Vector calculus

Vector calculus gy EIKagnaI I llOR6pp 16, 2011 pagcs 1 0. 8 0. 6 0. 4 z 0. 2 0-0. 2-0. 4 -S -10 -5y005S 10 10 x -10 Vector Calculus Michael Corral Schoolcraft College About the author: Michael Corral is an Adjunct Faculty member of the Department of Mathematics at Schoolcraft College. He received a g. A. in Mathematics from the University of California at Berkeley, and received an MA. in Mathematics and an M. S. in Industrial & Operations En ineerin from the University of Michigan. A This text was types in or306 using the G NU Emac Xt ed graphics were create ed MA-Script bundle, Llnux system. The nd Gnuplot.

Copyright c 2008 Michael Corral. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. 2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled «GNU Free Documentation License». 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 Swlpe to vlew nexr page III».

The prerequisites are the standard courses in single-variable calculus (a. k. a. Calculus I and II). 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 majorio,’ 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, would rate it as a 5.

There are 420 exercises throughout the text, which in my experience are more than nough for a semester course in this subject. There are exercises at the end of each section, 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 usually require sorne effort or insight to salve. A crude way of describing A, B and C would be «Easy’, «Moderate» and «‘Challenginã’, respectively. However, many of the g 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 program to solve some numerical approximation roblems (e. g. the Monte Carlo method for approximating multiple integrals, in Section 3. 4). The 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 chosen due to its ubiquity, relatively clear syntax, and easy availabiliry for multiple platforms. Answers and hints to most odd-numbered and some even-numbered exercises are provided in Appendix A. Appendix B contains a proof of the right- hand rule for the cross product, 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 iv matter, anyone can make as many copies of this book as de 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 http://www. mecmath. net). Feel free to contact me at mcorral@schoolcraft. edu for any questlons on this or any other matter involving the book (e. g. comments, suggestions, corrections, etc). welcome your input. Finally, I would like to hank my students in Math 240 for being the guinea pigs for the initial draft of this bock, and for finding the numerous errors and typos it contaned. January 2008 M ICHAEL C ORRAL Contents Preface 1 Vectors in Euclidean Space 1. 1 Introduction …… Vector Algebra 1. 3 Dot product 1. 4 Cross Product.. . 1. 6 Surfaces 1. 5 Cines and Planes . 1 . 7 Curvilinear Coordinates 1. 8 Vector-Valued Functions 1 Arc Length iii119152031 404751 59 65 65 71 75 7883 8996 101 101 105 110 113 117 124 128 1 Vectors in Euclidean Space 1. 1 Introduction In single-variable calculus, the functions that one encounters re functions of a variable (usually x or t) that varies over sorne subset of the real number line (which we denote by For such a function, say, y = f (x), the graph ofthe 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 2 ; the «2» represents the number of dimensions of the plane. The Euclidean plane has two perpendicular coordinate axes: the -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 3 . The graph off 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 blackboa ystem of Euclidean space can be represented on a flat surface, such as this page or a blackboard, 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). z c P(a, b, c) b Oaxxy xz-plane yz-plane yo xy-plane Figure 1. 1. 1 Figure 1. 1. 2 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 and, 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. . 3. Figure 1. 1. 3 Right-handed coordinate system An equivalent way of defining a right-handed system is [f you can point your thumb upward e z-axis direction while using the remaining four f e the x-axis towards the 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 ariables, the graphs exist in 4-dimensional space (i. e. 4 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 A BBOTr. 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 ‘n. One 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!

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. he derivative f’ (t) is just a number, which is positive if the object is moving in an agreedupon «positive» dire ‘ (t) is just a number, which is positive ifthe object is moving in an agreedupon «positive» direction, and negative if it moves in the pposite of that directicn.

So you can think of that number, which was called the velocity of the object, as having Ñ„’O 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 numbera O. 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. a 1-dimensional space, the velocities are also contained in that 1-dimensional space, slnce 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 multidimensional 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 (calle 306