edges visited in the path or cycle 1,2,5,2,3,4 walk of length 5 Walks and Paths 1,2,5,2,3,2,1 CW of length 6 1,2,3,4,6 path of length 4 Cycle  Cycle - closed path: cycle (a-b-c-d-a) , closed if x=y  Cycles denoted by Ck, where k is the number of nodes in the cycle C3 C4 C5 Shortest Path  Shortest Path is the path between two nodes that has the shortest length  Length – number of edges.  Distance between u and v is the length of a shortest path between them  The diameter of a graph is the length of the longest shortest path between any pairs of nodes in the graph
Revised second edition aligned for the 2008-2009 testing cycle, with a full index. REA's new Mathematics test prep for the required Texas Assessment of Knowledge and Skills (TAKS) high school exit-level exam provides all the instruction and practice students need to excel. The book's review features all test objectives, including Numbers and Operations; Equations and Inequalities; Functions; Geometry and Spatial Sense; Measurement; Data Analysis and Probability; and Problem Solving. Includes 2 full-length practice tests, detailed explanations to all answers, a study guide, and test-taking strategies to boost confidence.
Algebra and Trigonometry with Analytic Geometry Retains the elements that have made it so popular with instructors and students alike: clear exposition, an appealing and uncluttered layout, and ...Show synopsisRetains the elements that have made it so popular with instructors and students alike: clear exposition, an appealing and uncluttered layout, and applications-rich exercise sets. This title covers some more challenging topics, such as Descartes' Rule of Signs and the Theorems on Bounds685 Hardcover. May include moderately worn cover, writing,...Good. Hardcover. May include moderately worn cover, writing, markings or slight discoloration. SKU: 97808400685 Algebra and Trigonometry with Analytic Geometry This is a common secondary level text. Like most current texts of this type, it fails to give the reader any understanding of the breadth of this subject as developed over the last 150 years. There are a number of useful tricks of interest to engineers, computer graphics specialists, and other 3D modelers that this text does not even point to for additional reading. However, for its intended audience, it delivers adequately but still does not entice the imagination
Course Summary The Mathematics course has been created to offer all senior secondary students the opportunity to advance their mathematical skills, to build and use mathematical models, to solve problems, to learn how to conjecture and to reason logically, and to gain an appreciation of the elegance, beauty and creative nature of mathematics. Students use numbers and symbols to represent many situations in the world around them. They examine how mathematical methods associated with number, algebra and calculus allow for precise, strong conclusions to be reached, providing a form of argument not available to other disciplines. The Mathematics course allows for multiple entry points to accommodate the diversity of students' mathematics development at the point of entry into senior school as well as the diversity of post school destinations. Students can choose units based on their particular need: to develop their general mathematical skills for further training or employment, to enable university entry where further mathematics may not be essential, to prepare them for university courses where further mathematics studies is required or for preparation for higher level training in technical areas. Course documents—apart from any third party copyright material contained in them — may be freely copied, or communicated on an intranet, for non-commercial purposes in educational institutions, provided that the School Curriculum and Standards Authority is acknowledged as the copyright owner. Copying or communication for any other purpose can be done only within the terms of the Copyright Act or with prior written permission of the School Curriculum and Standards Authority. Copying or communication of any third party copyright material can be done only within the terms of the Copyright Act or with permission of the copyright owners.
Find a Clyde HillPrecalculus is an important step between Algebra II and Trigonometry and Calculus. The students refine their understanding of Algebra and Trigonometry and learn how to solve more complicated problems in preparation for Calculus. A good understanding of these courses is absolutely essential for success in Calculus
In the quarter of a century since three mathematicians and game theorists collaborated to create Winning Ways for Your Mathematical Plays, the book has become the definitive work on the subject of mathematical games. Now carefully revised and broken down into four volumes to accommodate new developments, the Second Edition retains the original's wealth of wit and wisdom. The authors' insightful strategies, blended with their witty and irreverent style, make reading a profitable pleasure. In Volume 4, the authors present a Diamond of a find, covering one-player games such as Solitaire. Uncover the secrets of the game industry's best programmers with the newest volume of the Game Programming Gems series With over 60 all new techniques, Game Programming Gems 4 continues to be the definitive resource for developers. Written by expert game developers who make today's amazing games, these articles not only provide quick solutions to cutting-edge problems, but they provide insights that you'll return to again and again. Modern and comprehensive, the new Fifth Edition of Zill's Advanced Engineering Mathematics, Fifth Edition provides an in depth overview of the many mathematical topics required for students planning a career in engineering or the sciences. A key strength of this best-selling text is Zill's emphasis on differential equations as mathematical models, discussing the constructs and pitfalls of each. The Fifth Edition is a full compendium of topics that are most often covered in the Engineering Mathematics course or courses, and is extremely flexible, to meet the unique needs of various course offerings ranging from ordinary differential equations to vector calculus. To experience the joy of mathematics is to realize that mathematics is not some isolated subject that has little relationship to the things around us other than to frustrate us with unbalanced checkbooks and complicated computations. Many of the phenomena around us can be described by mathematics. Mathematical concepts are even inherent in the structure of living cells Elements Of Modern Algebra, Eighth Edition, with its user-friendly format, provides you with the tools you need to succeed in abstract algebra and develop mathematical maturity as a bridge to higher-level mathematics courses. Strategy boxes give you guidance and explanations about techniques and enable you to become more proficient at constructing proofs. A summary of key words and phrases at the end of each chapter help you master the material. A reference section, symbolic marginal notes, an appendix, and numerous examples help you develop your problem-solving skills. Probability, Random Variables, and Random Processes is a comprehensive textbook on probability theory for engineers that provides a more rigorous mathematical framework than is usually encountered in undergraduate courses A groups-first option that enables those who want to cover groups before rings to do so easily. Proofs for beginners in the early chapters, which are broken into steps, each of which is explained and proved in detail. In the core course (chapters 1-8), there are 35% more examples and 13% more exercises. LEGO The Hobbit: The Video Game will be set around the first two films of Peter Jackson's The Hobbit trilogy, An Unexpected Journey and Desolation of Smaug, with the third being included as DLC. Playable characters include Bilbo Baggins and Gandalf the Grey, alongside all of the dwarves: Thorin, F li, K li, in, Gl in, Dwalin, Balin, Bifur, Bofur, Bombur, Dori, Nori and Ori. Warner Bros. adds that each dwarf will have his own unique ability, mentioning that Bombur can use his belly as a trampoline. Locations visited will include Bag End, Hobbiton, The Misty Mountains, Goblin-town, Mirkwood, Lake-Town, Dol Guldur and Rivendell. Players "will also be able to mine for gems, discover loot from enemies, and craft powerful magical items or build immense new LEGO structures," according to the game's press release, suggesting that Minecraft-esque elements could make their way into the next LEGO adventure.
The course Algebra 2 is part of the 16 online courses in MUST high school home study diploma program. It covers extensive subject knowledge and the course contents enrich the students' learning. It covers online study material that is free of cost for students at MUST. This online course of the home school diploma program is ahead of all other traditional & online courses. This course can be accessed in the MUST High School's online Classroom which covers both theory and practical aspects of the courses and includes case studies and real-life examples. Our online high school classes are very flexible. There is no fixed time for the high school classes online. You may enter into your online classroom 24x7 and can study according to your pace. So whether you are a student interested in online homeschooling diploma or a working adult interested in getting high school diploma from home or office, you may study English 9 course online with complete ease. To see why MUST's homeschool / online high school diploma is the first choice of working adults and homeschool students across the globe, please Click here. Topics Covered in This Course: Section 1 Equations And Inequalities Overview: In this section's topic 1, learn about real numbers, expressions, equations and linear equations on an advanced level. Linear Equations And Functions Overview: In this section's topic 2 learn about slope, linear equation, the rectangular coordinate system on an advanced level. Section 2 Linear Systems And Matrices Overview: In this section's topic 1 learn about the theories of linear system, systems of linear equations in three variables, systems of linear equations in variables and solving systems of linear equations by matrix methods. Quadratic Functions And Factoring Overview: The quadratic function set equal to zero results to a quadratic equation. The greatest common factor (GCF) is the largest term that is a factor of all terms in the polynomial. This topic elaborates on that. Polynomials And Polynomial Functions Overview: Polynomial is an expression constructed from one or more in determinates and constants, using the operations of addition, subtraction, multiplication, and raising to constant non-negative integer powers. This topic elaborates on that. Overview: Logarithms have been thought of as arithmetic sequences of numbers corresponding to geometric sequences of other (positive real) numbers, but are also the result of applying an analytic function. This topic elaborates on that Rational Functions Overview: This topic teaches you as to how a rational function can simply be the ratio of two polynomial functions. Section 4 Quadratic Relations And Conic Sections Overview: Circle, center, hyperbola and non-linear equation theories are discussed in depth in this topic. Counting Methods And Probability Overview: In this section you will thoroughly be taught the theories of interpretations, mathematical treatment, probability theory, applications and relation to randomness Data Analysis And Statistics Overview: This topic teaches you to describe data using statistical measures, transformations of data affect statistics, normal distribution, different sampling methods for collecting data and choosing the best model to represent a set of data. Section 5 Sequences And Series Overview: Recognize and write rules for number patterns, study arithmetic sequences and series, find the sum of infinite geometric series and use recursive rules for sequences while on this topic of Section 5. Trigonometric Ratios And Functions Overview: Trigonometry is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles). This topic elaborates on that. Trigonometric Graphs, Identities, And Equations Overview: Trigonometric identities, Pythagorean identities, sine, cosine, and tangent of a sum, half-angle identities, stereographic identities and triangle identities are the theories you get to learn in this topic. High School Diploma Program High school diploma program offered by MUST High School surpasses the similar traditional programs offered by world-class high schools across the globe in terms of ease & flexibility, affordability, quickness and quality of education. To read more about why MUST High School is the first choice of working adults and students across the globe, please Click here You can get credits for your prior learning. If you have strong command in some courses (through your work experience, prior knowledge or trainings, etc.) you can get their credits! So you won't have to study that course and both your tuition and time required will be reduced. Read more Credit Transfer If you have already studied some courses of high school diploma and have its transcript, we will accept the credit hours of those courses and reduce the equivalent from your program. So you won't have to study that course again. This will reduce both your tuition and time required to complete your education. Read more
What is Mathematics?: An Elementary Approach to Ideas and Methods The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ...Show synopsisThe teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but not real understanding or greater intellectual independence. The new edition of this classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. "Lucid . . . easily understandable".--Albert Einstein. 301 linecuts 1979-Paperback-Used-Acceptable--Shows substantial...Acceptable. 1979 study copy, shows heavy wear, text has markings, a...Fair. Good study copy, shows heavy wear, text has markings, a good study copy. We take great pride in accurately describing the condition of our books, ship within 48 hours and offer a 100% money back guarantee. Description:Fair. The book has some limited-writing in pencil, as well as...Fair. The book has some limited-writing in pencil, as well as general-coverwear (edges, corners, scuffs/scratches, and possibly creases What is Mathematics?: An Elementary Approach to Ideas and Methods For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning ofmathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant's and Herbert Robbins's classic work seeks to address thisproblem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from naturalnumbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving.With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat's LastTheorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to lifebefore your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view
Saxon Math Homeschool 5th Grade Saxon Math Homeschool 5th Grade by Stephen Hake Book Description The title of this book is Saxon Math Homeschool 5th Grade and is written by author Stephen Hake. The book Saxon Math Homeschool 5th Grade is published by Saxon Publishers. The ISBN of this book is 9781591413325 and the format is Paperback / softback. The publisher has not provided a book description for Saxon Math Homeschool 5th Grade by Stephen Hake. Includes facts and curios, prime number conjectures, the sieve of Eratosthenes, the Fibonacci series, and much more besides. This title features an approach which can appeal to recreational maths enthusiasts, puzzle solvers, and mathematicians of all ages.You'll the learn formulas and shortcuts to help in hundreds of everyday situations, from budgeting and paying bills to shopping, redecorating, preparing taxes, and evaluating loans and other financial instruments. With this easy-to-follow guide, you'll never get stuck on a math problem again
Elementary Algebra for College Students (8th Edition) 9780321620934 ISBN: 0321620933 Edition: 8 Pub Date: 2010 Publisher: Prentice Hall Summary: Angel, Allen R. is the author of Elementary Algebra for College Students (8th Edition), published 2010 under ISBN 9780321620934 and 0321620933. Four hundred sixty two Elementary Algebra for College Students (8th Edition) textbooks are available for sale on ValoreBooks.com, two hundred thirty five used from the cheapest price of $4.31, or buy new starting at $67.05the class that required me to use this book was math 101 at Rockland community college. the class was very effective especially with the professor who taught us each topic. it was a very cooperative class. there is nothing i would change about this book. it offered problems to do and even showed exactly how to do them with examples provided.
Beginning Algebra With Applications 9780618803590 ISBN: 0618803599 Pub Date: 2007 Publisher: Houghton Mifflin Summary: Intended for developmental math courses in beginning imm...ediate feedback, reinforcing the concept, identifying problem areas, and, overall, promoting student success."New!" "Interactive Exercises" appear at the beginning of an objective's exercise set (when appropriate), and provide students with guided practice on some of the objective's underlying principles."New!" "Think About It" Exercises are conceptual in nature and appear near the end of an objective's exercise set. They ask the students to think about the objective's concepts, make generalizations, and apply them to more abstract problems. The focus is on mental mathematics, not calculation or computation, and help students synthesize concepts."New!" "Important Points" have been highlighted to capture students' attention. With these signposts, students are able to recognize what is most important and to study more efficiently."New!" A Concepts of Geometry section has been added to Chapter 1."New!" Coverage of operations on fractions has been changed in Section 1.3 so that multiplication and division of rational numbers are presented first, followed by addition and subtraction"New!" A Complex Numbers section has been added to Chapter 11, "Quadratic Equations.""New Media!" Two key components have been added to the technology package: HM Testing (powered by Diploma) and, as part of the Eduspace course management tool, HM Assess, an online diagnostic assessment tool. Aufmann, Richard N. is the author of Beginning Algebra With Applications, published 2007 under ISBN 9780618803590 and 0618803599. Two hundred ninety five Beginning Algebra With Applications textbooks are available for sale on ValoreBooks.com, one hundred twenty three used from the cheapest price of $4.59, or buy new starting at $45.29.[read more] Ships From:Jackosnville, FLShipping:StandardComments:Book is in acceptable condition; cover shows signs of wear. Pages include markings from pencil/ p... [more]Book is in acceptable condition; cover shows signs of wear. Pages include markings from pencil/ pen/highlighter, but text is not obscured. Used stickers on binding and back cover. [less] Ships From:Castleton, NYShipping:StandardComments: 0618803599 AtAGlance Books--Orders ship next business day, with tracking numbers, from our wareh... [more] 0618803599 AtAGlance Books--Orders ship next business day, with tracking numbers, from our warehouse in upstate NY. This book is in brand new condition
Complex Analysis Complex Analysis This is a free online course offered by the Saylor Foundation.'... More This is a free online course offered by the Saylor Foundation. ' inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Simply put, you will be working with lines and sets and very specific functions on the complex plane—drawing pictures of them and teasing out all of their idiosyncrasies. You will again find yourself calculating line integrals, just as in multivariable calculus. However, the techniques you learn in this course will help you get past many of the seeming dead-ends you ran up against in calculus. Indeed, most of the definite integrals you will learn to evaluate in Unit 7 come directly from problems in physics and cannot be solved except through techniques from complex variables. We will begin by studying the minimal algebraically closed extension of real numbers: the complex numbers. The Fundamental Theorem of Algebra states that any non-constant polynomial with complex coefficients has a zero in the complex numbers. This makes life in the complex plane very interesting. We will also review a bit of the geometry of the complex plane and relevant topological concepts, such as connectedness. In Unit 2, we will study differential calculus in the complex domain. The concept of analytic or holomorphic function will be introduced as complex differentiability in an open subset of the complex numbers. The Cauchy-Riemann equations will establish a connection between analytic functions and differentiable functions depending on two real variables. In Unit 3, we will review power series, which will be the link between holomorphic and analytic functions. In Unit 4, we will introduce certain special functions, including exponentials and trigonometric and logarithmic functions. We will consider the Möbius Transformation in some detail. In Units 5, 6, and 7 we will study Cauchy Theory, as well as its most important applications, including the Residue Theorem. We will compute Laurent series, and we will use the Residue Theorem to evaluate certain integrals on the real line which cannot be dealt with through methods from real variables alone. Our final unit, Unit 8, will discuss harmonic functions of two real variables, which are functions with continuous second partial derivatives that satisfy the Laplace equation, conformal mappings, and the Open Mapping Theorem.'
Buy Used Textbook Buy New Textbook eTextbook We're Sorry Not Available More New and Used from Private Sellers Starting at $212/2001 presents the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a way that is both interesting and understandable. Using a wide range of examples and problems, this book focuses on the use of MathCAD functions and worksheets to illustrate the methods used when discussing the following concepts: solving linear and nonlinear equations, numerical linear algebra, numerical methods for data interpolation and approximation, numerical differentiation and integration, and numerical techniques for solving differential equations. For professionals in the fields of engineering, mathematics, computer science, and physical or life sciences who want to learn MathCAD functions for all major numerical methods. Table of Contents Preface xi Examples/Mathcad Functions/Algorithms xiii Foundations 1 (46) Sample Problems and Numerical Methods 4 (6) Roots of Nonlinear Equations 4 (1) Fixed-Point Iteration 5 (1) Linear Systems 6 (1) Gaussian Elimination 7 (1) Numerical Integration 8 (1) Trapezoid Rule 8 (2) Some Basic Issues 10 (18) Key Issues for Iterative Methods 10 (4) How Good Is the Result? 14 (8) Getting Better Results 22 (6) Getting Started in Mathcad 28 (19) Overview of the Mathcad Workspace 28 (3) Mathematical Computations 31 (1) Operators on the Math Toolbars 32 (3) Built-In Functions 35 (3) Programming in Mathcad 38 (9) Solving Equations of One Variable 47 (46) Bisection Method 50 (5) Step-by-Step Computation 50 (2) Mathcad Function for Bisection 52 (2) Discussion 54 (1) Regula Falsi and Secant Methods 55 (13) Step-by-Step Computation for Regula Falsi 56 (2) Mathcad Function for the Regula Falsi Method 58 (2) Step-by-Step Computation for the Secant Method 60 (2) Mathcad Function for the Secant Method 62 (2) Discussion 64 (4) Newton's Method 68 (7) Step-by-Step Computation 68 (2) Mathcad Function for Newton's Method 70 (2) Discussion 72 (3) Muller's Method 75 (6) Step-by-Step Computation for Muller's Method 76 (2) Mathcad Function for Muller's Method 78 (2) Discussion 80 (1) Mathcad's Methods 81 (12) Using the Built-In Functions 81 (3) Understanding the Algorithms 84 (9) Solving Systems of Linear Equations: Direct Methods 93 (38) Gaussian Elimination 96 (10) Using Matrix Notation 97 (1) Step-by-Step Procedure 98 (3) Mathcad Function for Basic Gaussian Elimination 101 (2) Discussion 103 (3) Gaussian Elimination with Row Pivoting 106 (7) Step-by-Step Computation 106 (4) Mathcad Function for Gaussian Elimination with Pivoting 110 (3) Discussion 113 (1) Gaussian Elimination for Tridiagonal Systems 113 (9) Step-by-Step Procedure 116 (2) Mathcad Function for the Thomas Method 118 (1) Discussion 119 (3) Mathcad's Methods 122 (9) Using the Built-In Functions 122 (1) Understanding the Algorithms 122 (9) Solving Systems of Linear Equations: Iterative Methods 131 (40) Jacobi Method 135 (9) Step-by-Step Procedure for Jacobi Iteration 136 (3) Mathcad Function for the Jacobi Method 139 (3) Discussion 142 (2) Gauss-Seidel Method 144 (7) Step-by-Step Computation for Gauss-Seidel Method 145 (3) Mathcad Function for Gauss-Seidel Method 148 (2) Discussion 150 (1) Successive Overrelaxation 151 (6) Step-by-Step Computation of SOR 152 (2) Mathcad Function for SOR 154 (1) Discussion 155 (2) Mathcad's Methods 157 (14) Using the Built-In Functions 157 (2) Understanding the Algorithms 159 (12) Systems of Equations and Inequalities 171 (30) Newton's Method for Systems of Equations 174 (7) Matrix-Vector Notation 176 (1) Mathcad Function for Newton's Method 177 (4) Fixed-Point Iteration for Nonlinear Systems 181 (6) Step-by-Step Computation 182 (1) Mathcad Function for Fixed-Point Iteration for Nonlinear Systems 182 (4) Discussion 186 (1) Minimum of Nonlinear Function 187 (5) Step-by-Step Computation of Minimization by Gradient Descent 187 (1) Mathcad Function for Minimization by Gradient Descent 188 (4) Mathcad's Methods 192 (9) Using the Built-In Functions 192 (1) Understanding the Algorithms 193 (8) LU Factorization 201 (32) LU Factorization from Gaussian Elimination 203 (4) A Step-by-Step Procedure for LU Factorization 204 (2) Mathcad Function for LU Factorization Using Gaussian Elimination 206 (1) LU Factorization of Tridiagonal Matrices 207 (2) Step-by-Step LU Factorization of a Tridiagonal Matrix 207 (1) Mathcad Function for LU Factorization of a Tridiagonal Matrix 208 (1) LU Factorization with Pivoting 209 (6) Step-by-Step Computation 209 (1) Mathcad Function for LU Factorization with Row Pivoting 210 (2) Discussion 212 (3) Direct LU Factorization 215 (4) Direct LU Factorization of a General Matrix 215 (2) LU Factorization of a Symmetric Matrix 217 (2) Applications of LU Factorization 219 (7) Solving a Tridiagonal System Using LU Factorization 222 (2) Determinant of a Matrix 224 (1) Inverse of a Matrix 224 (2) Mathcad's Methods 226 (7) Using the Built-In Functions 226 (1) Understanding the Algorithms 226 (7) Eigenvalues, Eigenvectors, and QR Factorization 233 (50) Power Method 236 (12) Basic Power Method 237 (5) Inverse Power Method 242 (5) Discussion 247 (1) QR Factorization 248 (19) Householder Transformations 248 (9) Givens Transformations 257 (4) Basic QR Factorization 261 (6) Finding Eigenvalues Using QR Factorization 267 (3) Basic QR Eigenvalue Method 267 (1) Better QR Eigenvalue Method 268 (2) Discussion 270 (1) Mathcad's Methods 270 (13) Using the Built-In Functions 270 (1) Understanding the Algorithms 271 (12) Interpolation 283 (66) Polynomial Interpolation 286 (24) Lagrange Interpolation Polynomials 286 (9) Newton Interpolation Polynomials 295 (11) Difficulties with Polynomial Interpolation 306 (4) Hermite Interpolation 310 (6) Rational Function Interpolation 316 (4) Spline Interpolation 320 (14) Piecewise Linear Interpolation 321 (1) Piecewise Quadratic Interpolation 322 (3) Piecewise Cubic Interpolation 325 (9) Mathcad's Methods 334 (15) Using the Built-In Functions 334 (1) Understanding the Algorithms 335 (14) Function Approximation 349 (44) Least Squares Approximation 352 (21) Linear Least-Squares Approximation 352 (7) Quadratic Least-Squares Approximation 359 (5) Cubic Least-Squares Approximation 364 (5) Least-Squares Approximation for Other Functional Forms 369 (4) Continuous Least-Squares Approximation 373 (8) Continuous Least-Squares with Orthogonal Polynomials 376 (1) Gram-Schmidt Process 376 (2) Legendre Polynomials 378 (1) Least-Squares Approximation with Legendre Polynomials 379 (2) Function Approximation at a Point 381 (4) Taylor Approximation 381 (1) Pade Function approximation 382 (3) Mathcad's Methods 385 (8) Using the Built-in Functions 385 (1) Understanding the Algorithms 386 (7) Fourier Methods 393 (43) Fourier Approximation and Interpolation 396 (11) Fast Fourier Transforms for n = 2r 407 (8) Discrete Fourier Transform 407 (1) Fast Fourier Transform 408 (7) Fast Fourier Transforms for General n 415 (8) Mathcad's Methods 423 (13) Using the Built-In Functions 423 (1) Understanding the Algorithms 424 (12) Numerical Differentiation and Integration 436 (41) Differentiation 436 (9) First Derivatives 436 (4) Higher Derivatives 440 (1) Partial Derivatives 441 (1) Richardson Extrapolation 442 (3) Basic Numerical Integration 445 (7) Trapezoid Rule 446 (2) Simpson Rule 448 (2) The Midpoint Formula 450 (2) Other Newton-Cotes Open Formulas 452 (1) Better Numerical Integration 452 (10) Composite Trapezoid Rule 453 (2) Composite Simpson's Rule 455 (3) Extrapolation Methods for Quadrature 458 (4) Gaussian Quadrature 462 (6) Gaussian Quadrature on [-1,1] 462 (2) Gaussian Quadrature on [a,b] 464 (4) Mathcad's Methods 468 (9) Using the Operators 468 (1) Understanding the Algorithms 469 (8) Ordinary Differential Equations: Initial-Value Problems 477 (52) Taylor Methods 479 (8) Euler's Method 479 (5) Higher-Order Taylor Methods 484 (3) Runge-Kutta Methods 487 (15) Midpoint Method 487 (5) Other Second-Order Runge-Kutta Methods 492 (2) Third-Order Runge-Kutta Methods 494 (1) Classic Runge-Kutta Method 495 (4) Other Runge-Kutta Methods 499 (2) Runge-Kutta-Fehlberg Methods 501 (1) Multistep Methods 502 (12) Adams-Bashforth Methods 502 (6) Adams-Moulton Methods 508 (1) Predictor-Corrector Methods 509 (5) Stability 514 (3) Mathcad's Methods 517 (12) Using the Built-In Functions 517 (3) Understanding the Algorithms 520 (9) Systems of Ordinary Differential Equations 529 (46) Higher-Order ODEs 532 (2) Systems of Two First-Order ODE 534 (7) Euler's Method for Solving Two ODE-IVPs 534 (3) Midpoint Method for Solving Two ODE-IVPs 537 (4) Systems of First-Order ODE-IVP 541 (16) Euler's Method for Solving Systems of ODEs 542 (2) Runge-Kutta Methods for Solving Systems of ODEs 544 (8) Multistep Methods for Systems 552 (5) Stiff ODE and Ill-Conditioned Problems 557 (2) Mathcad's Methods 559 (16) Using the Built-In Functions 559 (3) Understanding the Algorithms 562 (13) Ordinary Differential Equations-Boundary Value Problems 575 (34) Shooting Method for Solving Linear BVP 578 (7) Simple Boundary Conditions 578 (5) General Boundary Condition at x = b 583 (1) General Boundary Conditions at Both Ends of the Interval 584 (1) Shooting Method for Solving Nonlinear BVP 585 (7) Nonlinear Shooting Based on the Secant Method 585 (3) Nonlinear Shooting Using Newton's Method 588 (4) Finite-Difference Method for Solving Linear BVP 592 (7) Finite-Difference Method for Nonlinear BVP 599 (3) Mathcad's Methods 602 (7) Using the Built-In Functions 602 (2) Understanding the Algorithms 604 (5) Partial Differential Equations 609 (58) Classification of PDE 613 (1) Heat Equation: Parabolic PDE 614 (19) Explicit Method for Solving the Heat Equation 615 (8) Implicit Method for Solving the Heat Equation 623 (5) Crank-Nicolson Method for Solving the Heat Equation 628 (4) Heat Equation with Insulated Boundary 632 (1) Wave Equation: Hyperbolic PDE 633 (7) Explicit Method for Solving Wave Equations 634 (4) Implicit Method for Solving Wave Equation 638 (2) Poisson Equation: Elliptic PDE 640 (5) Finite-Element Method for Solving an Elliptic PDE 645 (13) Mathcad's Methods 658 (9) Using the Built-In Functions 658 (1) Understanding the Algorithms 659 (8) Bibliography 667 (6) Answers to Selected Problems 673 (22) Index 695 Excerpts The purpose of this text is to present the fundamental numerical techniques used in engineering, applied mathematics, computer science, and the physical and life sciences in a manner that is both interesting and understandable to undergraduate and beginning graduate students in those fields. The organization of the chapters, and of the material within each chapter, the use of Mathcad worksheets and functions to illustrate the methods, and the exercises provided are all designed with student learning as the primary objective.The first chapter sets the stage for the material in the rest of the text, by giving a brief introduction to the long history of numerical techniques, and a "preview of coming attractions" for some of the recurring themes of the remainder of the text. It also presents enough description of Mathcad to allow students to use the Mathcad functions presented for each of the numerical methods discussed in the other chapters. An algorithmic statement of each method is also included; the algorithm may be used as the basis for computations using a variety of types of technological support, ranging from paper and pencil, to calculators, Mathcad worksheets or developing computer programs.Each of the subsequent chapters begins with a one-page overview of the subject matter, together with an indication as to how the topics presented in the chapter are related to those in previous and subsequent chapters. Introductory examples are presented to suggest a few of the types of problems for which the topics of the chapter may be used. Following the sections in which the methods are presented, each chapter concludes with a summary of the most important formulas, a selection of suggestions for further reading, and an extensive set of exercises. The first group of problems provide fairly routine practice of the techniques; the second group are applications adapted from a variety of fields, and the final group of problems encourage students to extend their understanding of either the theoretical or the computational aspects of the methods.The presentation of each numerical technique is based on the successful teaching methodology of providing examples and geometric motivation for a method, and a concise statement of the steps to carry out the computation, before giving a mathematical derivation of the process or a discussion of the more theoretical issues that are relevant to the use and understanding of the topic. Each topic is illustrated by examples that range in complexity from very simple to moderate.Geometrical or graphical illustrations are included whenever they are appropriate. A simple Mathcad function is presented for each method, which also serves as a clear step-by-step description of the process; discussion of theoretical considerations is placed at the conclusion of the section. The last section of each chapter gives a brief discussion of Mathcad's built-in functions for solving the kinds of problems covered in the chapter.The chapters are arranged according to the following general areas: Chapters 2-5 deal with solving linear and nonlinear equations. Chapters 6 and 7 treat topics from numerical linear algebra. Chapters 8-10 cover numerical methods for data interpolation and approximation. Chapters 11 presents numerical differentiation and integration. Chapters 12-15 introduce numerical techniques for solving differential equations.For much of the material, a calculus sequence that includes an introduction to differential equations and linear algebra provides adequate background. For more in depth coverage of the topics from linear algebra (especially the QR method for eigenvalues) a linear algebra course would be an appropriate prerequisite. The coverage of Fourier approximation and FFT (Chapter 10) and partial differential equations (Chapter 15) also assumes that the students have somewhat more mathematical maturity than
K7-K9 mathematics pocket book features all the formulas and problem solving techniques in junior high school mathematics. The cards are designed to help users with the memorizing process. You can also read through this app quickly before the exam. Please note that K7-K9 mathematics pocket book is a Chinese-language (Traditional Chinese) application.
College Algebra - Student Solutions Manual - 2nd edition Summary: By following a distinctive approach in explaining algebra, College Algebra helps alleviate the readers anxiety toward math. New sections on modeling have been added at the end of each chapter. Sections have been included on Limits and Early Functions. There are also numerous examples integrated throughout the pages to assure that all problem types are represented. These examples contain more detailed annotations using everyday language. This approach encourages reade...show morers to develop sound study and problem solving
Tuesday, December 25, 2012 Funny, i have most of my g-grandfathers school books from the mid 1870s, among them a copy of Frenchs Common Arithmetic. Our district uses TERC Investiigations (trying to get rid of). Not only is the currciulum in the book published in 1869 clearer--my ggrandfather had a habit of working problems in the margins and end pages. Finally took it into a Board of Ed meeting one night and actually showed them how an 11 year old kid who went ot a one room school house on the Illinois prairie actually had better math fluency at the same age as his gg grandaughter, who attends a supposedly first class Westchester County NY public school. Were the problem sets not geared to a farmers offspring--lots in rods, furlongs, bushels etc--I'd just teach her from the old book. bky said... The newer problems are an example of what I see in my kids public middle school algebra: they take as much of the algebra out as they can. There is almost no manipulation of expressions except for the simplest equation solving. They introduce 2x2 systems like this, with both lines already in point-slope form and lots of questions about what you would do with a table? a graph? etc? It gets really depressing when they get to exponentials. Instead of algebra (manipulating expressions) they make tables and graphs, tables and graphs. They eliminate the handicraft aspect of algebra
Synopses & Reviews Publisher Comments: This book of problems has been designed to accompany an undergraduate course in probability. The only prerequisite is basic algebra and calculus. Each chapter is divided into three parts: Problems, Hints, and Solutions. To make the book self-contained all problem sections include expository material. Definitions and statements of important results are interlaced with relevant problems. The problems have been selected to motivate abstract definitions by concrete examples and to lead in manageable steps towards general results, as well as to provide exercises based on the issues and techniques introduced in each chapter. The book is intended as a challenge to involve students as active participants in the course. Synopsis: "Synopsis" by Springer,
College Algebra: Enhanced with Graphing Utilities Michael Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing ...Show synopsisMichael Sullivan??? s time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Series has evolved to meet today??? s course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors95649321795649-5-1Good. Hardcover. Missing components. May include moderately...Good. Hardcover. Missing components. May include moderately worn cover, writing, markings or slight discoloration. SKU: 978032183211532115The book is not that bad but the way it is organized leaves a lillte to be desired. I have found that the index is not correct when looking for particular items as they are off by a few pages and this has happened several times. Still, this is an OK book for my college
Naive Lie Theory - 10 edition Summary: In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called ''classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.This naive approach to Lie theory is originally due ...show moreto von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history.John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994). ...show less Edition/Copyright:10 Cover: Paperback Publisher:Springer-Verlag New York Published: 11/02/2010 International: No List Price: $54.95 Used Currently Sold Out New Currently Sold Out Rental $38.49 Due back 12/19/2014 Save $16.46 (30%) Free return shipping In stock 21-day satisfaction guarantee CDs or access codes may not be included Marketplace sellers starting at $37.69 16 more offers below. Additional Sellers for Naive Lie2010
Centreville, VA CalculusSo many times there are different ways to learn and understand Math and it can be as easy as looking at a tough problem from a new angle.Discrete Math (DM) deals with applying math to distinct/unique/separate values/objects and is used in areas such as graph theory, propositional logic, predicate...
Major Requirements Coursework Requirements Students majoring in mathematics must complete MATH 115 and one of 116/120 (or the equivalent) and at least eight units of 200-level and 300-level courses. These eight units must include 205, 206, 302, 305, and two additional 300-level courses. (Thus a student who places out of 115/116 and starts in 205 requires only eight courses.) At most two of 206, 210 and 215 may be counted towards the major. These courses must be completed for the mathematics major: Math 115: Calculus I and Math 116: Calculus II, or the equivalent Math 205: Multivariable Calculus Math 206: Linear Algebra Math 302: Elements of Analysis I Math 305: Abstract Algebra At least two elective 300-level courses not counting any of 350, 360, 370. A student may count Math 215/Phys 215 towards her mathematics major. However, she may count at most two of the course 206, 210, and 215 toward the major. Credit for Math 216/Phys 216 satisfies the requirement that a math major take 205, but cannot be counted as one of the 200- or 300-level units required for the major. Major Presentation Requirement Majors are also required to present one classroom talk in either their junior or senior year. This requirement can be satisfied with a presentation in the student seminar, but it can also be fulfilled by giving a talk in one of the courses whose catalog description says"Majors can fulfill the major presentation requirement in this course." In addition, a limited number of students may be able to fulfill the presentation requirement in other courses, with permission of the instructor
Elementary and Intermediate Algebra: A Combined Approach Master algebraic fundamentals with Kaufmann/Schwitters ELEMENTARY AND INTERMEDIATE ALGEBRA 5e. Learn from clear and concise explanations, multiple examples and numerous problem sets in an easy-to-read format. The text's 'learn, use & apply' formula helps you learn a skill, use the skill to solve equations, then apply it to solve application problems. With this simple, straightforward approach, you will grasp and apply key problem-solving skills necessary for success in future mathematics courses. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. About the author (2008
Archive This is a guest post by Nathan, who recently finished graduate school in math, and will begin a post-doc in the fall. He loves teaching young kids, but is still figuring out how to motivate undergraduates. The question Like most mathematicians in academia, I'm teaching calculus in the fall. I taught in grad school, but the syllabus and assignments were already set. This time I'll be in charge, so I need to make some design decisions, like the following: Are calculators/computers/notes allowed on the exams? Which purely technical skills must students master (by a technical skill I mean something like expanding rational functions into partial fractions: a task which is deterministic but possibly intricate)? Will students need to write explanations and/or proofs? I have some angst about decisions like these, because it seems like each one can go in very different directions depending on what I hope the students are supposed to get from the course. If I'm listing the pros and cons of permitting calculators, I need some yardstick to measure these pros and cons. My question is: what is the goal of a college calculus course? I'd love to have an answer that is specific enough that I can use it to make concrete decisions like the ones above. Part of my angst is that I've asked many people this question, including people I respect enormously for their teaching, but often end up with a muddled answer. And there are a couple stock answers that come to mind, but each one doesn't satisfy me for one reason or another. Here's what I have so far. The contenders. To teach specific tasks that are necessary for other subjects. These tasks would include computing integrals and derivatives, converting functions to power series or Fourier series, and so forth. Intuitive understanding of functions and their behavior. This is vague, so here's an example: a couple years ago, a friend in medical school showed me a page from his textbook. The page concerned whether a certain drug would affect heart function in one way or in the opposite way (it caused two opposite effects), and it showed a curve relating two involved parameters. It turned out that the essential feature was that this curve was concave down. The book did not use the phrase "concave down," though, and had a rather wordy explanation of the behavior. In this situation, a student who has a good grasp of what concavity is and what its implications are is better equipped to understand the effect described in the book. So if a student has really learned how to think about concavity of functions and its implications, then she can more quickly grasp the essential parts of this medical situation. To practice communicating with precision. I'm taking "communication" in a very wide sense here: carefully showing the steps in an integral calculation would count. Not Satisfied I have issues with each of these as written. I don't buy number 1, because the bread and butter of calculus class, like computing integrals, isn't something most doctors or scientists will ever do again. Number 2 is a noble goal, but it's overly idealistic; if this is the goal, then our success rate is less than 10%. Number 3 also seems like a great goal, relevant for most of the students, but I think we'd have to write very different sorts of assignments than we currently do if we really want to aim for it. I would love to have a clear and realistic answer to this question. What do you think? After recording my weekly Slate Money podcast this morning I will be off to the Clearwater Festival in Croton-on-Hudson. The weather's supposed to be gorgeous all weekend, which is good because I'm camping in a tent, and the last few times I went to bluegrass or folk festivals and camped in a tent it rained and I ended up sleeping in puddles. If you've never done that, let me tell you that there's something gross and creepy about wet pillows. My bandmate Jamie, who plays the mandolin and washboard, convinced me not only to go but to be a volunteer at this festival, which as it turns out means I'll be preparing food in the kitchen. There are 1,000 volunteers at this festival, so who knows how many people go; I'm preparing for a lot of diced carrots and onions no matter what. Or maybe I'll be doing dishes. I love doing dishes for some reason. So this Clearwater Festival was Pete Seeger's baby, he came every year, and since he passed away this past winter, the entire weekend will be a tribute to his life and his work. Some incredible musicians are going to be there to honor Pete, and I am hoping my kitchen duties don't conflict with my old favorite, Marty Sexton (Sunday at 4pm), as well as my new favorite, John Fullbright (Saturday at 2:30). No time for a post this morning but go read this post by Scott Aaronson on using a PageRank-like algorithm to understand human morality and decision making. The post is funny, clever, very thoughtful, and pretty long. To get a flavor of the exchange, we'll start with this from Andreessen: What never gets discussed in all of this robot fear-mongering is that the current technology revolution has put the means of production within everyone's grasp. It comes in the form of the smartphone (and tablet and PC) with a mobile broadband connection to the Internet. Practically everyone on the planet will be equipped with that minimum spec by 2020. versus this from Payne: If we're gonna throw around Marxist terminology, though, can we at least keep Karl's ideas intact? Workers prosper when they own the means of production. The factory owner gets rich. The line worker, not so much. Owning a smartphone is not the equivalent of owning a factory. I paid for my iPhone in full, but Apple owns the software that runs on it, the patents on the hardware inside it, and the exclusive right to the marketplace of applications for it. … You spent a lot of paragraphs on back-of-the-napkin economics describing the coming Awesome Robot Future, addressing the hypotheticals. What you left out was the essential question: who owns the robots? Namely, at some point we'll have all these robots doing stuff for us, but how are we going to spread that wealth around? Who owns the robots and when are they going to learn to share? In this vision of the distant future, that critical "singularity of moral enlightenment" (SME) is never explained. I wish I could ask Captain Picard how it all went down. It's one thing to lack an explanation for the SME, and to consider it an aspirational quasi-religious utopian goal, but it's another thing entirely to fail to acknowledge it. That someone as powerful and famous as Mark Andreessen, who is personally involved in the development and nurturing of so many technology platforms, has trouble seeing the logical inconsistency of his own rhetoric can only be explained by the fact that, as the controller of such platforms, it is he who reaps their benefits. It's yet another case of someone thinking "this system works for me therefore it is super awesome for everyone and everything, amen." I'm hoping Al3x's fine response will get Marc to consider how SME is gonna happen, and when. One of the reasons I enjoy my blog is that I get to try out an argument and then see if readers can 1) poke holes in my arguement, or 2) if they misunderstand my argument, or 3) if they misunderstand something tangential to my argument. The idea is this. Many mathematical models are meant to replace a human-made model that is deemed too expensive to work out at scale. Credit scores were like that; take the work out of the individual bankers' hands and create a mathematical model that does the job consistently well. The VAM was originally intended as such – in-depth qualitative assessments of teachers is expensive, so let's replace them with a much cheaper option. So all I'm asking is, how good a replacement is the VAM? Does it generate the same scores as a trusted, in-depth qualitative assessment? When I made the point yesterday that I haven't seen anything like that, a few people mentioned studies that show positive correlations between the VAM scores and principal scores. But here's the key point: positive correlation does not imply equality. Of course sometimes positive correlation is good enough, but sometimes it isn't. It depends on the context. If you're a trader that makes thousands of bets a day and your bets are positively correlated with the truth, you make good money. But on the other side, if I told you that there's a ride at a carnival that has a positive correlation with not killing children, that wouldn't be good enough. You'd want the ride to be safe. It's a higher standard. I'm asking that we make sure we are using that second, higher standard when we score teachers, because their jobs are increasingly on the line, so it matters that we get things right. Instead we have a machine that nobody understand that is positively correlated with things we do understand. I claim that's not sufficient. Let me put it this way. Say your "true value" as a teacher is a number between 1 and 100, and the VAM gives you a noisy approximation of your value, which is 24% correlated with your true value. And say I plot your value against the approximation according to VAM, and I do that for a bunch of teachers, and it looks like this: So maybe your "true value" as a teacher is 58 but the VAM gave you a zero. That would not just be frustrating to you, since it's taken as an important part of your assessment. You might even lose your job. And you might get a score of zero many years in a row, even if your true score stays at 58. It's increasingly unlikely, to be sure, but given enough teachers it is bound to happen to a handful of people, just by statistical reasoning, and if it happens to you, you will not think it's unlikely at all. In fact, if you're a teacher, you should demand a scoring system that is consistently the same as a system you understand rather than positively correlated with one. If you're working for a teachers' union, feel free to contact me about this. One last thing. I took the above graph from this post. These are actual VAM scores for the same teacher in the same year but for two different class in the same subject – think 7th grade math and 8th grade math. So neither score represented above is "ground truth" like I mentioned in my thought experiment. But that makes it even more clear that the VAM is an insufficient tool, because it is only 24% correlated with itself. I think I'm supposed to come away impressed, but that's not what happens. Let me explain. Their data set for students scores start in 1989, well before the current value-added teaching climate began. That means teachers weren't teaching to the test like they are now. Therefore saying that the current VAM works because an retrograded VAM worked in 1989 and the 1990′s is like saying I must like blueberry pie now because I used to like pumpkin pie. It's comparing apples to oranges, or blueberries to pumpkins. I'm surprised by the fact that the authors don't seem to make any note of the difference in data quality between pre-VAM and current conditions. They should know all about feedback loops; any modeler should. And there's nothing like telling teachers they might lose their job to create a mighty strong feedback loop. For that matter, just consider all the cheating scandals in the D.C. area where the stakes were the highest. Now that's a feedback loop. And by the way, I've never said the VAM scores are totally meaningless, but just that they are not precise enough to hold individual teachers accountable. I don't think Chetty et al address that question. So we can't trust old VAM data. But what about recent VAM data? Where's the evidence that, in this climate of high-stakes testing, this model is anything but random? If it were a good model, we'd presumably be seeing a comparison of current VAM scores and current other measures of teacher success and how they agree. But we aren't seeing anything like that. Tell me if I'm wrong, I've been looking around and I haven't seen such comparisons. And I'm sure they've been tried, it's not rocket science to compare VAM scores with other scores. The lack of such studies reminds me of how we never hear about scientific studies on the results of Weight Watchers. There's a reason such studies never see the light of day, namely because whenever they do those studies, they decide they're better off not revealing the results. And if you're thinking that it would be hard to know exactly how to rate a teacher's teaching in a qualitative, trustworthy way, then yes, that's the point! It's actually not obvious how to do this, which is the real reason we should never trust a so-called "objective mathematical model" when we can't even decide on a definition of success. We should have the conversation of what comprises good teaching, and we should involve the teachers in that, and stop relying on old data and mysterious college graduation results 10 years hence. What are current 6th grade teachers even supposed to do about studies like that? Note I do think educators and education researchers should be talking about these questions. I just don't think we should punish teachers arbitrarily to have that conversation. We should have a notion of best practices that slowly evolve as we figure out what works in the long-term. So here's what I'd love to see, and what would be convincing to me as a statistician. If we see all sorts of qualitative ways of measuring teachers, and see their VAM scores as well, and we could compare them, and make sure they agree with each other and themselves over time. In other words, at the very least we should demand an explanation of how some teachers get totally ridiculous and inconsistent scores from one year to the next and from one VAM to the next, even in the same year. The way things are now, the scores aren't sufficiently sound be used for tenure decisions. They are too noisy. And if you don't believe me, consider that statisticians and some mathematicians agree. There's been a movement to make primary and secondary education run more like a business. Just this week in California, a lawsuit funded by Silicon Valley entrepreneur David Welch led to a judge finding that student's constitutional rights were being compromised by the tenure system for teachers in California. The thinking is that tenure removes the possibility of getting rid of bad teachers, and that bad teachers are what is causing the achievement gap between poor kids and well-off kids. So if we get rid of bad teachers, which is easier after removing tenure, then no child will be "left behind." The problem is, there's little evidence for this very real achievement gap problem as being caused by tenure, or even by teachers. So this is a huge waste of time. As a thought experiment, let's say we did away with tenure. This basically means that teachers could be fired at will, say through a bad teacher evaluation score. An immediate consequence of this would be that many of the best teachers would get other jobs. You see, one of the appeals of teaching is getting a comfortable pension at retirement, but if you have no idea when you're being dismissed, then it makes no sense to put in the 25 or 30 years to get that pension. Plus, what with all the crazy and random value-added teacher models out there, there's no telling when your score will look accidentally bad one year and you'll be summarily dismissed. People with options and skills will seek other opportunities. After all, we wanted to make it more like a business, and that's what happens when you remove incentives in business! The problem is you'd still need teachers. So one possibility is to have teachers with middling salaries and no job security. That means lots of turnover among the better teachers as they get better offers. Another option is to pay teachers way more to offset the lack of security. Remember, the only reason teacher salaries have been low historically is that uber competent women like Laura Ingalls Wilder had no other options than being a teacher. I'm pretty sure I'd have been a teacher if I'd been born 150 years ago. So we either have worse teachers or education doubles in price, both bad options. And, sadly, either way we aren't actually addressing the underlying issue, which is that pesky achievement gap. People who want to make schools more like businesses also enjoy measuring things, and one way they like measuring things is through standardized tests like achievement scores. They blame teachers for bad scores and they claim they're being data-driven. I'm tempted to conclude that we should just go ahead and get rid of teacher tenure so we can wait a few years and still see no movement in the achievement gap. The problem with that approach is that we'll see great teachers leave the profession and no progress on the actual root cause, which is very likely to be poverty and inequality, hopelessness and despair. Not sure we want to sacrifice a generation of students just to prove a point about causation. On the other hand, given that David Welch has a lot of money and seems to be really excited by this fight, it looks like we might have no choice but to blame the teachers, get rid of their tenure, see a bunch of them leave, have a surprise teacher shortage, respond either by paying way more or reinstating tenure, and then only then finally gather the data that none of this has helped and very possibly made things worse. This is a great book. It's well written, clear, and it focuses on important issues. I did not check all of the claims made by the data but, assuming they hold up, the book makes two hugely important points which hopefully everyone can understand and debate, even if we don't all agree on what to do about them. First, the authors explain the insufficiency of monetary policy to get the country out of recession. Second, they suggest a new way to structure debt. To explain these points, the authors do something familiar to statisticians: they think about distributions rather than averages. So rather than talking about how much debt there was, or how much the average price of houses fell, they talked about who was in debt, and where they lived, and which houses lost value. And they make each point carefully, with the natural experiments inherent in our cities due to things like available land and income, to try to tease out causation. Their first main point is this: the financial system works against poor people ("borrowers") much more than rich people ("lenders") in times of crisis, and the response to the financial crisis exacerbated this discrepancy. The crisis fell on poor people much more heavily: they were wiped out by the plummeting housing prices, whereas rich people just lost a bit of their wealth. Then the government stepped in and protected creditors and shareholders but didn't renegotiate debt, which protected lenders but not borrowers. This is a large reason we are seeing so much increasing inequality and why our economy is stagnant. They make the case that we should have bailed out homeowners not only because it would have been fair but because it would have been helpful economically. The authors looked into what actually caused the Great Recession, and they come to a startling conclusion: that the banking crisis was an effect, rather than a cause, of enormous household debt and consumer pull-back. Their narrative goes like this: people ran up debt, then started to pull back, and and as a result the banking system collapsed, as it was utterly dependent on ever-increasing debt. Moreover, the financial system did a very poor job of figuring out how to allocate capital and the people who made those loans were not adequately punished, whereas the people who got those loans were more than reasonably punished. About half of the run-up of household debt was explained by home equity extraction, where people took out money from their home to spend on stuff. This is partly due to the fact that, in the meantime, wages were stagnant and home equity was a big thing and was hugely available. But the authors also made the case that, even so, the bubble wasn't directly caused by rising home valuations but rather to securitization and the creation of "financial innovation" which made investors believe they were buying safe products which were in fact toxic. In their words, securities are invented to exploit "neglected risks" (my experience working in a financial risk firm absolutely agrees to this; whenever you hear the phrase "financial innovation," please interpret it to mean "an instrument whose risk hides somewhere in the creases that investors are not yet aware of"). They make the case that debt access by itself elevates prices and build bubbles. In other words, it was the sausage factory itself, producing AAA-rated ABS CDO's that grew the bubble. Next, they talked about what works and what doesn't, given this distributional way of looking at the household debt crisis. Specifically, monetary policy is insufficient, since it works through the banks, who are unwilling to lend to the poor who are already underwater, and only rich people benefit from cheap money and inflated markets. Even at its most extreme, the Fed can at most avoid deflation but it not really help create inflation, which is what debtors need. Fiscal policy, which is to say things like helicopter money drops or added government jobs, paid by taxpayers, is better but it makes the wrong people pay – high income earners vs. high wealth owners – and isn't as directly useful as debt restructuring, where poor people get a break and it comes directly from rich people who own the debt. There are obstacles to debt restructuring, which are mostly political. Politicians are impotent in times of crisis, as we've seen, so instead of waiting forever for that to happen, we need a new kind of debt contract that automatically gets restructured in times of crisis. Such a new-fangled contract would make the financial system actually spread out risk better. What would that look like? The authors give two examples, for mortgages and student debt. The student debt example is pretty simple: how quickly you need to pay back your loans depends in part on how many jobs there are when you graduate. The idea is to cushion the borrower somewhat from macro-economic factors beyond their control. Next, for mortgages, they propose something the called the shared-responsibility mortgage. The idea here is to have, say, a 30-year mortgage as usual, but if houses in your area lost value, your principal and monthly payments would go down in a commensurate way. So if there's a 30% drop, your payments go down 30%. To compensate the lenders for this loss-share, the borrowers also share the upside: 5% of capital gains are given to the lenders in the case of a refinancing. In the case of a recession, the creditors take losses but the overall losses are smaller because we avoid the foreclosure feedback loops. It also acts as a form of stimulus to the borrowers, who are more likely to spend money anyway. If we had had such mortgage contracts in the Great Recession, the authors estimate that it would have been worth a stimulus of $200 billion, which would have in turn meant fewer jobs lost and many fewer foreclosures and a smaller decline of housing prices. They also claim that shared-responsibility mortgages would prevent bubbles from forming in the first place, because of the fear of creditors that they would be sharing in the losses. A few comments. First, as a modeler, I am absolutely sure that once my monthly mortgage payment is directly dependent on a price index, that index is going to be manipulated. Similarly as a college graduate trying to figure out how quickly I need to pay back my loans. And depending on how well that manipulation works, it could be a disaster. Second, it is interesting to me that the authors make no mention of the fact that, for many forms of debt, restructuring is already a typical response. Certainly for commercial mortgages, people renegotiate their principal all the time. We can address the issue of how easy it is to negotiate principal directly by talking about standards in contracts. Having said that I like the idea of having a contract that makes restructuring automatic and doesn't rely on bypassing the very real organizational and political frictions that we see today. Let me put it this way. If we saw debt contracts being written like this, where borrowers really did have down-side protection, then the people of our country might start actually feeling like the financial system was working for them rather than against them. I'm not holding my breath for this to actually happen. My schedule nowadays is to go to the Lede Program classes every morning from 10am until 1pm, then office hours, when I can, from 2-4pm. The students are awesome and are learning a huge amount in a super short time. So for instance, last time I mentioned we set up iPython notebooks on the cloud, on Amazon EC2 servers. After getting used to the various kinds of data structures in python like integers and strings and lists and dictionaries, and some simple for loops and list comprehensions, we started examining regular expressions and we played around with the old enron emails for things like social security numbers and words that had four or more vowels in a row (turns out that always means you're really happy as in "woooooohooooooo!!!" or really sad as in "aaaaaaarghghgh"). Then this week we installed git and started working in an editor and using the command line, which is exciting, and then we imported pandas and started to understand dataframes and series and boolean indexes. At some point we also plotted something in matplotlib. We had a nice discussion about unsupervised learning and how such techniques relate to surveillance. My overall conclusion so far is that when you have a class of 20 people installing git, everything that can go wrong does (versus if you do it yourself, then just anything that could go wrong might), and also that there really should be a better viz tool than matplotlib. Plus my Lede students are awesome. We moved to our apartment in New York almost exactly 9 years ago. I know that in part because I remember the date we moved in – June 4th, 2005 – but also because that first weekend we lived here, when we decided to try to buy some furniture for our nearly empty living room, we had to cross the Puerto Rican parade to get to Crate & Barrel on the east side of 5th Avenue. It was one of the most characteristic New York moments of my existence, and it made me feel like a real New Yorker. About two days after moving in I figured out with my friend Michael Thaddeus (who has guest blogged hugely successfuly before) that his apartment was within direct sight of mine. We could wave to each other from our windows across both 116th and Claremont! For a suburban girl like me this was a hoot. We decided to build a string telephone at some point. Well, we finally got around to doing it yesterday. I live on the 9th floor, and Thads lives on the 5th floor of his apartment, so there was no chance we could throw anything up to the window on the outside. Instead Thads came over with two balls of string and two cans. For each window we lowered the string to the street with the help of someone on the street who could guide the person in the window. I actually only saw the first half of this procedure because I was tasked with holding the string after the first window and waiting for the second string to be lowered. Then the idea was we'd tie the two strings together. So here I am, outside my building, holding a string in my hand that goes all the way up to a 9th floor building across the street. I'm also wearing my cowboy hat because it's sunny outside, but for some reason the combination made everyone walking by stop and ask me what the hell I'm doing. You see, there aren't many things that can make New Yorkers talk to each other on the street, but I've found that holding on to very very long strings whilst wearing a ridiculous hat does the trick. My favorite was when this middle aged Greek guy comes up to me and asks me what I'm doing, but he's clearly hoping it's mischievous, so I asked him to guess, and he says "You're pulling someone's tooth!!". After a while my neighbors noticed the string outside their window and got involved. And I noticed the security guard on the corner paying close attention, especially when we had both strings on the street and we were trying to tie them together, which took a while because they barely reached. There was even a cop car silently observing that part of the experiment, but it disappeared as soon as we got it connected and Johan pulled the string taut so it was above the tree line. After poking the strings into the cans, we tried our our string telephone. It was incredibly fun. Aunt Pythia is super glad to be here. It's a gorgeous day, Aunt Pythia has super fun plans that involve this place in Morristown, New Jersey, and the world is looking bright and colorful and happy. Aunt Pythia's usual skeptical gloom has given way to rainbows and puppies (Aunt Pythia is a dog person). Are you with me peoples?! Give it up for life! Give it up for humanity!! Having said that, Aunt Pythia has more than her usual number of slapdowns to administer today, as you will soon see below. Don't be intimidated, though, folks! After watching the abuse, do your best toSo yeah, shortest Aunt Pythia question ever. Turns out "this" is an article about yet another person who "hacked" OKCupid to find the love of their life. A male mathematician who dove headlong into the data mining of love. Ho hum. Please also see [another earlier article], where it was a woman instead of a man. I can't find it now because this article became so popular that it's cockblocking my google searches. Wait, I think she gave a TED talk as well. Oh yeah here she is! And she reverse-engineered the algorithm, too. And honestly she's telling her own story which is way more engaging than that article. Anyhoo, here's the thing. First of all, ew. He went on way too many dates too quickly. I'm glad he found love eventually, but let's face it, he was making himself less receptive, not more receptive, by going on all those dates. Plus he was posing artificially based on his "mathematical research," which came down to a clustering algorithm. Plus the woman he eventually proposed to FOUND HIM. Plus ew. "…the idea that math (or, more broadly, "formulas") can be used as a dating tactic is a surprisingly popular belief based on a number of very flawed premises, many of which reveal pickup artist-flavor misogynist attitudes among the nerdy white guys who champion them." Now given that I also have an example of a woman doing this, I'm not gonna claim it's all about sexism (although there's more than a veneer of nerdiness!). Rather, it's all about the weird non-human mindset. Here's another stab at what I'm talking about: "But much of the language used in the story reflects a weird mathematician-pickup artist-hybrid view of women as mere data points anyway, often quite literally: McKinlay refers to identity markers like ethnicity and religious beliefs as "all that crap"; his "survey data" is organized into a "single, solid gob"; unforeseen traits like tattoos and dog ownership are called "latent variables." By viewing himself as a developer, and the women on OkCupid as subjects to be organized and "mined," McKinlay places himself in a perceived greater place of power. Women are accessories he's entitled to. Pickup artists do this too, calling women "targets" and places where they live and hang out "marketplaces." It's a spectrum, to be sure, but McKinlay's worldview and the PUA worldview are two stops along it. Both seem to regard women as abstract prizes for clever wordplay or, as it may be, skilled coding. Neither seems particularly aware of, or concerned with, what happens after simply getting a woman to say yes." So, again, it's not just men who do this. Women who are ABSOLUTELY OBSESSED WITH FINDING MR. RIGHT also do this. They stop thinking about men as people and start thinking of them as bundles of attributes. You have to be tall! And weigh more than me! And culturally Jewish! If you want to think about this more, and how deeply damaging it is to society and our concepts of ourselves and our expectations of the future, not to mention how we perceive children, then take a look at the book Why Love Hurts: A Sociological Explanation. It's super fascinating. So there you go, a long answer to a short question. One last thing: I'm not saying that you should give up on your own algorithms and trust OKCupid's algorithms. Far from it! I just think that the key thing is to stay human. Plus all online dating sites are asking the wrong questions, as I mentioned here. Auntie P —— Dear Aunt Pythia, I'm about to start a PhD in Math at a top-ranked place. I'm pretty sure I won't end up in academia for a variety of personal reasons (mostly that my partner is a non-academic with a job that needs to be in New York, SF, or DC). What should I be doing my first year/summer to make sure I'm in a reasonably good place for a non-academic job hunt 5 or 6 years from now? (And to make matters more complicated, both finance and government creep me out morally, but I really want to end up somewhere with some fun, interesting mathematics.) Higher Education, Less Professionalism Dear HELP, Nice sign-off! Make sure you know how to code, make sure you know how it feels to work in a company, make sure you keep your eye on what makes you feel moral and useful and interested. Oh, and read my book! I wrote it for people like you. By the way, I'm hoping that, by the time you finish your Ph.D., there are better non-academic jobs out there for morally centered people with math skills. I'm just feeling optimistic today, I can't explain it. Aunt Pythia —— Dear Aunt Pythia, With data science hype at an all-time high (and rising), I've been hearing of more and more people who are deciding to make a career change to data science. These acquaintances are smart, science-minded people, but without any background in advanced math, statistics, or computer science. An example background would be a bachelors degree in Chemistry. They are planning to take a few online courses, or a semester-long course or two, and then enter the job market. My question is, do you think there's a place for "data scientists" like these? Who've learned all the programming/machine learning/statistics they can in 3 months part-time but nothing beyond that? As someone with a strong technical background, I am skeptical that data scientists can be successfully churned out so quickly. Then again, if the hype is all it's hyped up to be, maybe they'll all get great jobs. Wondering what your take is. Sincerely, Some Kooky Elitist Person Trying to Intuit Climate Dear SKEPTIC, Niiiiice sign-off! I am super proud. Two things. First, I certainly believe that anyone who has a high general level of intelligence and works hard can learn a new field diligently. So I don't doubt the intentions or efforts of our chemist friends. On the other hand, do data science jobs allow for follow-up training and – even more importantly – thinking? I'm guessing some do but most don't. So yes, I agree that for many of these people, it's a disappointment waiting to happen. And yes, certainly 3 months training does very little. At best you can start thinking a new way, but it's up to you to actually make things happen with that new mindset. They might find out their job is really nothing like the job they thought they had. They might end up being excel or SQL database monkeys, or they might find out their job is a front so that the company can claim to be doing "data science." Worst case they're asked to audit and approve models they don't understand which are being used in a predatory manner so they're on the hook when shit gets real. On the other hand, what are the options really? It's a new field and there's no major for it (UPDATE: there are post-bacc programs popping up everywhere, for example here and here). This is what new fields look like, a bunch of amateurs coming together trying to figure out what they're doing. Sometimes it works brilliantly and sometimes it produces frauds who ride the hype wave because they're good at that. In short, stay skeptical but don't presume that your friends and acquaintances have bad intent. Ask them probing questions, when you see them, about which above scenario they're in, it might help them figure it out for themselves. Unless that's creepy and/or obnoxious. Aunt Pythia —— Dear Aunt Pythia, How useful do you think "generate-and-test" results are? I am searching for good parameter settings using recent history from the last twelve days. For example, I just checked the report that is being generated and saw successful results eight times out of twelve. I actually could run a check against history, not including the last result and see how often the next result is good. Is this crazy or what? Sleepless in Mesquite Dear Sleepless, I have never heard of "generate and test" so I googled it and found this, which honestly seems ridiculous for the following reason: how will you ever know your "solution" works? So there is an example where it will work that illustrates my overall point. If you know that you have a line ("the solution") and you know two (different) points that are on that line, then once you find a line with those points you know you've found the solution, because it's unique. Similarly, if you know your solution is a quadratic equation, then all you need to do is test it on three (different) points and you know you're good. But in general, how do you "test" a solution? Unless you are given, a priori, the form of the solution, to test your solution in general you'd need to try it on every point in the universe where you care about the solution working. That doesn't sound like a useful approach. I know I'm talking abstractly here, but you gave me very little to work with. In any case 8 out of 12 doesn't sound very convincing, and 12 doesn't sound big enough for much of anything. That is, even if you got 12 out of 12 I still wouldn't be convinced you're done unless I know more information. I'm too busy this morning for a real post but I thought I'd share a few things I'm reading today. Matt This coming Sunday my friend Adam Reich is coming to Alternative Banking to talk about his work as the faculty director of a collaborative project this summer between Columbia's INCITE and the OUR Walmart campaign. The plan involves twenty students to scatter across the country, organizing and conducting oral history interviews alongside Walmart workers in five regions. It is also, not coincidentally, the 50th anniversary of the Freedom Summer of 1964, when a bunch of volunteers including students helped register black Mississippians to vote. I am now part of the administrative bloat over at Columbia. I am non-faculty administration, tasked with directing a data journalism program. The program is great, and I'm not complaining about my job. But I will be honest, it makes me uneasy. Although I'm in the Journalism School, which is in many ways separated from the larger university, I now have a view into how things got so bloated. And how they might stay that way, as well: it's not clear that, at the end of my 6-month gig, on September 16th, I could hand my job over to any existing person at the J-School. They might have to replace me, or keep me on, with a real live full-time person in charge of this program. There are good and less good reasons for that, but overall I think there exists a pretty sound argument for such a person to run such a program and to keep it good and intellectually vibrant. That's another thing that makes me uneasy, although many administrative positions have less of an easy sell attached to them. And studies suggest that administrative costs make up 20 to 30 percent of the United States health care bill, far higher than in any other country. American insurers, meanwhile, spent $606 per person on administrative costs, more than twice as much as in any other developed country and more than three times as much as many, according to a study by the Commonwealth Fund. A comprehensive study published by the Delta Cost Project in 2010 reported that between 1998 and 2008, America's private colleges increased spending on instruction by 22 percent while increasing spending on administration and staff support by 36 percent. Parents who wonder why college tuition is so high and why it increases so much each year may be less than pleased to learn that their sons and daughters will have an opportunity to interact with more administrators and staffers— but not more professors. There are similarities and there are differences between the university and the medical situations. A similarity is that people really want to be educated, and people really need to be cared for, and administrations have grown up around these basic facts, and at each stage they seem to be adding something either seemingly productive or vitally needed to contain the complexity of the existing machine, but in the end you have enormous behemoths of organizations that are much too complex and much too expensive. And as a reality check on whether that's necessary, take a look at hospitals in Europe, or take a look at our own university system a few decades ago. And that also points out a critical difference: the health care system is ridiculously complicated in this country, and in some sense you need all these people just to navigate it for a hospital. And ObamaCare made that worse, not better, even though it also has good aspects in terms of coverage. Whereas the university system made itself complicated, it wasn't externally forced into complexity, except if you count the US News & World Reports gaming that seems inescapable. What they term "unethical behavior" comes down to stuff like cutting off people and cars in an intersection, cheating in a game, and even stealing candy from a baby. The authors also show that rich people are more likely to think of greed as good, and that attitude is sufficient to explain their feelings of entitlement. Another way of saying this it that, once you "account for greed feelings," being rich doesn't make you more likely to cheat. I'd like to go one step further and ask, why do rich people think greed is good? A couple of things come to mind. First, rich people rarely get arrested, and even when they are arrested, their experiences are very different and much less likely to end up with a serious sentence. Specifically, the fees are not onerous for the rich, and fancier lawyers do better jobs for the rich (by the way, in Finland, speeding tickets are on a sliding scale depending on the income of the perpetrator). It's easy to think greed is good if you never get punished for cheating. Second, rich people are examples of current or legacy winners in the current system, and that feeling that they have won leaks onto other feelings of entitlement. They have faith in the system to keep them from having to deal with consequences because so far so good. Finally, some people deliberately judge that they can afford to be assholes. They are insulated from depending on other people because they have money. Who needs friends when you have resources? Of course, not all rich people are greed-is-good obsessed assholes. But there are some that specialize in it. They call themselves Libertarians. Paypal founder Peter Thiel is one of their heroes. Here's some good news: some of those people intend to sail off on a floating country. Thiel is helping fund this concept. The only problem is, they all are so individualistic it's hard for them to agree on ground rules and, you know, a process by which to decide things (don't say government!). This isn't a new idea, but for some reason it makes me very happy. I mean, wouldn't you love it if a good fraction of the people who cut you off in traffic got together and decided to leave town? I'm thinking of donating to that cause. Do they have a Kickstarter yet? You might notice that Aunt Pythia's advice is getting posted later than usual. That's because Aunt Pythia is a wee bit slow on the uptake this morning due to a mighty exciting and exhausting week followed by celebrations of said week. Please bear with her as she gives groggy, possibly irrelevant suggestions to your lovely, deeply and heartfelt questions. And please, after reading her worse-than-usual advice this morning/ afternoon,—— Dear Aunt Pythia, I seriously consider the "Ask Aunt Pythia" series on mathbabe.org as the greatest and bloggiest thing on the blogging planet (granted, I explored only a part of it, and this is only an individual opinion). Is this the right place to say it? Mount Trouillet With Love Dear MTWL, Why yes, yes it is. Thank you darling. Aunt Pythia —— Dear Aunt Pythia, As a grad student, I feel guilty constantly. Guilty that I am probably not spending enough time on my research, guilty that I don't spend enough time on teaching, guilty that I sleep too much… You get the idea. To have a successful academic career, how much should one be working, assuming average intelligence? Also, how should one avoid feeling guilty all the time? A Grad Student Who Loves To Sleep Dear AGSWLTS, Sleep sounds like a gooooooood idea right about now, I think I will. One of the things I don't miss about being an academic is the constant guilt I imposed upon myself. It was all me, and I can't blame anyone else. I can blame nothing except possibly the intense and competitive environment, which again, I chose to live in. It was, I guess, the internal drive to write papers and stay abreast of my field, and without it I might never have done those things, but it sucked. I don't even think I could summon up guilt feelings like that if I tried nowadays. Instead I do things out of sheer excitement about the ideas. I guess sometimes I feel frustrated that I haven't had time to do the stuff I want to, but that frustration is definitely preferable to the old guilt. And come to think of it, a much more efficient way to work too. My advice to you is to give yourself one day a week to do stuff that you just totally love, and banish guilt from your life. You might end up getting more done that way, and then you could expand it to two days a week, who knows. Tell me how that works for you! Auntie P p.s. Please work on your sign-offs. "AGSWLTS" means nothing to me. p.p.s. Never skimp on sleep. Skimp on reading Aunt Pythia, but never skimp on sleep. —— Dear Aunt Pythia, I have lived in a different country in each decade of my life and currently use three different languages on an every day basis. No language do I master well, especially in speaking and listening. The doctor says that I am healthy, and I try to study and practice as much as possible. But, I have communication difficulties in any language. Should a more drastic action be taken? For example, find a job that requires more oral communication. Or, move back to my mother tongue country and try to reactivate my native language ability? Regards, Smurf, or Schtroumpf Dear Smurf/Schtroumpf, I just wanna start this out by saying how very much I enjoyed the smurfs as a child. It was weird, the show was never very good but I always ascribed to those little blue creatures much more interesting lives than they seemed to have. At the end of each episode I remember thinking, "and now they'll go back to even more interesting things they do in their village in the woods with mushroom houses." I think that was their magic, in fact, to seem more interesting than they are. Smallish confession for Aunt Pythia readers: I have been doing my best to summon up a similar more-interesting-than-she-seems cachet pretty much all my life. That's right, everything I've ever done or ever will do goes back to my fascination with the smurfs, and especially papa smurf, who always seemed wiser than even Alan Greenspan back in the day ("NOT LONG NOW!"). As for your question, I'm of the opinion that people get good at what they focus on and what they are patient for. If you really want to focus on getting good at a given language, then you'll need to stop moving countries and just forgive yourself for not already knowing stuff you don't know, it will come with time. My husband, who is not particularly good with languages, has gotten really good at English since I met him 20 years ago. Stay blue! Aunt Pythia —— Dear Aunt Pythia, Your thoughts on the mathematical community being possibly less empathetic than average really hit home for me, because my experiences of being trans and attempting to do math have been really pretty miserable. So with that said, let's confront some cissexism: Plenty of human females have penises. Trans women are female. Plenty of human males have vaginas. Trans men are male. (and of course such porn exists) Talking about sexism in science is interesting. But we can (and should!) do it without erasing the experiences and existence of trans people, whose gender and sex are valid and real. I had a little secret about my survival in grad school, and that secret has a name, and that name is Jordan Ellenberg. We used to meet every Tuesday and Thursday to study schemes at the CallaLily Cafe a few blocks from the Science Center on Kirkland Street, and even though that sounds kind of dull, it was a blast. It was what kept me sane at Harvard. You see, Jordan has an infectious positivity about him, which balances my rather intense suspicions, and moreover he's hilariously funny. He's really somewhere between a mathematician and a stand-up comedian, and to be honest I don't know which one he's better at, although he is a deeply talented mathematician. The reason I'm telling you this is that he's written a book, called How Not To Be Wrong, and available for purchase starting today, which is a delight to read and which will make you understand why I survived graduate school. In fact nobody will ever let me complain again once they've read this book, because it reads just like Jordan talks. In reading it, I felt like I was right back at CallaLily, singing Prince's "Sexy MF" and watching Jordan flirt with the cashier lady again. Aaaah memories. So what's in the book? Well, he talks a lot about math, and about mathematicians, and the lottery, and in fact he has this long riff which starts out with lottery math, then goes to error-correcting codes and then to made-up languages and then to sphere packing and then arrives again at lotteries. And it's brilliant and true and beautiful and also funny. I have a theory about this book that you could essentially open it up to any page and begin to enjoy it, since it is thoroughly enjoyable and the math is cumulative but everywhere so well explained that it wouldn't take long to follow along, and pretty soon you'd be giggling along with Jordan at every ridiculous footnote he's inserted into his narrative. In other words, every page is a standalone positive and ontological examination of the beauty and surprise of mathematical discovery. And so, if you are someone who shares with Jordan a love for mathematics, you will have a consistently great time with this book. In fact I'm imagining that you have an uncle or a mom who loves math or science, in which case this would be a seriously perfect gift to them, but of course you could also give that gift to yourself. I mean, this is a guy who can make nazi jokes funny, and he does. Having said that, the magic of the book is that it's not just a collection of wonderful mathy tidbits. Jordan also has a point about the act of scrutinizing something in a logical and mathematical fashion. That act itself is courageous and should be appreciated, and he explains why, and he tells us how much we've already benefited from people in the past who have had the bravery to do so. He appreciates them and we should too. And yet, he also sends the important message that it's not an elitist crew of the usual genius suspects, that in fact we can all do this in our own capacity. It's a great message and, if it ends up allowing people to re-examine their need for certainty in an uncertain world, then Jordan will really end up doing good. Fingers crossed. That's not to say it's a perfect book, and I wanted to argue with points on basically every other page, but mostly in a good, friendly, over-drinks kind of way, which is provocative but not annoying. One exception I might make came on page 256: no, Jordan, municipal bonds do not always get paid back, and no, stocks do not always go up, not even in expectation. In fact to the extent that both of those statements seem true to many people is the result of many cynical political acts and is damaging, mostly to people like retired civil servants. Don't go there! Another quibble: Jordan talks about how public policy makers make proclamations in the face of uncertainty, and he has a lot of sympathy and seems to think the should keep doing this. I'm on the other side on this one. Telling people to avoid certain foods and then changing stances seems more damaging than helpful and it happens constantly. And it's often tied to industry and money, which also doesn't impress. Even so, even when I strongly disagree with Jordan, I always want to have the conversation. He forces that on the reader because he's so darn positive and open-minded. A few more goodies that I wanted to adore without giving too much away. Jordan does a great job with something he calls "The Great Square of Men" and Berkson's Fallacy: it will explain to many many women why they are not finding the man they're looking for. He also throws out a bone to nerds like me when he almost proves that every pig is yellow, and he absolutely kills it, stand-up comedian style, when comparing Ross Perot to a small dark pile of oats. Holy crap he was on a roll there. So here's one thing I've started doing since reading the book. When I give my 5-year-old son his dessert, it's in the form of Hershey Drops, which are kind of like fat M&M's. I give him 15 and I ask him to count them to make sure I got it right. Sometimes I give him 14 to make sure he's paying attention. But that's not the new part. The new part is something I stole from Jordan's book. The new part is that some days I ask him, "do you want me to give you 3 rows of 5 drops?" And I wait for him to figure out that's enough and say "yes!" And the other days I ask him "do you want me to give you 5 rows of 3 drops?" and I again wait. And in either case I put the drops out in a rectangle. And last night, for the first time, he explained to me in a slightly patronizing voice that it doesn't matter which way I do it because it ends up being the same, because of the rectangle formation and how you look at it. And just to check I asked him which would be more, 10 rows of 7 drops or 7 rows of 10 drops, and he told me, "duh, it would be the same because it couldn't be any different." Yesterday was the first day of the Lede Program and so far so awesome. After introducing ourselves – and the 17 students are each amazing – we each fired up an EC2 server on the Amazon cloud (in North Virginia) and cloning a pre-existing disc image, we got an inspiration speech from Matt Jones about technological determinism and the ethical imperative of reproducibility. Then Adam Parrish led the class in a fun "Hello, world!" exercise on the iPython notebook. In other words, we rocked out. Today we'll hear from Soma about some bash command line stuff, file systems, and some more basic python. I can't wait. Our syllabi are posted on github. Have you seen Obama's latest response to the student debt crisis (hat tip Ernest Davis)? He's going to rank colleges based on some criteria to be named later to decide whether a school deserves federal loans and grants. It's a great example of a mathematical model solving the wrong problem. Now, I'm not saying there aren't nasty leeches who are currently gaming the federal loan system. For example, take the University of Phoenix. It's not a college system, it's a business which extracts federal and private loan money from unsuspecting people who want desperately to get a good job some day. And I get why Obama might want to put an end to that gaming, and declare the University of Phoenix and its scummy competitors unfit for federal loans. I get it. And state funding for public schools has decreased while tuition has increased especially since the financial crisis: The bottomline is that we – and especially our children – need more state school funding much more than we need a ranking algorithm. The best way to bring down tuition rates at private schools is to give them competition at good state schools.
Topology Now! - 06 edition Summary: Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make this material accessible to undergraduate students who may be at the beginning of their study of upper-level mathematics and who may not have covered the extensive prerequisites required for a traditional course in topology. The approach is to cultivate the intuitive ideas of continu...show moreity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces, presents students with the exciting geometrical ideas of topology now(!) rather than later. Anyone using this book should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs. Multivariable calculus, linear algebra, and one further proof-oriented mathematics courses are suitable preparation. ...show less 2006 Hardcover Fair PLEASE NOTE: Book is new and unread but was BOUND UPSIDE DOWN. Still perfectly readable and usable. Save money and amaze your friends! Buy with confidence-Satisfaction Guarantee...show mored! ...show less $43.98 +$3.99 s/h Good Texas Texts Flower Mound, TX Good No underlining or highlighting. Book and pages have some normal shelf wear. Cover may have bent or dinged corners. We ship all books within 24hrs. Books purchased on the weekend ship first thin...show moreg Monday morning. ...show less $50.50 +$3.99 s/h New Sequitur Books Boonsboro, MD Brand new. We distribute directly for the publisher. Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The...show more authors have written this textbook to make this material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces exposes students to the exciting geometrical ideas of topology now(!) rather than later.Students using this textbook should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs. Multivariable calculus, linear algebra, and one further proof-oriented mathematics course are suitable preparation. ...show less $135
This institute builds upon the foundational math content knowledge of teachers at the upper elementary, middle and high school level in the context of standards-based instruction. This course will address the patterns, relations, and algebra strand of the MA Mathematics Curriculum Framework, placing particular emphasis on variables, equations and functions. Participants will gain an understanding of how these important mathematics concepts are developed across grades 4-10, how classic misconceptions of variable and equivalence impact students' algebraic understanding, and how the development of algebraic thinking habits of mind can contribute to student conceptual understanding and problem solving in this strand. Throughout the course participants will do and discuss mathematics; analyze classroom artifacts and discuss subsequent actions; develop questioning strategies that focus, assess and advance students' algebraic thinking; design an algebra pre-assessment; and develop high cognitive demand lessons that support students' prior knowledge and level of understanding relative to articulated standards-based objectives.
Trigonometry provides you with all you need to know to understand the basic concepts of trigonometry — whether you need a supplement to your textbook and classes or an at-a-glance reference. Trigonometry isn't just measuring angles; it has many applications in the real world, such as in navigation, surveying, construction, and many other branches of science, including mathematics and physics. As you work your way through this review, you'll be ready to tackle such concepts as Trigonometric functions, such as sines and cosines Graphs and trigonometric identities Vectors, polar coordinates, and complex numbers Inverse functions and equations You can use CliffsQuickReview Trigonometry in any way that fits your personal style for study and review — you decide what works best with your needs. You can read the book from cover to cover or just look for the information you want and put it back on the shelf for later. Here are just a few ways you can search for topics: With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades. Editorial Reviews From the Back Cover Leading educators help you succeed When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core trigonometer concepts — from trigonometric functions and trigonometric identities to vectors, polar coordinates, and complex numbers — and get the best possible grade. At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented writers who know how to cut to the chase—and zero in on the essential information you need to succeed. Master the basics—fast Complete coverage of core concepts Accessible, topic-by-topic organization Free pocket guide for easy reference About the Author DAVE KAY is a writer, engineer, and aspiring naturalist and artist. He has written or cowritten more than a dozen computer books. Most Helpful Customer Reviews This book is good for one thing, a Quick Review. I suppose I can't blame CliffsNotes, since that is given in the title of the book. I bought this book when I was beginning to learn Trigonometry, along with another textbook. I found that a lot of the things the textbook went over was not even mentioned in this Quick Review book. If you have taken Trigonometry in the past and want to refresh some of the basic trigonometry concepts, then this book is for you. However, if you know nothing about Trigonometry and want to learn about it, this book will do nothing but CONFUSE you. Most people that will look into these books are people who want to learn the subject for their first time, so take my advice: instead of buying this book, save that money towards a better, complete basic trigonometry textbook. I'm a (returning :P) university Freshman preparing for the College Board CLEP tests. I was already familiar with the material covered in this book, but needed to refresh my memory. This review turned out to be exactly what I needed. The author's ability to explain the material to the student are just shy of enlightening. The discussions & theorem proofs are written in a very concise, clear style. I'm a big advocate of the Cliff's QuickReview series. Intended as a course supplement, these books are also GREAT for students wanting to refine their skills. Most of them are also very accessible to students with less familiarity on the subject; trying to learn it for the first time. After reading this, I bought the Calculus & Differential Equations QuickReviews & I'm looking forward to reading them! After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain." Trig is something you always have to practice if you want to remain competent. When practicing basic trig problems (identities, equations, vectors, graphs, angles, cmplx.#'s....) this book gives me just enough explanation with the info I need. But like others said, you should already have some trig under your belt before purchasing it.
Geometry with Geometry Explorer combines a discovery-based geometry text with powerful integrated geometry software. This combination allows for the deep exploration of topics that would be impossible without well-integrated technology, such as hyperbolic geometry, and encourages the kind of experimentation and self-discovery needed for students to develop a natural intuition for various topics in geometry.
Functions, Equations, and Systems Reviews and extends student ability to recognize, describe, and use functional relationships among quantitative variables, with special emphasis on relationships that involve two or more independent variables. Topics include: Direct and inverse variation and joint variation; power functions; linear equations in standard form; and systems of two linear equations with two variables, including solution by graphing, substitution, and elimination. Unit 2 Matrix Methods Develops student understanding of matrices and ability to use matrices to represent and solve problems in a variety of real-world and mathematical settings. Topics include: Representing two-dimensional figures and modeling situations with coordinates, including computer-generated graphics; distance in the coordinate plane, midpoint of a segment, and slope; coordinate and matrix models of rigid transformations (translations, rotations, and line reflections), of size transformations, and of similarity transformations; animation effects. Unit 4 Regression and Correlation Develops student understanding of the characteristics and interpretation of the least squares regression equation and of the use of correlation to measure the strength of the linear association between two variables. Nonlinear Functions and Equations Introduces function notation, reviews and extends student ability to construct and reason with functions that model parabolic shapes and other quadratic relationships in science and economics, with special emphasis on formal symbolic reasoning methods, and introduces common logarithms and algebraic methods for solving exponential equations. Trigonometric Methods Develops student understanding of trigonometric functions and the ability to use trigonometric methods to solve triangulation and indirect measurement problems. Topics include: Sine, cosine, and tangent functions of measures of angles in standard position in a coordinate plane and in a right triangle; indirect measurement; analysis of variable-sided triangle mechanisms; Law of Sines and Law of Cosines.
Shipping prices may be approximate. Please verify cost before checkout. About the book: In real-world problems related to finance, business, and management, mathematicians and economists frequently encounter optimization problems. In this classic book, George Dantzig looks at a wealth of examples and develops linear programming methods for their solutions. He begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them. Treatments of the price concept, the transportation problem, and matrix methods are also given, and key mathematical concepts such as the properties of convex sets and linear vector spaces are covered. George Dantzig is properly acclaimed as the "father of linear programming." Linear programming is a mathematical technique used to optimize a situation. It can be used to minimize traffic congestion or to maximize the scheduling of airline flights. He formulated its basic theoretical model and discovered its underlying computational algorithm, the "simplex method," in a pathbreaking memorandum published by the United States Air Force in early 1948. Linear Programming and Extensions provides an extraordinary account of the subsequent development of his subject, including research in mathematical theory, computation, economic analysis, and applications to industrial problems. Dantzig first achieved success as a statistics graduate student at the University of California, Berkeley. One day he arrived for a class after it had begun, and assumed the two problems on the board were assigned for homework. When he handed in the solutions, he apologized to his professor, Jerzy Neyman, for their being late but explained that he had found the problems harder than usual. About six weeks later, Neyman excitedly told Dantzig, "I've just written an introduction to one of your papers. Read it so I can send it out right away for publication." Dantzig had no idea what he was talking about. He later learned that the "homework" problems had in fact been two famous unsolved problems in statistics691080003 Publisher: Princeton University Press, 1963 Usually ships in 1-2 business days Hardcover, ISBN 0691080003 Publisher: Princeton University Press,1080003 Publisher: Princeton University Press691080003 Publisher: Princeton University Press 66 # Bookseller Notes Price 1. Elderberrybooks via United States Hardcover, ISBN 0691080003 Publisher: Princeton University Press, 1963 Used - Good. B000NW5WH6 No DJ. Cloth hardcover has edge and corner wear. Name written on first page and it is separating from front cover. Hardcover, ISBN 0691080003 Publisher: Princeton University Press, 1963 hardcover Edition: Condition: Very Good Description: Previous owner's name on the first free end paper. Fast shipping with free delivery confirmation. Hardcover, ISBN 0691080003 Publisher: Princeton University Press, 1963 Used - Good, Usually ships in 1-2 business days, Lacks dust jacket, name inked out on the end page, light edge-wear, the text is clean and the binding is secure. Hardcover, ISBN 0691080003 Publisher: Princeton University Press, 1963 cloth Edition: First Edition Condition: Very Good Description: Vg/vg first edition. Dust jacket has some wear on folds and at spine, has shallow price clippings. Very light foxing to page edges. 625p. Hardcover, ISBN 0691080003 Publisher: Princeton University Press, 1963 Used - Very Good. 0691080003 Publisher: Princeton University PressDate of Publication: 1963Binding: hard coverEdition: Condition: Very Good/Very GoodDescription: Very nice clean used book with no marks. DJ has light edge/corner wear but no serious splits. Hardcover, ISBN 0691080003 Publisher: Princeton University Press, 1963 Used - Very Good, Usually ships in 1-2 business days, Publisher: Princeton University PressDate of Publication: 1963Binding: hard coverEdition: Condition: Very Good/Very GoodDescription: Very nice clean used book with no marks. DJ has light edge/corner wear but no serious splits.
Provides a number of additional challenging problems for students to solve that are drawn from real-world applications. Problems are keyed to each chapter and are designed to highlight and emphasize key concepts. More editions of Critical Thinking Workbook Student Edition for use with Elementary Statistics: A Brief Version: "Elementary Statistics: A Brief Version", is a shorter version of the popular text "Elementary Statistics: A Step by Step Approach". This softcover edition includes all the features of the longer book, but it is designed for a course in which the time available limits the number of topics covered. It MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing technologies commonly used in such courses. "Elementary Statistics: A Step by Step Approach" Minitab, and the TI-83 Plus and TI 84-Plus graphing calculators, computing technologies commonly used in such courses.
i really need your help in this. i have a project due on the 1st June which is on "linear algebra application". I am basically a business student and we are given a project to apply linear systems, echelon forms,determinants, subspaces, inverses,null spaces, column spaces, linear transformation(anything that comes under linear algebra)on business. How can we make use of the knowledge of linear algebra while being in business? What can we make out of it? What products can we make? What ideas can we generate? basically how can it be used in anything related to business. it's simply answering the question of WHY DO WE HAVE TO STUDY LINEAR ALGEBRA WHEN WE ARE IN A BUSINESS SCHOOL? please help me with this.. i have no time left.. PLZ PLZ HELP ME!!
Students will solve problems by writing and solving polynomial equations. 10: Quadratic Equations and Functions: Students will be able to graph and solve quadratic equations while comparing linear, exponential, and quadratic models. 13: Probability and Data Analysis: Students will be able to fi...
How to order your own hardcover copy Wouldn't you rather have a bound book instead of 640 loose pages? Your laser printer will thank you! Order from Amazon.com. Chapter 30: Complex Numbers Complex numbers are an extension of the ordinary numbers used in everyday math. They have the unique property of representing and manipulating two variables as a single quantity. This fits very naturally with Fourier analysis, where the frequency domain is composed of two signals, the real and the imaginary parts. Complex numbers shorten the equations used in DSP, and enable techniques that are difficult or impossible with real numbers alone. For instance, the Fast Fourier Transform is based on complex numbers. Unfortunately, complex techniques are very mathematical, and it requires a great deal of study and practice to use them effectively. Many scientists and engineers regard complex techniques as the dividing line between DSP as a tool, and DSP as a career. In this chapter, we look at the mathematics of complex numbers, and elementary ways of using them in science and engineering. The following three chapters discuss important techniques based on complex numbers: the complex Fourier transform, the Laplace transform, and the z-transform. These complex transforms are the heart of theoretical DSP. Get ready, here comes the math!
$42State-of-the-art analysis of geological structures has become increasingly quantitative but traditionally, graphical methods are used in teaching. This innovative lab book provides a unified methodology for problem-solving in structural geology using linear algebra and computation. Assuming only limited mathematical training, the book begins with classic orientation problems and progresses to more fundamental topics of stress, strain and error propagation. It introduces linear algebra methods as the foundation for understanding vectors and tensors, and demonstrates the application of geometry and kinematics in geoscience without requiring students to take a supplementary mathematics course. All algorithms are illustrated with a suite of online MATLAB functions, allowing users to modify the code to solve their own structural problems. Containing 20 worked examples and over 60 exercises, this is the ideal lab book for advanced undergraduates or beginning graduate students. It will also provide professional structural geologists with a valuable reference and refresher for calculations. Relates basic topics such as fold geometry to more complicated concepts such as tensors, allowing students to develop an intuitive feel for vectors and tensors before applying them in the context of stress and strain Provides a concise review of error analysis, which is an increasingly important topic for structural analysis Supported by a full suite of MATLAB codes available online, which can be modified and developed for solving other structural problems Reviews & endorsements "I highly recommend this book to all structural geology students and practitioners, as well as to earth scientists from a wide range of fields, who will benefit from this clear introduction of the principles and application of linear algebra in the analysis of commonly encountered vector and tensor quantities." Roland Bürgmann, Mathematical Geosciences "The book is suitable for numerate researchers and advanced undergraduates who are reasonably comfortable with mathematics … it is essential in the twenty-first century that we have numerate geoscientists trained in quantitative techniques of structural geology … The authors take care to describe the basics of tensor algebra as well as its application; this book is a solid foundation for understanding the mathematical analysis of how the Earth deforms." John Wheeler, American Mineralogist
Thus, I can not only explain linear algebra to students, but also provide examples of how it is used in real-life applications. I am a PhD Computational Scientist, and in the course of my research I have written hundreds of Perl programs to solve computational problems. I have an excellent knowledge of the language, and I have experience training graduate students to program in Perl.
This book presents a complete and accurate study of algebraic circuits, digital circuits whose performance can be associated with any algebraic structure. The authors distinguish between basic algebraic circuits, such as Linear Feedback Shift Registers (LFSRs) and?cellular automata and algebraic circuits, such as finite fields or Galois fields.The... more... In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the... more... A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music... more... The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems and examples. ?The aim is to collect material that will help mathematicians already working or wishing to work... more... Ever since Lorensen and Cline published their paper on the Marching Cubes algorithm, isosurfaces have been a standard technique for the visualization of 3D volumetric data. Yet there is no book exclusively devoted to isosurfaces. Isosurfaces: Geometry, Topology, and Algorithms represents the first book to focus on basic algorithms for isosurface... more... Trigonometry: A Complete Introduction is the most comprehensive yet easy-to-use introduction to Trigonometry. Written by a leading expert, this book will help you if you are studying for an important exam or essay, or if you simply want to improve your knowledge. The book covers all areas of trigonometry including the theory and equations of tangent,... more... Presented in an easy-to-follow, step-by-step tutorial format, Puppet 3.0 Beginner?s Guide will lead you through the basics of setting up your Puppet server with plenty of screenshots and real-world solutions.This book is written for system administrators and developers, and anyone else who needs to manage computer systems. You will need to be able... more... This book is concerned with one of the most fundamental questions of mathematics: the relationship between algebraic formulas and geometric images.At one of the first international mathematical congresses (in Paris in 1900), Hilbert stated a special case of this question in the form of his 16th problem (from his list of 23 problems left over from the... more... This welcome boon for students of algebraic topology cuts a much-needed central path between other texts whose treatment of the classification theorem for compact surfaces is either too formalized and complex for those without detailed background knowledge, or too informal to afford students a comprehensive insight into the subject. Its dedicated,... more...
Mathematics Finite Mathematics 1 Semester / 1 Credit(s) This course expands students' mathematical reasoning and problem-solving skills as they cover topics such as mathematics of voting, weighted voting systems, fair division, apportionment, Euler circuits, Hamilton circuits, mathematics of networks, and game theory. The course will encourage students to make mathematical connections from the classroom to the world after high school, while learning the importance of mathematics in everyday life. This course is offered as an addition to Pre-Calculus/Trigonometry, not a replacement. A SCIENTIFIC CALCULATOR IS REQUIRED. Probability & Statistics 1 Semester / 1 Credit(s) This course introduces and examines the statistical topics that are applied during the decision-making process. Topics include: descriptive statistics, probability, and statistical inference. Techniques investigated include: data collection through experiments or surveys, data organization, sampling theory and making inferences from samples. Computers are used for data analysis and data presentation. This course should not be taken as a replacement for Pre-Calculus/Trigonometry in a college preparatory course of study. A SCIENTIFIC CALCULATOR IS REQUIRED. Honors Probability & Statistics 1 Semester / 1 Credit(s) This course introduces students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. This course is required for students taking MAT 230 in the spring. Students complete a rigorous study of the following concepts: describing patterns and departures from patterns, planning and conducting a statistical study, exploring random phenomena using probability and simulation, and estimating population parameters and testing hypotheses. A GRAPHING CALCULATOR IS REQUIRED. AP Calculus 2 Semesters / 2 Credit(s) This course is intended for students who have a thorough knowledge of college preparatory mathematics. It covers both the theoretical basis for and applications of differentiation and integration. Concepts and problems are approached graphically, numerically, analytically and verbally. All students enrolled in this course will take the AP Calculus (AB) Exam. A GRAPHING CALCULATOR IS REQUIRED. Honors Pre-Calculus/Trigonometry 2 Semesters / 2 Credit(s) This course covers the same topics as Pre-Calculus/Trigonometry listed above. Greater emphasis is placed on applications and developing the depth of understanding and skills necessary for success in AP Calculus. This course is required for students who plan to take AP Calculus. A GRAPHING CALCULATOR IS REQUIRED. This course expands and develops the topics learned in Honors Algebra I. Content areas include the topics listed for Algebra II with greater emphasis on preparation for upper level mathematics content. The course is required for students who plan to take AP Calculus, and it is recommended that this course be taken at the same time as Honors Geometry unless Honors Geometry was taken as a freshman. A GRAPHING CALCULATOR IS REQUIRED. Algebra II 2 Semesters / 2 Credit(s) This course further develops the topics learned in Algebra I with extensive work on learning to graph equations and inequalities in the Cartesian coordinate system. Topics include: relations and functions, systems of equations and inequalities, conic sections, polynomials, algebraic fractions, logarithmic and exponential functions, sequences and series, and counting principles and probability. A GRAPHING CALCULATOR IS REQUIRED. Honors Geometry 2 Semesters / 2 Credit(s) This course covers the same topics as Geometry, but with greater emphasis on complex direct deductive proof and indirect proof and on utilization of more advanced algebraic techniques. Content is extended to include topics such as analytic geometry and the interrelationships of inscribed polyhedra. A GRAPHING CALCULATOR IS REQUIRED. Geometry (1&2) 1 Semester / 2 Credit(s) This Geometry course is designed for those students who did not receive a C- average or better in Geometry. This course will meet everyday. Geometry 2 Semesters / 2 Credit(s) The purpose of Geometry is to use logical thought processes to develop spatial skills. Students work with figures in one, two- and three-dimensional Euclidean space. The interrelationships of the properties of figures are studied through visualization, using computer drawing programs and constructions, as well as through formal proof and algebraic applications. A GRAPHING CALCULATOR IS REQUIRED. Honors Algebra I 2 Semesters / 2 Credit(s) The same topics as in Algebra I are covered with more emphasis on problem solving and critical thinking skills in order to challenge the mathematically talented student. Projects are incorporated into the lessons for the purpose of applying the mathematical concepts. A GRAPHING CALCULATOR IS REQUIRED. Algebra I (1&2) 1 Semester / 2 Credit(s) This course is designed for those students who did not receive a B- average or better in Algebra I or did not pass the ISTEP+ Algebra I Graduation Exam. This course will meet everyday. Algebra Enrichment is a mathematics support course for Algebra I. The course provides students with additional time to build the foundations necessary for high school math courses, while concurrently having access to rigorous, grade-level appropriate courses. The five critical areas of Algebra Enrichment align with the critical areas of Algebra I: Relationships between Quantities and Reasoning with Equations; Linear and Exponential Relationships; Descriptive Statistics; Expressions and Equations; and Quadratic Functions and Modeling. However, whereas Algebra I contains exclusively grade-level content, Algebra Enrichment combines standards from high school courses with foundational standards from the middle grades. This course counts as a Mathematics Course for the General Diploma only or as an Elective for the Core 40, Core 40 with Academic Honors and Core 40 with Technical Honors diplomas. Algebra Enrichment is designed as a support course for Algebra I. As such, a student taking Algebra Enrichment should also be enrolled in Algebra I during the same academic year. A GRAPHING CALCULATOR IS REQUIRED. EventsAug13 First Day of School 7:55 AM - 2
A Level Maths A Mathematics A Level can lead to any number of educational and career opportunities. Learning maths is so valuable because mathematics forms the basis of so many systems and processes. Many occupational fields require advanced study of maths, making the A Level Mathematics course one of the most versatile you can study. The A Level Mathematics goes beyond the basics of addition and subtraction into the worlds of algebra, geometry, trigonometry, mathematical modelling and more. The A Level Maths course will develop your existing knowledge of mathematics into a range of more advanced maths study areas. The distance learning course will introduce you to the possibilities offered by algebra, trigonometry, geometry, differentiation and integration. Your maths study will then allow you to take your knowledge into the world of physics, examining the ways in which maths influences a number of processes. You'll study mathematical modelling, kinematics, physical forces and momentum, Newton's laws of motion, circular motion and the application of differential equations. Because learning maths is integral to so many different fields of study and work, your Mathematics A Level will be a hugely versatile qualification and an asset in whatever you go on to do. An A Level in Mathematics can lead to university studies and a wide variety of careers, from science-related roles to business and teaching. If you want to enter or progress in employment, you'll find your Mathematics A Level will demonstrate to employers that you have the ability to commit to learning, and have acquired good reasoning and analytical skills - essential in practically every walk of life
Discrete Mathematics For Computer Science 9781930190863 ISBN: 1930190867 Pub Date: 2005 Publisher: Key College Publishing Summary: "Discrete Mathematics for Computer Science" is the perfect text to combine the fields of mathematics and computer science. Written by leading academics in the field of computer science, readers will gain the skills needed to write and understand the concept of proof. This text teaches all the math, with the exception of linear algebra, that is needed to succeed in computer science. The book explores the topics of bas...ic combinatorics, number and graph theory, logic and proof techniques, and many more. Appropriate for large or small class sizes or self study for the motivated professional reader. Assumes familiarity with data structures. Early treatment of number theory and combinatorics allow readers to explore RSA encryption early and also to encourage them to use their knowledge of hashing and trees (from CS2) before those topics are covered in this course. Bogart, Kenneth P. is the author of Discrete Mathematics For Computer Science, published 2005 under ISBN 9781930190863 and 1930190867. Two hundred twenty six Discrete Mathematics For Computer Science textbooks are available for sale on ValoreBooks.com, ten used from the cheapest price of $8.59, or buy new starting at $15.35.[read more]
Editorial Reviews From the Publisher From the reviews: "This is an introductory book on geometry, easy to read, written in an engaging style. The author's goal is … to increase one's overall understanding and appreciation of the subject. … Along the way, he presents elegant proofs of well-known theorems … . The advantage of the author's approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006) "The pillars of the title are … Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. … The writing style is both student-friendly and deeply informed. The pleasing brevity of the book … makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors … . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e) "The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. … Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. … The book can be recommended to be used in undergraduate courses on geometry … ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006) "Any new mathematics textbook by John Stillwell is worth a serious look. Stillwell is the prolific author of more than half a dozen textbooks … . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an in-depth, semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007) "The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. … the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007) "This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. … The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007) "The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. … The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. … The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view." (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15
The Math Center is a resource center that provides individual and group assistance in mathematics. The Math Center also facilitates Directed Learning Activities. Faculty instructors, instructional assistants, and Student tutors are available to assist students with challenging topics, answer questions, encourage understanding, and provide support for all math students. Students also have access to textbooks, graphing calculators, instructional videos, and computer programs. Math Center's Goals To help some students further develop basic skills in mathematics and keep them coming to school. To assist other students to further sharpen their pre-existing math skills and advance through math courses. To guide all students toward success in math and encourage them to excel through their scholastic endeavors and beyond. Math 81 requires that students spend one hour per week in the Math Center working on a particular DLA. Some of these activities will cover typically challenging math topics and will be completed through the interaction with the Math Center Instructors and Staff. Others consist of developing a student's study skills and will be completed using one of the Math Center computers. It is vital that students log in to and out of the DLA attendance computer in order to receive DLA credit for their course. WARNING! 1-HOUR-PER-WEEK DLA REQUIREMENT You are REQUIRED to attend at least 1 hour in the Math Center during the 1st week of classes AND at least 1 hour during the 2nd week. FAILURE TO ATTEND EITHER OF THESE WEEKS WILL RESULT IN YOU BEING DROPPED FROM THE CLASS. * REPEAT: You will be DROPPED as of Monday, Feb 24th if you have failed to attend 1 DLA hour each weekduring the first 2 weeks of school. The Math Center Management does not drop students. Your instructor is required to drop their students who do not complete 1-hour-per-week DLA requirement. Classes listed below. How much does it cost? It is completely free. Where? The session will be held at the SAC Math Center L-204. Publishing Page Content 2: Start the Semester Off Right! For students who are entering into Math 70, 80/81, 160, 170, 180, 185 or Bridge to Engineering Math 78 160-Trigonometry, Math 170-PreCalculus, or Math 180 Calculus 1.
ANSWER KEY EXAM 114 1. Answer: B Math explains the logic of and relationship between numbers. It is used every day in countless ways. In order to minimize potential math phobia, teachers need to make the subject relevant to the students' lives and use examples with which they are familiar and that make sense to them. In order to do that, learning the basics is critical because all math concepts are built on addition, division, fractions and shapes. All mathematical relationships flow from these concepts. It is imperative students understand one concept before moving on to the next. If they fail to grasp the basics, students become confused as they progress to higher levels because they are unable to apply applicable background knowledge when introduced to geometry, algebra, probability and statistics. 2. Answer: C Mathematics is a formal science of structure, order and relationships and is considered the basic language and foundation of all the other sciences. It evolved from counting, measuring and describing shapes. Some areas and their definitions: Arithmetic: A system to count numbers using addition, subtraction, multiplication and division Algebra: An abstract form of arithmetic using symbols to represent numbers Geometry: The relationship of points, lines, angles, surfaces and solids Probability: The chance random events will occur Statistics: The collection, organization and analysis of data Trigonometry: The relationship of the sides and angles of triangles Calculus: The limits, differentiation and integration of the functions of variables 3. Answer: B There are basic concepts in algebra that allow generalizations about "unknowns." Patterns and functions represent change and relationships. Repeating patterns show the same unit over and over again. In growth patterns, each unit is dependent upon the one before it as well as its position in the pattern. The function is the relationship between values, i.e., the second depends on the first. Once functional relationships are understood, symbols are used as an abstract stand-in for the relationships. Equivalence and balance are critical concepts in understanding algebraic equations. The equal sign represents some type of relationship between the numbers and symbols on each side of the sign. If a calculation is performed on one side, the same calculation must be performed on the other side. Each side is equal and they must balance. 4. Answer: B Adolescents come to school with background knowledge and a basic understanding of how things work. They have reached conclusions based on their perception of the physical world and what they learned in previous classes. A wise teacher uses students' knowledge and natural curiosity when introducing and explaining complicated scientific concepts. He/she builds on ideas already known and corrects any misconceptions. Teachers should explain that science has a history. Students need to be familiar with the socio-economic environment in which a theory was introduced in order to truly understand why something did or did not work, why it may have been proven wrong or why a better way was discovered with later experimentation. 5. Answer: D Natural Science is concerned with the natural world. Social Science studies human behavior. Both are based on empirical evidence, which is observable data that can be verified by other scientists working in similar situations under the same conditions. Formal Science is the systematic study of a specific area. It is essential to developing hypotheses, theories and laws used in other scientific disciplines, i.e., describing how things work (natural science) and how people think and why they do what they do individually and as a society (social sciences). It is based on a priori evidence, which proceeds from a theory or assumption rather than observable phenomena. Applied Science is using the results of scientific research in any of the natural, social and formal sciences and adapting it to address human needs. 6. Answer: A Scientific Method is a set of procedures used to study natural phenomena. It provides guidelines with which to pose questions, analyze data and reach conclusions. It is used to investigate an event, gain knowledge or correct earlier conclusions about the occurrence and integrate the new information with previously learned data. Researchers pose hypotheses and design experiments and studies to test them. The process must be objective, documented and shared with other researchers, so the results can be verified by replicating the study in similar situations under the same conditions. Scientific method rarely follows a predictable path. The testing of one hypothesis usually leads to other questions, which leads to the formation of other hypotheses. 7. Answer: C The steps described are not necessarily used in exactly the same way in all sciences. Sometimes they happen at the same time or in a different order and may be repeated during the course of the study, but should be applied with intelligence, imagination and creativity. The following sequence is the one used most of the time: A question is asked about a natural phenomenon. It should be stated in specific language to focus the inquiry. The subject is thoroughly researched. Previous test results are studied. It is important to understand what the earlier experiment(s) proved or disproved. With information gleaned from researching the topic, a hypothesis is formed about a cause or effect of the event or its relationship to other occurrences. An experiment is designed and conducted to test the hypothesis and gather information. The resulting data is analyzed to determine if they support or refute the hypothesis. 8. Answer: C Life science or biology is the study of living organisms, their structure, function, growth, origin, evolution and distribution. The word biology is Greek. "Bio" means life. "Logos" means speech. "Biology" literally means, "to talk about life." This science studies how living things began, divides them into species, describes what they do and how they interact with and relate to each other and the rest of the natural world. The disciplines in the life sciences are grouped by the organisms they study: Botany studies plants; zoology studies animals and microbiology studies microorganisms. These groups are further divided into smaller, specialized categories based on the level at which they are studied and the methods used to study them, i.e. biochemistry studies the chemistry of life while ecology studies how organisms interrelate in the natural world. Applied fields of the life sciences such as medicine and genetic research combine multiple specialized categories. 9. Answer: D Cell Theory: The cell is the basic building block of all living things. It is the smallest unit of life able to function on its own. There are two kinds of cells: Prokaryotic which are present only in bacteria and eukaryotic found in all other life forms. New cells form by dividing from existing cells. Evolution: As a result of natural selection and changes in the gene pool (genetic drift), inherited traits morph from one generation to the next. Gene Theory: The traits of all living organisms are encoded in their DNA; the chromosome component that carries genetic information. Biochemical characteristics are capable of adapting to changes in the environment, but the only way these adaptations can be transferred to the genes is through evolution. Homeostasis: A self-regulating, physiological process that keeps biological systems stable and in proper balance internally, no matter what is happening in the external environment. 10. Answer: C The U.S. Department of Education established criteria for testing comprehension of math and science concepts using recommendations from the National Assessment of Educational Progress. Students in both disciplines are required to not only know facts but also need to be able to integrate those facts into previously acquired information by using critical thinking skills developed through studying these subjects. In other words, students need to be able to use the facts in practical applications found in the real world. The assessments developed by educators, curriculum specialists and the business community emphasize the importance of assessing students' ability to reason, understand concepts, solve problems, evaluate results and communicate knowledge of the subject matter. The tests attempt to measure whether students can take cognitive skills learned in math and science, apply them in other disciplines and use them outside of school in meaningful ways.
College Algebra with Modeling and Visualization - 4th edition Summary: Gary Rockswold teaches algebra in context, answering the question, ldquo;Why am I learning this?rdquo; By experiencing math through applications, students see how it fits into their lives, and they become motivated to succeed. Rockswoldrsquo;s focus on conceptual understanding helps students make connections between the concepts and as a result, students see the bigger picture of math and are prepared for future courses. Introduction to Functions and Graphs; Linear Functions and E...show morequations; Quadratic Functions and Equations; More Nonlinear Functions and Equations; Exponential and Logarithmic Functions; Trigonometric Functions; Trigonometric Identities and Equations; Further Topics in Trigonometry; Systems of Equations and Inequalities; Conic Sections; Further Topics in Algebra For all readers interested in college algebra. ...show less Dr. Gary Rockswold has been teaching mathematics for 25 years at all levels from seventh grade to graduate school, including junior high and high school students, talented youth, vocational, undergraduate and graduate students, and adult education classes. He is currently employed at Minnesota State University, Mankato, where he is a full professor of mathematics and the chair of the mathematics department. He graduated with majors in mathematics and physics from St. Olaf College in Northfield, Minnesota, where he was elected to Phi Beta Kappa. He received his Ph.D. in applied mathematics from Iowa State University. He has an interdisciplinary background and has also taught physical science, astronomy, and computer science. Outside of mathematics, he enjoys spending time with his wife and two children
Buy Article: Abstract: Designing an optimal Norman window is a standard calculus exercise. How much more difficult (or interesting) is its generalization to deploying multiple semicircles along the head (or along head and sill, or head and jambs)? What if we use shapes beside semi-circles? As the number of copies of the shape increases and the optimal Norman windows approach a rectangular shape, what proportions arise? How does the perimeter of the limiting rectangle compare to the limit of the perimeters? These questions provide challenging optimization problems for students and the graphical depiction of the geometry of these window sequences illustrates more vividly than sequences of numbers, the concept of limit. The College Mathematics Journal is designed to enhance classroom learning and stimulate thinking regarding undergraduate mathematics. CMJ publishes articles, short Classroom Capsules, problems, solutions, media reviews and other pieces. All are aimed at the college mathematics curriculum with emphasis on topics taught in the first two years.
Search Results About this title: This leading mathematics text for elementary and middle school educators helps readers quickly develop a true understanding of mathematical concepts. It integrates rich problem-solving strategies with relevant topics and extensive opportunities for hands-on
to view additional materials from 7 00:00:13 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:22 at ocw.mit.edu. 9 00:00:22 --> 00:00:28 PROFESSOR STRANG: Just to give an overview in three lines: the 10 00:00:28 --> 00:00:32 text is the book of that name, Computational Science 11 00:00:32 --> 00:00:33 and Engineering. 12 00:00:33 --> 00:00:38 That was completed just last year, so it really ties 13 00:00:38 --> 00:00:41 pretty well with the course. 14 00:00:41 --> 00:00:43 I don't cover everything in the book, by all means. 15 00:00:43 --> 00:00:47 And I don't, certainly, don't stand here and read the book. 16 00:00:47 --> 00:00:50 That would be no good. 17 00:00:50 --> 00:00:55 But you'll be able, if you miss a class -- well, 18 00:00:55 --> 00:00:56 don't miss a class. 19 00:00:56 --> 00:01:01 But if you miss a class, you'll be able, probably, 20 00:01:01 --> 00:01:05 to see roughly what we did. 21 00:01:05 --> 00:01:08 OK, so the first part of the semester is applied 22 00:01:08 --> 00:01:10 linear algebra. 23 00:01:10 --> 00:01:13 And I don't know how many of you have had a linear algebra 24 00:01:13 --> 00:01:16 course, and that's why I thought I would start 25 00:01:16 --> 00:01:19 with a quick review. 26 00:01:19 --> 00:01:23 And you'll catch on. 27 00:01:23 --> 00:01:26 I want matrices to come to life, actually. 28 00:01:26 --> 00:01:31 You know, instead of just being a four by four array of 29 00:01:31 --> 00:01:34 numbers, there are four by four, or n by n or m by n 30 00:01:34 --> 00:01:36 array of special numbers. 31 00:01:36 --> 00:01:38 They have a meaning. 32 00:01:38 --> 00:01:41 When they multiply a vector, they do something. 33 00:01:41 --> 00:01:47 And so it's just part of this first step is just, like, 34 00:01:47 --> 00:01:50 getting to recognize, what's that matrix doing? 35 00:01:50 --> 00:01:52 Where does it come from? 36 00:01:52 --> 00:01:53 What are its properties? 37 00:01:53 --> 00:01:57 So that's a theme at the start. 38 00:01:57 --> 00:02:04 Then differential equations, like Laplace's equation, 39 00:02:04 --> 00:02:06 are beautiful examples. 40 00:02:06 --> 00:02:11 So here we get, especially, to numerical methods; 41 00:02:11 --> 00:02:14 finite differences, finite elements, above all. 42 00:02:14 --> 00:02:17 So I think in this class you'll really see how finite elements 43 00:02:17 --> 00:02:20 work, and other ideas. 44 00:02:20 --> 00:02:21 All sorts of ideas. 45 00:02:21 --> 00:02:25 And then the last part of the course is about Fourier. 46 00:02:25 --> 00:02:29 That's Fourier series, that you may have seen, and 47 00:02:29 --> 00:02:30 Fourier integrals. 48 00:02:30 --> 00:02:34 But also, highly important, Discrete Fourier 49 00:02:34 --> 00:02:36 Transform, DFT. 50 00:02:36 --> 00:02:40 That's a fundamental step for understanding what 51 00:02:40 --> 00:02:42 a signal contains. 52 00:02:42 --> 00:02:46 Yeah, so that's great stuff, Fourier. 53 00:02:46 --> 00:02:52 OK, what else should I say before I start? 54 00:02:52 --> 00:02:56 I said this was my favorite course, and maybe I 55 00:02:56 --> 00:03:01 elaborate a little. 56 00:03:01 --> 00:03:06 Well, I think what I want to say is that I really feel my 57 00:03:06 --> 00:03:12 life is here to teach you and not to grade you. 58 00:03:12 --> 00:03:15 I'm not going to spend this semester worrying about 59 00:03:15 --> 00:03:18 grades, and please don't. 60 00:03:18 --> 00:03:19 They come out fine. 61 00:03:19 --> 00:03:22 We've got lots to learn. 62 00:03:22 --> 00:03:26 And I'll do my very best to explain it clearly. 63 00:03:26 --> 00:03:30 And I know you'll do your best. 64 00:03:30 --> 00:03:31 I know from experience. 65 00:03:31 --> 00:03:36 This class goes for it and does it right. 66 00:03:36 --> 00:03:40 So that's what makes it so good. 67 00:03:40 --> 00:03:41 OK. 68 00:03:41 --> 00:03:46 Homeworks, by the way, well, the first homework will simply 69 00:03:46 --> 00:03:50 be a way to get a grade list, a list of everybody 70 00:03:50 --> 00:03:52 taking the course. 71 00:03:52 --> 00:03:55 They won't be graded in great detail. 72 00:03:55 --> 00:03:59 Too large a class. 73 00:03:59 --> 00:04:03 And you're allowed to talk to each other about homework. 74 00:04:03 --> 00:04:05 So homework is not an exam at all. 75 00:04:05 --> 00:04:09 So let me just leave any discussion of exams and 76 00:04:09 --> 00:04:12 grades for the future. 77 00:04:12 --> 00:04:14 I'll tell you, you'll see how informally the 78 00:04:14 --> 00:04:18 first homework will be. 79 00:04:18 --> 00:04:21 And I hope it'll go up on the website. 80 00:04:21 --> 00:04:23 The first homework will be for Monday. 81 00:04:23 --> 00:04:29 So it's a bit early, but it's pretty open-ended. 82 00:04:29 --> 00:04:33 If you could take three problems from 1.1, the first 83 00:04:33 --> 00:04:38 section of the book, any three, and any three problems from 84 00:04:38 --> 00:04:45 1.2, and print your name on the homework -- because we're going 85 00:04:45 --> 00:04:48 to use that to create the grade list -- I'll 86 00:04:48 --> 00:04:50 be completely happy. 87 00:04:50 --> 00:04:52 Well, especially if you get them right and do them 88 00:04:52 --> 00:04:53 neatly and so on. 89 00:04:53 --> 00:04:59 But actually we won't know. 90 00:04:59 --> 00:05:02 So that's for Monday. 91 00:05:02 --> 00:05:03 OK. 92 00:05:03 --> 00:05:05 And we'll talk more about it. 93 00:05:05 --> 00:05:11 I'll announce the TA on the website and the TA hours, the 94 00:05:11 --> 00:05:12 office hours, and everything. 95 00:05:12 --> 00:05:17 There'll be a Friday afternoon office hour, because homeworks 96 00:05:17 --> 00:05:20 will typically come Monday. 97 00:05:20 --> 00:05:20 OK. 98 00:05:20 --> 00:05:27 Questions about the course before I just start? 99 00:05:27 --> 00:05:30 OK. 100 00:05:30 --> 00:05:31 Another time for questions, too. 101 00:05:31 --> 00:05:41 OK, so can we just start with that matrix? 102 00:05:41 --> 00:05:45 So I said about matrices, I'm interested in their properties. 103 00:05:45 --> 00:05:47 Like, I'm going to ask you about that. 104 00:05:47 --> 00:05:51 And then, I'm interested in their meaning. 105 00:05:51 --> 00:05:53 Where do they come from? 106 00:05:53 --> 00:05:56 You know, why that matrix instead of some other? 107 00:05:56 --> 00:06:01 And then, the numerical part is how do we deal with them? 108 00:06:01 --> 00:06:05 How do we solve a linear system with that coefficient matrix? 109 00:06:05 --> 00:06:07 What can we say about the solution? 110 00:06:07 --> 00:06:09 So the purpose. 111 00:06:09 --> 00:06:10 Right. 112 00:06:10 --> 00:06:15 OK, now help me out. 113 00:06:15 --> 00:06:18 So I guess my plan with the video taping is, whatever 114 00:06:18 --> 00:06:20 you say, I'll repeat. 115 00:06:20 --> 00:06:27 So say it as clearly as possible, and it's fantastic 116 00:06:27 --> 00:06:30 to have discussion, conversation here. 117 00:06:30 --> 00:06:34 So I'll just repeat it so that it safely gets on the tape. 118 00:06:34 --> 00:06:35 So tell me its properties. 119 00:06:35 --> 00:06:41 Tell me the first property that you notice about that matrix. 120 00:06:41 --> 00:06:41 Symmetric. 121 00:06:41 --> 00:06:42 Symmetric. 122 00:06:42 --> 00:06:44 Right. 123 00:06:44 --> 00:06:46 I could have slowed down a little and everybody probably 124 00:06:46 --> 00:06:48 would have said that at once. 125 00:06:48 --> 00:06:51 So that's a symmetric matrix. 126 00:06:51 --> 00:06:54 Now we might as well pick up some matrix notation. 127 00:06:54 --> 00:06:58 How do I express the fact that this a symmetric matrix? 128 00:06:58 --> 00:07:03 In simple matrix notation, I would say that K is 129 00:07:03 --> 00:07:07 the same as K transpose. 130 00:07:07 --> 00:07:11 The transpose, everybody knows, it comes from -- oh, I 131 00:07:11 --> 00:07:14 shouldn't say this -- flipping it across the diagonal. 132 00:07:14 --> 00:07:17 That's not a very "math" thing to do. 133 00:07:17 --> 00:07:21 But that's the way to visualize it. 134 00:07:21 --> 00:07:27 And let me use a capital T for transpose. 135 00:07:27 --> 00:07:30 So it's symmetric. 136 00:07:30 --> 00:07:31 Very important. 137 00:07:31 --> 00:07:32 Very, very important. 138 00:07:32 --> 00:07:34 That's the most important class of matrices, 139 00:07:34 --> 00:07:35 symmetric matrices. 140 00:07:35 --> 00:07:39 We'll see them all the time, because they come from 141 00:07:39 --> 00:07:41 equilibrium problems. 142 00:07:41 --> 00:07:44 They come from all sorts of -- they come everywhere 143 00:07:44 --> 00:07:47 in applications. 144 00:07:47 --> 00:07:49 And we will be doing applications. 145 00:07:49 --> 00:07:55 The first week or week and a half, you'll see pretty much 146 00:07:55 --> 00:08:00 discussion of matrices and the reasons, what their meaning is. 147 00:08:00 --> 00:08:03 And then we'll get to physical applications; 148 00:08:03 --> 00:08:05 mechanics and more. 149 00:08:05 --> 00:08:06 OK. 150 00:08:06 --> 00:08:08 All right. 151 00:08:08 --> 00:08:11 Now I'm looking for properties, other 152 00:08:11 --> 00:08:13 properties, of that matrix. 153 00:08:13 --> 00:08:18 Let me write "2" here so that you got a spot to put it. 154 00:08:18 --> 00:08:21 What are you going to tell me next about that matrix? 155 00:08:21 --> 00:08:22 Periodic. 156 00:08:22 --> 00:08:23 Well, okay. 157 00:08:23 --> 00:08:25 Actually, that's a good question. 158 00:08:25 --> 00:08:31 Let me write periodic down here. 159 00:08:31 --> 00:08:36 You're using that word, because somehow that pattern is 160 00:08:36 --> 00:08:37 suggesting something. 161 00:08:37 --> 00:08:43 But you'll see I have a little more to add before I would 162 00:08:43 --> 00:08:44 use the word periodic. 163 00:08:44 --> 00:08:47 So that's great to see that here. 164 00:08:47 --> 00:08:47 What else? 165 00:08:47 --> 00:08:50 Somebody else was going to say something. 166 00:08:50 --> 00:08:51 Please. 167 00:08:51 --> 00:08:52 Sparse! 168 00:08:52 --> 00:08:53 Oh, very good. 169 00:08:53 --> 00:08:54 Sparse. 170 00:08:54 --> 00:08:59 That's also an obvious property that you see from 171 00:08:59 --> 00:09:01 looking at the matrix. 172 00:09:01 --> 00:09:03 What does sparse mean? 173 00:09:03 --> 00:09:05 Mostly zeros. 174 00:09:05 --> 00:09:07 Well that isn't mostly zeros, I guess. 175 00:09:07 --> 00:09:11 I mean, that's got what, out of sixteen entries, 176 00:09:11 --> 00:09:13 it's got six zeros. 177 00:09:13 --> 00:09:14 That doesn't sound like sparse. 178 00:09:14 --> 00:09:19 But when I grow the matrix -- because this is 179 00:09:19 --> 00:09:21 just a four by four. 180 00:09:21 --> 00:09:24 I would even call this one K_4. 181 00:09:24 --> 00:09:30 When the matrix grows to 100 by 100, then you really 182 00:09:30 --> 00:09:31 see it as sparse. 183 00:09:31 --> 00:09:35 So if that matrix was 100 by 100, how many 184 00:09:35 --> 00:09:37 non-zeros would it have? 185 00:09:37 --> 00:09:44 So if n is 100, then the number of non-zeros -- wow, that's the 186 00:09:44 --> 00:09:46 first MATLAB command I've written. 187 00:09:46 --> 00:09:51 A number of non-zeros of K would be -- anybody 188 00:09:51 --> 00:09:53 know what it would be? 189 00:09:53 --> 00:10:00 I'm just asking to go up to five by five. 190 00:10:00 --> 00:10:03 I'm asking you to keep that pattern alive. 191 00:10:03 --> 00:10:08 Twos on the diagonal, minus ones above and below. 192 00:10:08 --> 00:10:13 So yeah, so 298, would it be? 193 00:10:13 --> 00:10:20 A hundred diagonal entries, 99 and 99, maybe 298? 194 00:10:20 --> 00:10:27 298 out of 100 by 100 would be what? 195 00:10:27 --> 00:10:29 It's been a long summer. 196 00:10:29 --> 00:10:32 Yeah, a lot of zeros. 197 00:10:32 --> 00:10:32 A lot. 198 00:10:32 --> 00:10:33 Right. 199 00:10:33 --> 00:10:37 Because the matrix has got what 100 x 100, 10,000 entries. 200 00:10:37 --> 00:10:39 Out of 10,000. 201 00:10:39 --> 00:10:42 So that's sparse. 202 00:10:42 --> 00:10:46 But we see those all the time, and fortunately we do. 203 00:10:46 --> 00:10:48 Because, of course, this matrix, or even 100 204 00:10:48 --> 00:10:53 by 100, we could deal with if it was dense. 205 00:10:53 --> 00:10:58 But 10,000, 100,000, or 1 million, which happens 206 00:10:58 --> 00:11:02 all the time now in scientific computation. 207 00:11:02 --> 00:11:05 A million by million dense matrix is not a nice 208 00:11:05 --> 00:11:07 thing to think about. 209 00:11:07 --> 00:11:13 A million by million matrix like this is a cinch. 210 00:11:13 --> 00:11:14 OK. 211 00:11:14 --> 00:11:15 So sparse. 212 00:11:15 --> 00:11:18 What else do you want to say? 213 00:11:18 --> 00:11:19 Toeplitz. 214 00:11:19 --> 00:11:22 Holy Moses. 215 00:11:22 --> 00:11:23 Exactly right. 216 00:11:23 --> 00:11:29 But I want to say, before I use that word, so that'll be 217 00:11:29 --> 00:11:30 my second MATLAB command. 218 00:11:30 --> 00:11:31 Thanks. 219 00:11:31 --> 00:11:33 Toeplitz. 220 00:11:33 --> 00:11:35 What's that mean? 221 00:11:35 --> 00:11:40 So this matrix has a property that we see 222 00:11:40 --> 00:11:46 right away, which is? 223 00:11:46 --> 00:11:51 I want to stay with Toeplitz but everybody tell me something 224 00:11:51 --> 00:11:54 more about properties of that matrix. 225 00:11:54 --> 00:11:56 Tridiagonal. 226 00:11:56 --> 00:12:02 Tridiagonal, so that's almost a special subcase of sparse. 227 00:12:02 --> 00:12:05 It has just three diagonals. 228 00:12:05 --> 00:12:08 Tridiagonal matrices are truly important. 229 00:12:08 --> 00:12:11 They come in all the time, we'll see that they come from 230 00:12:11 --> 00:12:14 second order differential equations, which are, thanks 231 00:12:14 --> 00:12:17 to Newton, the big ones. 232 00:12:17 --> 00:12:23 Ok, now it's more than tridiagonal and what more? 233 00:12:23 --> 00:12:26 So what further, we're getting deeper now. 234 00:12:26 --> 00:12:32 What patterns do you see beyond just tridiagonal, because 235 00:12:32 --> 00:12:35 tridiagonal would allow any numbers there but those are 236 00:12:35 --> 00:12:39 not, there's more of a pattern than just three 237 00:12:39 --> 00:12:42 diagonals, what is it? 238 00:12:42 --> 00:12:45 Those diagonals are constant. 239 00:12:45 --> 00:12:48 If I run down each of those three diagonals, 240 00:12:48 --> 00:12:50 I see the same number. 241 00:12:50 --> 00:12:53 Twos, minus ones, minus ones, and that's what 242 00:12:53 --> 00:12:55 the word Toeplitz means. 243 00:12:55 --> 00:13:05 Toeplitz is constant diagonal. 244 00:13:05 --> 00:13:05 Ok. 245 00:13:05 --> 00:13:09 And that kind of matrix is so important. 246 00:13:09 --> 00:13:18 It corresponds, yeah, if we were in EE, I would use the 247 00:13:18 --> 00:13:23 words time invariant filter, linear time invariant. 248 00:13:23 --> 00:13:27 So it's linear because we're dealing with a matrix. 249 00:13:27 --> 00:13:31 And it's time invariant, shift invariant. 250 00:13:31 --> 00:13:36 I just use all these equivalent words to mean that we're seeing 251 00:13:36 --> 00:13:41 the same thing row by row, except of course, at shall 252 00:13:41 --> 00:13:44 I call that the boundary? 253 00:13:44 --> 00:13:46 That's like, the end of the system and this is like the 254 00:13:46 --> 00:13:51 other end and there it's chopped off. 255 00:13:51 --> 00:13:56 But if it was ten by ten I would see that row eight times. 256 00:13:56 --> 00:13:58 100 by 100 I'd see it 98 times. 257 00:13:58 --> 00:14:05 So it's constant diagonals and the guy who first studied 258 00:14:05 --> 00:14:08 that was Toeplitz. 259 00:14:08 --> 00:14:14 And we wouldn't need that great historical information except 260 00:14:14 --> 00:14:18 that MATLAB created a command to create that matrix. 261 00:14:18 --> 00:14:25 K, MATLAB is all set to create Toeplitz matrices. 262 00:14:25 --> 00:14:30 Yeah, so I'll have to put what MATLAB would put. 263 00:14:30 --> 00:14:37 I realize I'm already using the word MATLAB. 264 00:14:37 --> 00:14:42 I think that MATLAB language is really convenient to 265 00:14:42 --> 00:14:44 talk about linear algebra. 266 00:14:44 --> 00:14:46 And how many know MATLAB or have used it? 267 00:14:46 --> 00:14:49 Yeah. 268 00:14:49 --> 00:14:51 You know it better than I. 269 00:14:51 --> 00:14:56 I talk a good line with MATLAB but I, the code never runs. 270 00:14:56 --> 00:14:58 Never! 271 00:14:58 --> 00:15:02 I always forget some stupid semicolon. 272 00:15:02 --> 00:15:04 You may have had that experience. 273 00:15:04 --> 00:15:11 And I just want to say it now that there are other languages, 274 00:15:11 --> 00:15:14 and if you want to do homeworks and want to do your own work 275 00:15:14 --> 00:15:18 in other languages, that makes sense. 276 00:15:18 --> 00:15:23 So the older established alternatives were Mathematica 277 00:15:23 --> 00:15:29 and Maple and those two have symbolic, they can deal with 278 00:15:29 --> 00:15:33 algebra as well as numbers. 279 00:15:33 --> 00:15:34 But there are newer languages. 280 00:15:34 --> 00:15:37 I don't know if you know them. 281 00:15:37 --> 00:15:41 I just know my friends say, Yes they're terrific. 282 00:15:41 --> 00:15:46 Python is one. 283 00:15:46 --> 00:15:47 And R. 284 00:15:47 --> 00:15:51 I've just had a email saying, Tell your class about R. 285 00:15:51 --> 00:15:53 And others. 286 00:15:53 --> 00:15:59 Ok, so but we'll use MATLAB language because that's really 287 00:15:59 --> 00:16:01 a good common language. 288 00:16:01 --> 00:16:03 Ok, so what is a Toeplitz matrix? 289 00:16:03 --> 00:16:06 A Toeplitz matrix is one with constant diagonals. 290 00:16:06 --> 00:16:09 You could use the word time invariant, linear time 291 00:16:09 --> 00:16:11 invariant filter. 292 00:16:11 --> 00:16:16 And to create K, this is an 18.085 command. 293 00:16:16 --> 00:16:19 It's just set up for us. 294 00:16:19 --> 00:16:26 I can create K by telling the system the first row. 295 00:16:26 --> 00:16:32 Two, minus one, zero, zero. 296 00:16:32 --> 00:16:37 That would, then if it wasn't symmetric I would have to 297 00:16:37 --> 00:16:40 give the first column also. 298 00:16:40 --> 00:16:42 Toeplitz would be constant diagonal, it doesn't 299 00:16:42 --> 00:16:44 have to be symmetric. 300 00:16:44 --> 00:16:47 But if it's symmetric, then the first row and first column are 301 00:16:47 --> 00:16:50 the same vector, so I just have to give that vector. 302 00:16:50 --> 00:16:55 Okay, so that's the quickest way to create K. 303 00:16:55 --> 00:17:00 And of course, if it was bigger then I would, rather than 304 00:17:00 --> 00:17:08 writing 100 zeros, I could put zeros of 98 and one. 305 00:17:08 --> 00:17:09 Wouldn't I have to say that? 306 00:17:09 --> 00:17:11 Or is it one and 98? 307 00:17:11 --> 00:17:15 You see why it doesn't run. 308 00:17:15 --> 00:17:18 Well I guess I'm thinking of that as a row. 309 00:17:18 --> 00:17:19 I don't know. 310 00:17:19 --> 00:17:24 Anyway. 311 00:17:24 --> 00:17:27 I realize getting this videotaped means I'm supposed 312 00:17:27 --> 00:17:28 to get things right! 313 00:17:28 --> 00:17:31 Usually it's like, we'll get it right later. 314 00:17:31 --> 00:17:36 But anyway, that might work. 315 00:17:36 --> 00:17:37 Okay. 316 00:17:37 --> 00:17:39 So there's a command that you know. 317 00:17:39 --> 00:17:44 Zeros that creates a matrix of this size with all zeros. 318 00:17:44 --> 00:17:45 Okay. 319 00:17:45 --> 00:17:48 That would create the 100 by 100. 320 00:17:48 --> 00:17:48 Good. 321 00:17:48 --> 00:17:50 Ok. 322 00:17:50 --> 00:17:52 Oh, by the way, as long as we're speaking about 323 00:17:52 --> 00:17:55 computation I've gotta say something more. 324 00:17:55 --> 00:17:59 We said that the matrix is sparse. 325 00:17:59 --> 00:18:02 And this 100 by 100 matrix is certainly sparse. 326 00:18:02 --> 00:18:07 But if I create it this way, I've created all those zeros 327 00:18:07 --> 00:18:13 and if I ask MATLAB to work with that matrix, to square it 328 00:18:13 --> 00:18:19 or whatever, it would carry all those zeros and do all 329 00:18:19 --> 00:18:21 those zero computations. 330 00:18:21 --> 00:18:25 In other words, it would treat K like a dense matrix and it 331 00:18:25 --> 00:18:27 would just, it wouldn't know the zeros were there 332 00:18:27 --> 00:18:29 until it looked. 333 00:18:29 --> 00:18:33 So I just want to say that if you have really big systems 334 00:18:33 --> 00:18:38 Sparse MATLAB is the way to go. 335 00:18:38 --> 00:18:42 Because Sparse MATLAB keeps track only of the non-zeros. 336 00:18:42 --> 00:18:45 So it knows-- and their locations, of course. 337 00:18:45 --> 00:18:47 What the numbers are and their location. 338 00:18:47 --> 00:18:50 So I could create a sparse matrix out of that, 339 00:18:50 --> 00:18:53 like KS for K sparse. 340 00:18:53 --> 00:18:59 I think if I just did sparse(K) that would 341 00:18:59 --> 00:19:01 create a sparse matrix. 342 00:19:01 --> 00:19:07 And then if I do stuff to it, MATLAB would automatically know 343 00:19:07 --> 00:19:11 those zeros were there and not spend it's time multiplying by 344 00:19:11 --> 00:19:15 zero But of course, this isn't perfect because I've created 345 00:19:15 --> 00:19:17 the big matrix before sparsifying it. 346 00:19:17 --> 00:19:20 And better to have created it in the first place 347 00:19:20 --> 00:19:22 as a sparse matrix. 348 00:19:22 --> 00:19:27 Ok. 349 00:19:27 --> 00:19:32 So those were properties that you could see. 350 00:19:32 --> 00:19:36 Now I'm looking for little deeper. 351 00:19:36 --> 00:19:39 What's the first question I would ask about a matrix if I 352 00:19:39 --> 00:19:43 have to solve a system of equations, say KU=F 353 00:19:43 --> 00:19:46 or something. 354 00:19:46 --> 00:19:53 I got a 4 by 4 matrix, four equations, four unknowns. 355 00:19:53 --> 00:19:57 What would I want to know next? 356 00:19:57 --> 00:19:59 Is it invertible? 357 00:19:59 --> 00:20:04 Is the matrix invertible? 358 00:20:04 --> 00:20:07 And that's an important question and how do you 359 00:20:07 --> 00:20:10 recognize an invertible matrix? 360 00:20:10 --> 00:20:12 This one is invertible. 361 00:20:12 --> 00:20:15 So let me say K is invertible. 362 00:20:15 --> 00:20:17 And what does that mean? 363 00:20:17 --> 00:20:21 That means that there's another matrix, K inverse such that 364 00:20:21 --> 00:20:27 K times K inverse is the identity matrix. 365 00:20:27 --> 00:20:33 The identity matrix in MATLAB would be eye(n) and it's 366 00:20:33 --> 00:20:35 the diagonal matrix of one. 367 00:20:35 --> 00:20:39 It's the unit matrix is the matrix that doesn't do 368 00:20:39 --> 00:20:43 anything to a vector. 369 00:20:43 --> 00:20:48 So this K has an inverse. 370 00:20:48 --> 00:20:49 But how do you know? 371 00:20:49 --> 00:20:53 How can you recognize that a matrix is invertible? 372 00:20:53 --> 00:20:56 Because obviously that's a critical question and many, 373 00:20:56 --> 00:21:00 many-- since our matrices are not-- a random matrix would be 374 00:21:00 --> 00:21:06 invertible, for sure, but our matrices have patterns, they're 375 00:21:06 --> 00:21:11 created out of a problem and the question of whether that 376 00:21:11 --> 00:21:13 matrix is invertible is fundamental. 377 00:21:13 --> 00:21:18 I mean finite elements has these, zero energy modes that 378 00:21:18 --> 00:21:24 you have to watch out for because, what are they? 379 00:21:24 --> 00:21:28 They produce non-invertible stiffness matrix. 380 00:21:28 --> 00:21:28 Ok. 381 00:21:28 --> 00:21:31 So how did we know, or how could we know that 382 00:21:31 --> 00:21:34 this K is invertible? 383 00:21:34 --> 00:21:37 Somebody said invertible and I wrote it down. 384 00:21:37 --> 00:21:39 Yeah? 385 00:21:39 --> 00:21:41 Well ok. 386 00:21:41 --> 00:21:45 Now I get to make a speech about determinants. 387 00:21:45 --> 00:21:46 Don't deal with them! 388 00:21:46 --> 00:21:49 Don't touch determinants. 389 00:21:49 --> 00:21:54 I mean this particular four by four happens to have 390 00:21:54 --> 00:21:56 a nice determinant. 391 00:21:56 --> 00:21:58 I think it's five. 392 00:21:58 --> 00:22:04 But if it was a 100 by 100 how would we show that the 393 00:22:04 --> 00:22:06 matrix was invertible? 394 00:22:06 --> 00:22:10 And what I mean by this is the whole family is invertible. 395 00:22:10 --> 00:22:13 All sizes are invertible. 396 00:22:13 --> 00:22:17 K_ n is invertible for every n, not just this particular guy, 397 00:22:17 --> 00:22:20 whose determinant we could take. 398 00:22:20 --> 00:22:24 But as five by five, six by six, we would be up in the-- 399 00:22:24 --> 00:22:28 but you're completely right. 400 00:22:28 --> 00:22:33 The determinant is a test. 401 00:22:33 --> 00:22:35 Alright. 402 00:22:35 --> 00:22:40 But I guess I'm saying that it's not the test 403 00:22:40 --> 00:22:45 that I would use. 404 00:22:45 --> 00:22:49 So what I do? 405 00:22:49 --> 00:22:52 I would row reduce. 406 00:22:52 --> 00:22:58 That's the default option in linear algebra. 407 00:22:58 --> 00:23:01 If you don't know what to do with a matrix, if you want to 408 00:23:01 --> 00:23:03 see what's going on, row reduce. 409 00:23:03 --> 00:23:04 What does that mean? 410 00:23:04 --> 00:23:09 That means, shall I try it? 411 00:23:09 --> 00:23:21 So let me just start it just so I'm not using a word 412 00:23:21 --> 00:23:24 that we don't need. 413 00:23:24 --> 00:23:25 Ok. 414 00:23:25 --> 00:23:29 And actually, maybe the third lecture, maybe next Monday 415 00:23:29 --> 00:23:33 we'll come back to row reduce. 416 00:23:33 --> 00:23:38 So I won't make heavy weather of that, certainly not now. 417 00:23:38 --> 00:23:43 So what is row reduce, just so you know. 418 00:23:43 --> 00:23:46 I want to get that minus one to be a zero. 419 00:23:46 --> 00:23:50 I'm aiming for a triangular matrix. 420 00:23:50 --> 00:23:55 I want to clean out below the diagonal because if my matrix 421 00:23:55 --> 00:23:59 is triangular then I can see immediately everything. 422 00:23:59 --> 00:24:01 Right? 423 00:24:01 --> 00:24:06 Ultimately I'll reach a matrix U that'll be upper triangular 424 00:24:06 --> 00:24:11 and that first row won't change but the second row will change. 425 00:24:11 --> 00:24:13 And what does it change to? 426 00:24:13 --> 00:24:17 How do I clean out, get a zero in that where the 427 00:24:17 --> 00:24:21 minus one is right now? 428 00:24:21 --> 00:24:29 Well I want to use the first row, the first equation. 429 00:24:29 --> 00:24:32 I want to add some multiple of the first 430 00:24:32 --> 00:24:36 row to the second row. 431 00:24:36 --> 00:24:38 And what should that multiple be? 432 00:24:38 --> 00:24:41 I want to multiply that row by something. 433 00:24:41 --> 00:24:43 And I'll say "add" today. 434 00:24:43 --> 00:24:47 Later I'll say "subtract." But what shall I do? 435 00:24:47 --> 00:24:50 Just tell me what the heck to do. 436 00:24:50 --> 00:24:53 I've got that row and I want to use it, I want to take a 437 00:24:53 --> 00:24:55 combination of these two rows. 438 00:24:55 --> 00:24:59 This row and some multiple of this one that'll 439 00:24:59 --> 00:25:00 produce a zero. 440 00:25:00 --> 00:25:02 This is called the pivot. 441 00:25:02 --> 00:25:04 That's the first pivot P-I-V-O-T. 442 00:25:04 --> 00:25:07 Pivot. 443 00:25:07 --> 00:25:11 And then that's the pivot row. 444 00:25:11 --> 00:25:14 And what do I do? 445 00:25:14 --> 00:25:15 Tell me what to do. 446 00:25:15 --> 00:25:18 Add half this row to this one. 447 00:25:18 --> 00:25:21 When I add half of that row to that one, what do I get? 448 00:25:21 --> 00:25:22 I get that zero. 449 00:25:22 --> 00:25:26 What do I get here for the second pivot? 450 00:25:26 --> 00:25:27 What is it? 451 00:25:27 --> 00:25:30 1.5, 3/2. 452 00:25:30 --> 00:25:32 Because half of that is, so 3/2. 453 00:25:32 --> 00:25:39 And the rest won't change. 454 00:25:39 --> 00:25:43 So I'm happy with that zero. 455 00:25:43 --> 00:25:48 Now I've got a couple more entries below that first pivot, 456 00:25:48 --> 00:25:49 but they're already zero. 457 00:25:49 --> 00:25:52 That's where the sparseness pays off. 458 00:25:52 --> 00:25:54 The tridiagonal really pays off. 459 00:25:54 --> 00:25:59 So those zeros say the first column is finished. 460 00:25:59 --> 00:26:02 So I'm ready to go on to the second column. 461 00:26:02 --> 00:26:08 It's like I got to this smaller problem with the 3/2 here. 462 00:26:08 --> 00:26:12 And a zero there. 463 00:26:12 --> 00:26:13 What do I do now? 464 00:26:13 --> 00:26:16 There is the second pivot, 3/2. 465 00:26:16 --> 00:26:17 Below it is a non-zero. 466 00:26:17 --> 00:26:20 I gotta get rid of it. 467 00:26:20 --> 00:26:23 What do I multiply by now? 468 00:26:23 --> 00:26:24 2/3. 469 00:26:24 --> 00:26:28 2/3 of that new, second row added to the third row will 470 00:26:28 --> 00:26:30 clean out the third row. 471 00:26:30 --> 00:26:32 This was already cleaned out. 472 00:26:32 --> 00:26:34 This is already a zero. 473 00:26:34 --> 00:26:38 But I want to have 2/3 of this row added to this one so 474 00:26:38 --> 00:26:41 what's my new third row? 475 00:26:41 --> 00:26:43 Starts with zero and what's the third pivot now? 476 00:26:43 --> 00:26:46 You see the pivots appearing? 477 00:26:46 --> 00:26:51 The third pivot will be 4/3 because I've got 2/3 this 478 00:26:51 --> 00:26:56 minus one and two is 6/3 so I have 6/3. 479 00:26:56 --> 00:27:00 I'm taking 2/3 away, I get 4/3 and that minus 480 00:27:00 --> 00:27:01 one is still there. 481 00:27:01 --> 00:27:07 So you see that I'm-- this is fast. 482 00:27:07 --> 00:27:08 This is really fast. 483 00:27:08 --> 00:27:11 And the next step, maybe you can see the beautiful 484 00:27:11 --> 00:27:13 patterns that are coming. 485 00:27:13 --> 00:27:16 Do you want to just guess the fourth pivot? 486 00:27:16 --> 00:27:21 5/4, good guess, right. 487 00:27:21 --> 00:27:24 5/4. 488 00:27:24 --> 00:27:29 Now this is actually how MATLAB would find the determinant. 489 00:27:29 --> 00:27:32 It would do elimination. 490 00:27:32 --> 00:27:36 I call that elimination because it eliminated all those numbers 491 00:27:36 --> 00:27:39 below the diagonal and got zeros. 492 00:27:39 --> 00:27:42 Now what's the determinant? 493 00:27:42 --> 00:27:45 If I asked you for the determinant, and I will very 494 00:27:45 --> 00:27:51 rarely use the word determinant, but I guess I'm 495 00:27:51 --> 00:27:55 into it now, so tell me the determinant. 496 00:27:55 --> 00:27:58 Five. 497 00:27:58 --> 00:27:59 Why's that? 498 00:27:59 --> 00:28:01 I guess I did say five earlier. 499 00:28:01 --> 00:28:06 But how do you know it's five? 500 00:28:06 --> 00:28:10 Whatever the determinant of that matrix is, why is it five? 501 00:28:10 --> 00:28:12 Because it's a triangular matrix. 502 00:28:12 --> 00:28:16 Triangular matrices, you've got all these zeros. 503 00:28:16 --> 00:28:18 You can see what's happening. 504 00:28:18 --> 00:28:21 And the determinant of a triangular matrix is just the 505 00:28:21 --> 00:28:24 product down the diagonal. 506 00:28:24 --> 00:28:25 The product of these pivots. 507 00:28:25 --> 00:28:29 The determinant is the product of the pivots. 508 00:28:29 --> 00:28:32 And that's how MATLAB would compute a determinant. 509 00:28:32 --> 00:28:36 And it would take two times 3/2 times 4/3 times 5/4 and it 510 00:28:36 --> 00:28:40 would give answer five. 511 00:28:40 --> 00:28:45 My friend Alan Edelman told me something yesterday. 512 00:28:45 --> 00:28:54 MATLAB computes in floating point. 513 00:28:54 --> 00:29:02 So 4/3, that's 1.3333, etc. 514 00:29:02 --> 00:29:06 So MATLAB would not, when it does that multiplication, 515 00:29:06 --> 00:29:08 get a whole number. 516 00:29:08 --> 00:29:09 Right? 517 00:29:09 --> 00:29:14 Because in MATLAB that would be 1.333 and probably it would 518 00:29:14 --> 00:29:18 make that last pivot a decimal, a long decimal. 519 00:29:18 --> 00:29:22 And then when it multiplies that it gets whatever it gets. 520 00:29:22 --> 00:29:25 But it's not exactly five I think. 521 00:29:25 --> 00:29:30 Nevertheless MATLAB will print the answer five. 522 00:29:30 --> 00:29:31 It's cheated actually. 523 00:29:31 --> 00:29:36 It's done that calculation and I don't know if it takes the 524 00:29:36 --> 00:29:42 nearest integer when it knows that the-- I shouldn't tell 525 00:29:42 --> 00:29:46 you this, this isn't even interesting. 526 00:29:46 --> 00:29:50 If the determinant of an integer matrix, whole number is 527 00:29:50 --> 00:29:54 a whole number, so MATLAB says, Better get a whole number. 528 00:29:54 --> 00:29:58 And somehow it gets one. 529 00:29:58 --> 00:30:01 Actually, it doesn't always get the right one. 530 00:30:01 --> 00:30:09 So maybe later I'll know the matrix whose determinant 531 00:30:09 --> 00:30:11 might not come out right. 532 00:30:11 --> 00:30:15 But ours is right, five. 533 00:30:15 --> 00:30:19 Now where was this going? 534 00:30:19 --> 00:30:23 It got thrown off track by the determinant. 535 00:30:23 --> 00:30:25 What's the real test? 536 00:30:25 --> 00:30:28 Well so I said there are two ways to see that a 537 00:30:28 --> 00:30:30 matrix is invertible. 538 00:30:30 --> 00:30:32 Or not invertible. 539 00:30:32 --> 00:30:34 Here we're talking about the first way. 540 00:30:34 --> 00:30:38 How do I know that this matrix-- I've got an 541 00:30:38 --> 00:30:39 upper triangular matrix. 542 00:30:39 --> 00:30:41 When is it invertible? 543 00:30:41 --> 00:30:47 When is an upper triangular matrix invertible? 544 00:30:47 --> 00:30:48 Upper triangular is great. 545 00:30:48 --> 00:30:50 When you've got it in that form you should 546 00:30:50 --> 00:30:51 be able to see stuff. 547 00:30:51 --> 00:30:58 So this key question of invertible, which is not 548 00:30:58 --> 00:31:04 obvious for a typical matrix is obvious for 549 00:31:04 --> 00:31:06 a triangular matrix. 550 00:31:06 --> 00:31:06 And why? 551 00:31:06 --> 00:31:10 What's the test? 552 00:31:10 --> 00:31:12 Well, we could do the determinant but we can say it 553 00:31:12 --> 00:31:15 without using that long word. 554 00:31:15 --> 00:31:18 The diagonal is non-zero. 555 00:31:18 --> 00:31:22 K as invertible because the diagonal-- no, it's got 556 00:31:22 --> 00:31:24 a full set of pivots. 557 00:31:24 --> 00:31:26 It's got four non-zero pivots. 558 00:31:26 --> 00:31:28 That's what it takes. 559 00:31:28 --> 00:31:31 That's what it's going to take to solve systems. 560 00:31:31 --> 00:31:33 So this is the first step in solving this system. 561 00:31:33 --> 00:31:38 In other words, to decide if a matrix is invertible, you 562 00:31:38 --> 00:31:41 just go ahead and use it. 563 00:31:41 --> 00:31:45 You don't stop first necessarily to check 564 00:31:45 --> 00:31:46 invertibility. 565 00:31:46 --> 00:31:49 You go forward, you get to this point and you see non-zeros 566 00:31:49 --> 00:31:53 there and then you're practically got to 567 00:31:53 --> 00:31:55 the answer here. 568 00:31:55 --> 00:32:00 I'll leave for another day the final back to going back 569 00:32:00 --> 00:32:03 upwards that gives you the answer. 570 00:32:03 --> 00:32:05 So K is invertible. 571 00:32:05 --> 00:32:15 That means full set of pivots. n non-zero pivots. 572 00:32:15 --> 00:32:20 And here they are, two, 3/2, 4/3 and 5/4. 573 00:32:20 --> 00:32:23 Worth knowing because this matrix K is so important. 574 00:32:23 --> 00:32:24 We'll see it over and over again. 575 00:32:24 --> 00:32:33 Part of my purpose today is to give some matrices a name 576 00:32:33 --> 00:32:35 because we'll see them again and you'll know them and 577 00:32:35 --> 00:32:38 you'll recognize them. 578 00:32:38 --> 00:32:44 While I'm on this invertible or not invertible business I 579 00:32:44 --> 00:32:48 want to ask you to change K. 580 00:32:48 --> 00:32:52 To make it not invertible. 581 00:32:52 --> 00:32:54 Change that matrix. 582 00:32:54 --> 00:32:56 How could I change that matrix? 583 00:32:56 --> 00:32:58 Well, of course, many ways. 584 00:32:58 --> 00:33:01 But I'm interested in another matrix and this'll be 585 00:33:01 --> 00:33:04 among my special matrices. 586 00:33:04 --> 00:33:07 And it will start out the same. 587 00:33:07 --> 00:33:14 It'll have these same diagonals. 588 00:33:14 --> 00:33:16 It'll be Toeplitz. 589 00:33:16 --> 00:33:23 I'm going to call it C and I want to say the reason I'm 590 00:33:23 --> 00:33:25 talking about it now is that it's not going to 591 00:33:25 --> 00:33:29 be invertible. 592 00:33:29 --> 00:33:38 And I'm going to tell you a C and see if you can tell me 593 00:33:38 --> 00:33:40 why it is not invertible. 594 00:33:40 --> 00:33:42 So here's the difference; I'm going to put minus 595 00:33:42 --> 00:33:45 one in the corners. 596 00:33:45 --> 00:33:49 Still zeros there. 597 00:33:49 --> 00:33:56 So that matrix C still has that pattern. 598 00:33:56 --> 00:33:58 It's still a Toeplitz matrix, actually. 599 00:33:58 --> 00:34:04 That would still be the matrix Toeplitz of two, minus 600 00:34:04 --> 00:34:04 one, zero, minus one. 601 00:34:04 --> 00:34:07 602 00:34:07 --> 00:34:14 I claim that matrix is not invertible and I claim that we 603 00:34:14 --> 00:34:19 can see that without computing determinants, we can see it 604 00:34:19 --> 00:34:22 without doing elimination, too. 605 00:34:22 --> 00:34:24 MATLAB would see it by doing elimination. 606 00:34:24 --> 00:34:30 We can see it by just human intelligence. 607 00:34:30 --> 00:34:33 Now why? 608 00:34:33 --> 00:34:39 How do I recognize a matrix that's not invertible? 609 00:34:39 --> 00:34:44 And then, by converse, how a matrix that is invertible. 610 00:34:44 --> 00:34:49 I claim-- and let may say first, let me say 611 00:34:49 --> 00:34:51 why that letter C. 612 00:34:51 --> 00:35:00 That letter C stands for circulant. it's because this 613 00:35:00 --> 00:35:03 word circulant, why circulant, it's because that diagonal 614 00:35:03 --> 00:35:09 which only had three guys circled around to the fourth. 615 00:35:09 --> 00:35:12 This diagonal that only had three entries circled around 616 00:35:12 --> 00:35:14 to the fourth entry. 617 00:35:14 --> 00:35:16 This diagonal with two zeros circled around to 618 00:35:16 --> 00:35:17 the other two zeros. 619 00:35:17 --> 00:35:22 The diagonal are not only constant, they loop around. 620 00:35:22 --> 00:35:24 And you use the word periodic. 621 00:35:24 --> 00:35:29 Now for me, that's the periodic matrix. 622 00:35:29 --> 00:35:35 See, a circulant matrix comes from a periodic problem. 623 00:35:35 --> 00:35:38 Because it loops around. 624 00:35:38 --> 00:35:42 It brings numbers, zero is the same as number 625 00:35:42 --> 00:35:45 four or something. 626 00:35:45 --> 00:35:51 And why is that not invertible? 627 00:35:51 --> 00:35:55 The thing is can you find a vector? 628 00:35:55 --> 00:35:57 Because matrices multiply vectors, that's 629 00:35:57 --> 00:35:59 their whole point. 630 00:35:59 --> 00:36:03 Can you see a vector that it takes to zero? 631 00:36:03 --> 00:36:05 Can you see a solution to Cu=0? 632 00:36:06 --> 00:36:11 I'm looking for a u with four entries so that 633 00:36:11 --> 00:36:18 I get four zeros. 634 00:36:18 --> 00:36:20 Do you see it? 635 00:36:20 --> 00:36:21 All ones. 636 00:36:21 --> 00:36:23 All ones. 637 00:36:23 --> 00:36:25 That will do it. 638 00:36:25 --> 00:36:33 So that's a nice, natural entry, a constant. 639 00:36:33 --> 00:36:37 And do you see why when I-- we haven't spoken about 640 00:36:37 --> 00:36:42 multiplying matrices times vectors. 641 00:36:42 --> 00:36:44 And most people will do it this way. 642 00:36:44 --> 00:36:46 And let's do this one this way. 643 00:36:46 --> 00:36:48 You take row one times that, you get two, minus 644 00:36:48 --> 00:36:49 one, zero, minus one. 645 00:36:51 --> 00:36:53 You get the zero because of that new number. 646 00:36:53 --> 00:36:58 Here we always got zero from the all ones vector and now 647 00:36:58 --> 00:37:04 over here that minus one, you see it's just right. 648 00:37:04 --> 00:37:09 If all the rows add to zero then this vector of all ones 649 00:37:09 --> 00:37:14 will be, I would use the word in the null space if you 650 00:37:14 --> 00:37:18 wanted a fancy word, a linear algebra word. 651 00:37:18 --> 00:37:19 What does that mean? 652 00:37:19 --> 00:37:21 It solves Cu=0. 653 00:37:21 --> 00:37:24 654 00:37:24 --> 00:37:29 And why does that show that the matrix isn't invertible? 655 00:37:29 --> 00:37:31 Because that's our point here. 656 00:37:31 --> 00:37:32 I have a solution to Cu=0. 657 00:37:35 --> 00:37:39 I claim that the existence of such a solution has wiped out 658 00:37:39 --> 00:37:45 the possibility that the matrix is invertible because if it 659 00:37:45 --> 00:37:49 was invertible, what would this lead to? 660 00:37:49 --> 00:37:56 If invertible, if C inverse exists what would I do to that 661 00:37:56 --> 00:38:04 equation that would show me that C inverse can't exist? 662 00:38:04 --> 00:38:08 Multiply both sides by C inverse. 663 00:38:08 --> 00:38:11 So you're seeing, just this first day you're seeing some 664 00:38:11 --> 00:38:14 of the natural steps of linear algebra. 665 00:38:14 --> 00:38:17 Row reduction, multiply when you want to see what's 666 00:38:17 --> 00:38:21 happening, multiply both sides by C inverse. 667 00:38:21 --> 00:38:25 That's the same as in ordinary language, Do the same thing 668 00:38:25 --> 00:38:27 to all the equations. 669 00:38:27 --> 00:38:30 So I multiply both sides by the same matrix. 670 00:38:30 --> 00:38:31 And here I would get (C inverse)(Cu)=(C inverse)(0). 671 00:38:31 --> 00:38:36 672 00:38:36 --> 00:38:40 So what does that tell me? 673 00:38:40 --> 00:38:43 I made it long, I threw in this extra step. 674 00:38:43 --> 00:38:51 You were going to jump immediately to C inverse C is I 675 00:38:51 --> 00:38:54 is the identity matrix and when the identity matrix multiplies 676 00:38:54 --> 00:38:57 a vector u, you get u. 677 00:38:57 --> 00:39:00 And on the right side, C inverse, whatever it is if 678 00:39:00 --> 00:39:05 it existed, times zero would have to be zero. 679 00:39:05 --> 00:39:09 So this would say that if C inverse exists, then the only 680 00:39:09 --> 00:39:13 solution is u equals u. 681 00:39:13 --> 00:39:15 That's a good way to recognize invertible matrices. 682 00:39:15 --> 00:39:20 If it is invertible then the only solution to Cu=0 u=0. 683 00:39:21 --> 00:39:24 And that wasn't true here. 684 00:39:24 --> 00:39:28 So we conclude C is not invertible. 685 00:39:28 --> 00:39:32 C is therefore not invertible. 686 00:39:32 --> 00:39:36 Now can I even jump in. 687 00:39:36 --> 00:39:38 I've got two more matrices that I want to tell you 688 00:39:38 --> 00:39:43 about that are also close cousins of K and C. 689 00:39:43 --> 00:39:50 But let me just explain physically a little 690 00:39:50 --> 00:39:54 bit about where these matrices are coming from. 691 00:39:54 --> 00:39:59 So maybe next to K-- so I'm not going to put periodic there. 692 00:39:59 --> 00:40:01 Right? 693 00:40:01 --> 00:40:03 That's the one that I would call periodic. 694 00:40:03 --> 00:40:08 This one is fixed at the ends. 695 00:40:08 --> 00:40:13 Can I draw a little picture that aims to show that? 696 00:40:13 --> 00:40:18 Aims to show where this is coming from. 697 00:40:18 --> 00:40:22 It's coming from I think of this as controlling 698 00:40:22 --> 00:40:23 like four masses. 699 00:40:23 --> 00:40:28 Mass one, mass two, mass three and mass four with springs 700 00:40:28 --> 00:40:40 attached and with endpoints fixed. 701 00:40:40 --> 00:40:47 So if I put some weights on those masses-- we'll do this; 702 00:40:47 --> 00:40:51 masses and springs is going to be the very first application 703 00:40:51 --> 00:40:55 and it will connect to all these matrices. 704 00:40:55 --> 00:41:05 And all I'm doing now is just asking to draw the system. 705 00:41:05 --> 00:41:06 Draw the mechanical system. 706 00:41:06 --> 00:41:09 Actually I'll usually draw it vertically. 707 00:41:09 --> 00:41:14 But anyway, it's got four masses and the fact that this 708 00:41:14 --> 00:41:19 minus one here got chopped off, what would I call that end? 709 00:41:19 --> 00:41:21 I'd call that a fixed end. 710 00:41:21 --> 00:41:25 So this is a fixed, fixed matrix. 711 00:41:25 --> 00:41:28 Both ends or fixed. 712 00:41:28 --> 00:41:32 And it's the matrix that would govern and the springs and 713 00:41:32 --> 00:41:36 masses all the same is what tells me that the 714 00:41:36 --> 00:41:38 thing is Toeplitz. 715 00:41:38 --> 00:41:42 Now what's the picture that goes with C? 716 00:41:42 --> 00:41:46 What's the picture with C? 717 00:41:46 --> 00:41:49 Do you have an instinct of that? 718 00:41:49 --> 00:41:52 So C is periodic. 719 00:41:52 --> 00:41:57 So again we've got four masses connected by springs. 720 00:41:57 --> 00:42:03 But what's up with those masses to make the problem cyclic, 721 00:42:03 --> 00:42:07 periodic, circular, whatever word you like. 722 00:42:07 --> 00:42:13 They're arranged in a ring. 723 00:42:13 --> 00:42:16 The fourth guy comes back to the first one. 724 00:42:16 --> 00:42:22 So the four masses would be, so in some kind of a ring, the 725 00:42:22 --> 00:42:27 springs would connect them. 726 00:42:27 --> 00:42:31 I don't know if that's suggestive, but I hope so. 727 00:42:31 --> 00:42:37 And what's the point of, can we just speak about 728 00:42:37 --> 00:42:39 mechanics one moment? 729 00:42:39 --> 00:42:46 How does that system differ from this fixed system? 730 00:42:46 --> 00:42:53 Here the whole system can't move, right? 731 00:42:53 --> 00:42:55 If there no force, then nothing can happen. 732 00:42:55 --> 00:43:00 Here the whole system can turn. 733 00:43:00 --> 00:43:03 They can all displace the same amount and just turn without 734 00:43:03 --> 00:43:06 any compression of the springs, without any force 735 00:43:06 --> 00:43:08 having to do anything. 736 00:43:08 --> 00:43:12 And that's why the solution that kills this matrix 737 00:43:12 --> 00:43:13 is one, one, one, one. 738 00:43:15 --> 00:43:19 So one, one, one, one would describe a case where all the 739 00:43:19 --> 00:43:21 displacements were equal. 740 00:43:21 --> 00:43:25 In a way it's like the arbitrary constant in calculus. 741 00:43:25 --> 00:43:29 You're always adding plus C. 742 00:43:29 --> 00:43:35 So here we've got a solution of all ones that produces zero the 743 00:43:35 --> 00:43:38 way the derivative of a constant function is 744 00:43:38 --> 00:43:41 the zero function. 745 00:43:41 --> 00:43:49 So this is just like an indication. 746 00:43:49 --> 00:43:51 Yes, perfect. 747 00:43:51 --> 00:43:52 I've got two more matrices. 748 00:43:52 --> 00:43:58 Are you okay for two more? 749 00:43:58 --> 00:44:03 Yes okay, what are they? 750 00:44:03 --> 00:44:10 Okay a different blackboard for the last two. 751 00:44:10 --> 00:44:17 So one of them is going to come by freeing up this end. 752 00:44:17 --> 00:44:24 So I'm going to take that support away. 753 00:44:24 --> 00:44:29 And you might imagine like a tower oscillating up and down 754 00:44:29 --> 00:44:34 or you might turn it upside down and like a hanging spring, 755 00:44:34 --> 00:44:39 or rather four springs with four masses hanging onto them. 756 00:44:39 --> 00:44:43 But this end is fixed and this is not fixed anymore, 757 00:44:43 --> 00:44:46 this is now free. 758 00:44:46 --> 00:44:50 And can I tell you the matrix, the free-fixed matrix. 759 00:44:50 --> 00:44:53 Free-fixed. 760 00:44:53 --> 00:44:56 Because it's the top end that I changed, I'm 761 00:44:56 --> 00:44:58 going to call it T. 762 00:44:58 --> 00:45:10 So all the other guys are going to be the same but the top one, 763 00:45:10 --> 00:45:15 the top row, the boundary row, boundary conditions are always 764 00:45:15 --> 00:45:20 the tough part, the tricky part, the key part of a model, 765 00:45:20 --> 00:45:23 and here the natural boundary condition is 766 00:45:23 --> 00:45:25 to have a one there. 767 00:45:25 --> 00:45:34 That two changed to a one. 768 00:45:34 --> 00:45:37 Now if I asked you for the properties of that matrix-- 769 00:45:37 --> 00:45:41 so that's the third. shall I do the fourth one? 770 00:45:41 --> 00:45:44 So you have them all, you'll have the whole picture. 771 00:45:44 --> 00:45:45 The fourth one, well you can guess. 772 00:45:45 --> 00:45:48 What's the fourth? 773 00:45:48 --> 00:45:51 What am I going to do? 774 00:45:51 --> 00:45:53 Free up the other end. 775 00:45:53 --> 00:45:59 So this guy had one free end and the other guy 776 00:45:59 --> 00:46:01 has B for both ends. 777 00:46:01 --> 00:46:04 B for both ends are going to be free. 778 00:46:04 --> 00:46:06 So this is free-fixed. 779 00:46:06 --> 00:46:08 This'll be free-free. 780 00:46:08 --> 00:46:13 So that means I have this free end, the usual stuff in the 781 00:46:13 --> 00:46:23 middle, no change, and the last row is what? 782 00:46:23 --> 00:46:25 What am I going to put in the last row? 783 00:46:25 --> 00:46:26 Minus one, one. 784 00:46:26 --> 00:46:26 Minus one, one. 785 00:46:26 --> 00:46:29 786 00:46:29 --> 00:46:34 So I've changed the diagonal. 787 00:46:34 --> 00:46:38 There I put a single one in because I freed up one end. 788 00:46:38 --> 00:46:41 With B I freed both ends and I got two minus ones. 789 00:46:41 --> 00:46:44 Now what do you think? 790 00:46:44 --> 00:46:52 So we've drawn the free-fixed one and what's your guess? 791 00:46:52 --> 00:46:55 They're all symmetric. 792 00:46:55 --> 00:46:57 That's no accident. 793 00:46:57 --> 00:47:00 They're all tridiagonal, no accident again. 794 00:47:00 --> 00:47:02 Why are they tridiagonal? 795 00:47:02 --> 00:47:06 Physically they're tridiagonal because that mass is only 796 00:47:06 --> 00:47:08 connected to it's two neighbors, it's not 797 00:47:08 --> 00:47:10 connected to that mass. 798 00:47:10 --> 00:47:16 That's why we get a zero in the two, four position. 799 00:47:16 --> 00:47:19 Because two is not connected to four. 800 00:47:19 --> 00:47:21 So it's tridiagonal. 801 00:47:21 --> 00:47:25 And it's not Toeplitz anymore, right? 802 00:47:25 --> 00:47:27 Toeplitz says constant diagonals and these are 803 00:47:27 --> 00:47:29 not quite constant. 804 00:47:29 --> 00:47:34 I would create K, I would take T equal K if I was going to 805 00:47:34 --> 00:47:37 create this matrix and then I would say T of one, 806 00:47:37 --> 00:47:40 one equal one. 807 00:47:40 --> 00:47:49 That command would fix up the first entry. 808 00:47:49 --> 00:47:50 Yeah, that's a serious question. 809 00:47:50 --> 00:47:53 Maybe, can I hang on until Friday, and 810 00:47:53 --> 00:47:54 even maybe next week. 811 00:47:54 --> 00:47:56 Because it's very important. 812 00:47:56 --> 00:48:00 When I said boundary conditions are the key to 813 00:48:00 --> 00:48:02 problems, I'm serious. 814 00:48:02 --> 00:48:07 If I had to think okay, what do people come in my office ask 815 00:48:07 --> 00:48:09 about questions, I say right away, What's the 816 00:48:09 --> 00:48:10 boundary condition? 817 00:48:10 --> 00:48:12 Because I know that's where the problem is. 818 00:48:12 --> 00:48:16 And so here we'll see these guys clearly. 819 00:48:16 --> 00:48:23 Fixed and free, very important. 820 00:48:23 --> 00:48:26 But also let me say two more words, I never can resist. 821 00:48:26 --> 00:48:30 So fixed means the displacement is zero. 822 00:48:30 --> 00:48:32 Something was set to zero. 823 00:48:32 --> 00:48:36 The fifth guy, the fifth over here, that fifth column 824 00:48:36 --> 00:48:39 was knocked out. 825 00:48:39 --> 00:48:44 Free means that in here it could mean that the fifth guy 826 00:48:44 --> 00:48:49 is the same as the fourth. 827 00:48:49 --> 00:48:52 The slope is zero. 828 00:48:52 --> 00:48:55 Fixed is u is zero. 829 00:48:55 --> 00:48:59 Free is slope is zero. 830 00:48:59 --> 00:49:05 So here I have a slope of zero at that end, here 831 00:49:05 --> 00:49:05 I have it at both ends. 832 00:49:05 --> 00:49:09 So maybe that's a sort of part answer. 833 00:49:09 --> 00:49:12 Now I wanted to get to the difference between 834 00:49:12 --> 00:49:15 these two matrices. 835 00:49:15 --> 00:49:19 And the main properties. 836 00:49:19 --> 00:49:19 So what are we see? 837 00:49:19 --> 00:49:23 Symmetric again, tridiagonal again, not quite Toeplitz, 838 00:49:23 --> 00:49:27 but almost, sort of morally Toeplitz. 839 00:49:27 --> 00:49:32 But then the key question was invertible or not. 840 00:49:32 --> 00:49:34 Key question was invertible or not. 841 00:49:34 --> 00:49:35 Right. 842 00:49:35 --> 00:49:37 And what's your guess on these two? 843 00:49:37 --> 00:49:41 Do you think that one's invertible or not? 844 00:49:41 --> 00:49:41 Make a guess. 845 00:49:41 --> 00:49:46 You're allowed to guess. 846 00:49:46 --> 00:49:47 Yeah it is. 847 00:49:47 --> 00:49:48 Why's that? 848 00:49:48 --> 00:49:52 Because this thing has still got a support. 849 00:49:52 --> 00:49:56 It's not free to shift forever. 850 00:49:56 --> 00:49:57 It's held in there. 851 00:49:57 --> 00:50:01 So that gives you a hint about this guy. 852 00:50:01 --> 00:50:04 Invertible or not for B? 853 00:50:04 --> 00:50:06 No. 854 00:50:06 --> 00:50:09 And now prove that it's not. 855 00:50:09 --> 00:50:14 Physically you were saying, well this free guy with this 856 00:50:14 --> 00:50:19 thing gone now, this is now free-free. 857 00:50:19 --> 00:50:21 Physically we're saying the whole thing can move, 858 00:50:21 --> 00:50:24 there's nothing holding it. 859 00:50:24 --> 00:50:27 But now, for linear algebra, that's not the proper language. 860 00:50:27 --> 00:50:31 You have to say something about that matrix. 861 00:50:31 --> 00:50:37 Maybe tell me something about Bu=0. u What are 862 00:50:37 --> 00:50:38 you going to take for u? 863 00:50:38 --> 00:50:39 Yeah. 864 00:50:39 --> 00:50:41 Same u. 865 00:50:41 --> 00:50:45 We're lucky in this course, u equal is the 866 00:50:45 --> 00:50:48 guilty main vector many times. 867 00:50:48 --> 00:50:55 Because again the rows are all adding to zero and the all ones 868 00:50:55 --> 00:51:02 vector is in the null space. 869 00:51:02 --> 00:51:06 If I could just close with one more word. 870 00:51:06 --> 00:51:07 Because it's the most important. 871 00:51:07 --> 00:51:10 Two words, two words. 872 00:51:10 --> 00:51:12 Because they're the most important words, they're the 873 00:51:12 --> 00:51:15 words that we're leading to in this chapter. 874 00:51:15 --> 00:51:18 And I'm assuming that for most people they will be 875 00:51:18 --> 00:51:21 new words, but not for all. 876 00:51:21 --> 00:51:24 It's a further property of this matrix. 877 00:51:24 --> 00:51:25 So we've got, how many? 878 00:51:25 --> 00:51:27 Four properties, or five? 879 00:51:27 --> 00:51:29 I'm going to go for one more. 880 00:51:29 --> 00:51:33 And I'm just going to say that name first so 881 00:51:33 --> 00:51:36 you know it's coming. 882 00:51:36 --> 00:51:38 And then I'll say, I can't resist saying 883 00:51:38 --> 00:51:41 a tiny bit about it. 884 00:51:41 --> 00:51:45 I'll use a whole blackboard for this. 885 00:51:45 --> 00:51:54 So I'm going to say that K and T are, here it comes, take 886 00:51:54 --> 00:52:07 a breath; positive definite matrices. 887 00:52:07 --> 00:52:10 So if you don't know what that means, I'm happy. 888 00:52:10 --> 00:52:10 Right? 889 00:52:10 --> 00:52:14 Because well, I can tell you one way to recognize a 890 00:52:14 --> 00:52:16 positive definite matrix. 891 00:52:16 --> 00:52:21 And while we're at it, let me tell you about C and B. 892 00:52:21 --> 00:52:30 Those are positive semi-definite because 893 00:52:30 --> 00:52:32 they hit zero somehow. 894 00:52:32 --> 00:52:35 Positive means up there, greater than zero. 895 00:52:35 --> 00:52:40 And what is greater than zero that we've already seen? 896 00:52:40 --> 00:52:42 And we'll say more. 897 00:52:42 --> 00:52:44 The pivots were. 898 00:52:44 --> 00:52:50 So if I have a symmetric matrix and the pivots are all positive 899 00:52:50 --> 00:52:55 then that matrix is not only invertible, because I'm in good 900 00:52:55 --> 00:52:59 shape, the determinant isn't zero, I can go backwards and do 901 00:52:59 --> 00:53:03 everything, those positive numbers are telling me that 902 00:53:03 --> 00:53:07 more than that, the matrix is positive definite. 903 00:53:07 --> 00:53:11 So that's a test. 904 00:53:11 --> 00:53:14 We'll say more about positive definite, but one way to 905 00:53:14 --> 00:53:18 recognize it is compute the pivots by elimination. 906 00:53:18 --> 00:53:20 Are they positive? 907 00:53:20 --> 00:53:23 We'll see that all the eigenvalues are positive. 908 00:53:23 --> 00:53:27 The word positive definite just brings the whole of 909 00:53:27 --> 00:53:29 linear algebra together. 910 00:53:29 --> 00:53:33 It connects to pivots, it connects to eigenvalues, it 911 00:53:33 --> 00:53:36 connects to least squares, it's all over the place. 912 00:53:36 --> 00:53:39 Determinants too. 913 00:53:39 --> 00:53:42 Questions or discussion. 914 00:53:42 --> 00:53:44 It's a big class and we're just meeting for the first time 915 00:53:44 --> 00:53:49 but there's lots of time to, chance to ask me. 916 00:53:49 --> 00:53:52 I'll always be here after class. 917 00:53:52 --> 00:53:53 So shall we stop today? 918 00:53:53 --> 00:53:58 I'll see you Friday or this afternoon. 919 00:53:58 --> 00:54:03 If this wasn't familiar, this afternoon would be a good idea. 920 00:54:03 --> 00:54:05 Thank you.
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More About This Textbook Overview This text covers college-level algebra and trigonometry and is appropriate for a one-or two-term course in precalculus mathematics. Its approach is more interactive than most precalculus texts, and is designed to help students achieve a greater understanding of sophisticated mathematical concepts than they might be able to achieve with other texts. Our goal is to provide as much support and help for students as we can, in order to ease the difficult transition into their college-level mathematics course. At the same time, we want to give students solid preparation for calculus and maintain the appropriate topical coverage and level of presentation for a precalculus
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Undergraduate Algebraic Geometry 9780521356626 ISBN: 0521356628 Pub Date: 1988 Publisher: Cambridge University Press Summary: Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics... and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory. Reid, Miles is the author of Undergraduate Algebraic Geometry, published 1988 under ISBN 9780521356626 and 0521356628. Five hundred fourteen Undergraduate Algebraic Geometry textbooks are available for sale on ValoreBooks.com, one hundred ten used from the cheapest price of $18.68, or buy new starting at $38
Abstract Algebra - 3rd edition Summary: Widely acclaimed algebra text. This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader's understanding. In this way, readers gain an appreciation for how mathematical structures and their ...show moreinterplay lead to powerful results and insights in a number of different settings. ...show less61.45 +$3.99 s/h VeryGood ocbookstx Richardson, TX 0471433349641433347-5-0 $67.42 +$3.99 s/h VeryGood Penntext Downingtown, PA Excellent Condition. No wear/tear. Please contact us if you have any Questions. $6745 +$3.99 s/h VeryGood Bookbyte-OR Salem, OR Has minor wear and/or markings. SKU:9780471433347-3-0 $73.4776.90 +$3.99 s/h Good TextbookBarn Woodland Hills, CA 0471433349 $79.50 +$3.99 s/h New Payless Textbook Rowland Heights, CA Brand new, hardcover. Isbn: 9780471433347, receive the book in 2-5 days
A Brief Version is written for students in the beginning statistics course whose mathematical background is limited to basic algebra. The book uses a nontheoretical approach in which concepts are explained intuitively and supported by examples for your student. There are no formal proofs in the book. The applications are general in nature and the exercises include problems from agriculture, biology, business, economics, education, psychology, engineering, medicine, sociology, and computer science. The learning system found in Elementary Statistics: A Brief Version provides your student with a valuable framework in which to learn and apply concepts!
Elementary Algebrais typically a 1-semester course that provides a solid foundation in algebraic skills and reasoning for students who have little or no previous experience with the topic. The goal is to effectively prepare students to transition into Intermediate Algebra.
Synopses & Reviews Publisher Comments: This ingenious, user-friendly introduction to calculus recounts adventures that take place in the mythical land of Carmorra. As the story's narrator meets Carmorra's citizens, they confront a series of practical problems, and their method of working out solutions employs calculus. As readers follow their adventures, they are introduced to calculating derivatives; finding maximum and minimum points with derivatives; determining derivatives of trigonometric functions; discovering and using integrals; working with logarithms, exponential functions, vectors, and Taylor series; using differential equations; and much more. This introduction to calculus presents exercises at the end of each chapter and gives their answers at the back of the book. Step-by-step worksheets with answers are included in the chapters. Computers are used for numerical integration and other tasks. The book also includes graphs, charts, and whimsicalSynopsis: Synopsis: (back cover) Calculus makes sense when you approach them the E-Z way! Open this book for a clear, concise, step-by-step review of: About the Author Douglas Downing earned B.S. and Ph.D. degrees from Yale, and has taught economics at Seattle Pacific University since 1983. Calculus the Easy Way is part of a trilogy with Algebra the Easy Way and Trigonometry the Easy Way, all written be Douglas Downing and available from Barron's. "Synopsis" by Netread,"Synopsis" by Netread, (back cover) Calculus makes sense when you approach them the E-Z way! Open this book for a clear, concise, step-by-step review
Mathematics for High School Teachers An Advanced Perspective 9780130449412 ISBN: 0130449415 Pub Date: 2002 Publisher: Prentice Hall Summary: Mathematics for High School Teachers-An Advanced Perspectiveis intended as a text for mathematics courses for prospective or experienced secondary school mathematics teachers and all others who wish to examine high school mathematics from a higher point of view. Preliminary versions of the book have been used in a variety of ways, ranging from junior and senior (capstone) or graduate mathematics courses for pre-servi...ce secondary mathematics education majors to graduate professional development courses for teachers. Some courses included both undergraduate and graduate students and practicing teachers with good success. There is enough material in this book for at least a full year (two semesters) of study under normal conditions, even if only about half of the problems are assigned. With a few exceptions, the chapters are relatively independent and an instructor may choose from them. However, some chapters contain more sophisticated content than others. Here are four possible sequences for a full semester's work: Algebra emphasis: Chapters 1-6 Geometry emphasis: Chapters 1, 7-11 Introductory emphasis: Chapters 1, 3, 4, 7, 8,10 More advanced emphasis: Chapters 1, 2, 5, 6, 9,11. In each sequence we suggest beginning with Chapter 1 so that students are aware of the features of this book and of some of the differences between it and other mathematics texts they may have used. More information and suggestions in this regard can be found in the Instructor's Notes. Additional instructional resources are also at the web site . The presentation assumes the student has had at least one year of calculus and a post-calculus mathematics course (such as real analysis, linear algebra, or abstract algebra) in which proofs were required and algebraic structures were discussed. The term "from an advanced standpoint" is taken to mean that the text examines high school mathematical ideas from a perspective appropriate for college mathematics majors, and makes use of the kind of mathematical knowledge and sophistication the student is gaining or has gained in other courses. Two basic characteristics ofMathematics for High School Teachers-An Advanced Perspective,taken together, distinguish courses taught from this book from many current courses. First, the material is rooted in the core mathematical content and problems of high school mathematics courses before calculus. Specifically, the development emanates from the major concepts found in high school mathematics: numbers, algebra, geometry, and functions. Second, the concepts and problems are treated from a mathematically advanced standpoint, and differ considerably from materials designed for high school students. The authors feel that the mathematical content in this book lies in an area of mathematics that is of great benefit to all those interested in mathematics at the secondary school level, but is rarely seen by them. Specifically, we have endeavored to include: analyses of alternate definitions, language, and approaches to mathematical ideas extensions and generalizations of familiar theorems discussions of the historical contexts in which concepts arose and have changed over time applications of the mathematics in a wide range of settings analyses of common problems of high school mathematics from a deeper mathematical level demonstrations of alternate ways of approaching problems, including ways with and without calculator and computer technology connections between ideas that may have been studied separately in different courses relationships of ideas studied in school to ideas students may encounter in later study. There are many reasons why we believe a teacher or other person interested in high school mathematics Usiskin, Zalman is the author of Mathematics for High School Teachers An Advanced Perspective, published 2002 under ISBN 9780130449412 and 0130449415. Four hundred twenty two Mathematics for High School Teachers An Advanced Perspective textbooks are available for sale on ValoreBooks.com, one hundred thirty six used from the cheapest price of $46.59, or buy new starting at $98 gives readers a comprehensive look at the most important concepts in the mathematics taught in grades 9-12. Real numbers, functions, congruence, similarity, area [more] This book gives readers a comprehensive look at the most important concepts in the mathematics taught in grades 9-12. Real numbers, functions, congruence, similarity, area and volume, trigonometry and more. For high sch
Seki Takakazu calculusBranch of mathematics concerned with the calculation of instantaneous rates of change (differential calculus) and the summation of infinitely many small factors to determine some whole (integral calculus).... determinantIn linear and multilinear algebra, a value, denoted det A, associated with a square matrix A of n rows and n columns. Designating any element of the matrix by the symbol a r c (the subscript r identifies... mathematicsThe science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation,...
Geometry : From Euclid to Knots - 03 edition Summary: The main purpose of this book is to inform the reader about the formal, or axiomatic, development of Euclidean geometry. It follows Euclid's classic text Elements very closely, with an excellent organization of the subject matter, and over 1,000 practice exercises provide the reader with hands-on experience in solving geometrical problems. Providing a historical perspective about the study of plane geometry, this book covers such topics as other geometries, the neutr...show moreal geometry of the triangle, non-neutral Euclidean geometry, circles and regular polygons, projective geometry, symmetries, inversions, informal topology, graphs, surfaces, and knots and links. ...show less 0130329274 Item in very good condition and at a great price! Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!! $5.7335joe10861 Hialeah, FL Hardcover Good 0130329274 1A2 Very Good, Slight cover wear, marks, but in overall good shape, 99% of inner pages are clean and free of marks. International shipping is available. Descriptions are a...show moreccurate, buy with confidence. ...show less Buy with Confidence. Excellent Customer Support. We ship from multiple US locations. No CD, DVD or Access Code Included. $29.95 +$3.99 s/h New Edge Bookstore Diamond Bar, CA New New as pictured-clean, pristine condition-Ships from legendary independent online bookstore in Murrieta, California. Thousands of satisfied customers. We ship promptly and Worldwide. We work har...show mored30.18 +$3.99 s/h Good Big Planet Books Burbank, CA 2002-08-10
More About This Textbook Overview Produced by the award-winning maranGraphics Group, Maran Illustrated Effortless Algebra is a valuable resource to a wide range of readers-from people first being introduced to algebra to those studying for their SATs or GEDs. Maran Illustrated Effortless Algebra shows the reader the best way to perform each task, while the full-color examples and clear, step-by-step instructions walk the reader through each task from beginning to end. Thorough topic introductions and useful tips provide additional information and exercises to help enhance the readers' algebra experience. Maran Illustrated Effortless Algebra is packed with essential information for those who are learning algebra for the first time, and will provide more experienced readers with a refresher course on the basics and the opportunity to gain more advanced skills. Maran Illustrated Effortless Algebra will cost less than the price of one private tutoring session, and will provide years of valuable reference. Related Subjects Meet the Author maranGraphics Development Group combines the efforts of many talented people. An industry expert in the field of each book and Maran's own writers work together to produce books that are highly visual, technically sound, easy for readers to understand, and adhere to maranGraphics' standards of structuring to provide the best
1 Program Components Introduction ... The Honors Gold Series helps students develop a deep understanding of mathematics through thinking, reasoning, ... This workbook contains daily lesson support with Think About a Plan, Practice, and Mathematics (in 2007) and the Mathematical Association of America Allegheny Mountain Section Mentoring Award (in 2009). Professor Sellers has enjoyed many interactions at the high school and middle school levels. ... LESSON 1 An Introduction to Algebra II ... Multilingual Workbook For Mathematics Crs 1 (P) By Download Full Version Of this Book Download Full PDF Version of This Book This is the only site that you can get the free pdf version of this book, enjoy!
The AP Course Audit will begin accepting submissions for new courses offered in the 2014-15 school year. School administrators can begin finalizing Course Audit forms for new courses and for those recently transferred to their schools by new teachers. The AP Program unequivocally supports the principle that each individual school must develop its own curriculum for courses labeled "AP." Rather than mandating any one curriculum for AP courses, the AP Course Audit instead provides each AP teacher with a set of expectations that college and secondary school faculty nationwide have established for college-level courses. More AP teachers are encouraged to develop or maintain their own curriculum that either includes or exceeds each of these expectations; such courses will be authorized to use the "AP" designation. Credit for the success of AP courses belongs to the individual schools and teachers that create powerful, locally designed AP curricula. The AP Calculus BC course should be designed by your school to provide students with a learning experience equivalent to that of a full-year college course in single-variable calculus. Your Calculus BC course needs to develop students' concepts of calculus and provide experience with its methods and applications. The course should emphasize a multirepresentational approach to calculus, with concepts, results and problems being expressed graphically, numerically, analytically and verbally. In addition, the connections among these representations should be highlighted. Before studying calculus, students should complete four years of secondary mathematics designed for college-bound students; in these courses, they should study algebra, geometry, trigonometry, analytic geometry and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions and the graphs of functions. Students must also understand the language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts and so on) and know the values of the trigonometric functions of the numbers 0, π/6, π/4, π/3, π/2 and their multiples. All students who are willing and academically prepared to accept the challenge of a rigorous academic curriculum should be considered for admission to AP courses. The College Board encourages the elimination of barriers that restrict access to AP courses for students from ethnic, racial and socioeconomic groups that have been traditionally underrepresented in the AP Program. Schools should make every effort to ensure that their AP classes reflect the diversity of their student population. High schools offering this exam must provide the exam administration resources described in the AP Coordinator's Manual. Course and Exam Description Describes in detail the AP course and exam. Includes the curriculum framework and a representative sample of exam questions. Review this resource to establish your understanding of the objectives and expectations of the AP course and exam. Curricular/Resource Requirements Identifies the set of curricular and resource expectations that college faculty nationwide have established for a college-level course. Example Textbook List Includes a sample of AP college-level textbooks that meet the content requirements of the AP course. Syllabus Development Guide Includes the guidelines reviewers use to evaluate syllabi along with three samples of evidence for each requirement. This guide also specifies the level of detail required in the syllabus to receive course authorization. Four Annotated Sample Syllabi Provide examples of how the curricular requirements can be demonstrated within the context of actual syllabi
... More About This Book republication of the classic 1914 edition. 74 figures. Index. Product Details Table of Contents 1. The Earliest Period 2. The Second Period 3. The Development of the Soroban 4. The Sangi Applied to Algebra 5. The Third Period 6. Seki Kowa 7. Seki's Contemporaries and Possible Western Influences 8. The Yenri or Circle Principle 9. The Eighteenth Century 10. Ajima Chokuyen 11. The Opening of the Nineteenth Century 12. Wada Nei 13. The Close of the Old Wasan 14. The Introduction of Occidental Mathematics
Synopses & Reviews Publisher Comments: 'Many mathematics students have trouble the first time they take a course, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems. This textbook will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed \"scratchwork\" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. Numerous exercises give students the opportunity to construct their own proofs. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.' Synopsis: Teaches the techniques needed to read and write proofs. Synopsis: "Synopsis" by Ingram, Teaches the techniques needed to read and write proofs. "Synopsis" by Gardners,
08359358 BASIC MATH PACEMAKER THIRD EDITION WKB 2000C Pacemaker Basic Math is a comprehensive program that provides a solid, well-balanced approach to teaching math content and building math skills in whole numbers, basic arithmetic operations, and mastery of simple geometry and algebra as it prepares students for the rigors of difficult standards and proficiency tests. This program provides educators with tools to meet the needs of diverse classrooms, keep learning up-to-date and relevant, and create supportive learning environments for a range of learning styles. Correlated to the NCTM standards, the materials and techniques used in the program are accessible, predictable, age-appropriate, and relevant as it bridges the gap between varied abilities of students and the ladder to success in algebra. Visual learners and struggling readers are supported with photographs, charts, graphs, and illustrations, and high-interest projects gear up students for lessons. To view sample lessons and pages, click on Download a Brochure to the left. For ISBNs and prices, click on Program Components

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