Vector Calculus - mecmath

1 CorralVector CalculusMichael CorralSchoolcraft CollegeAbout the author:Michael Corral is an Adjunct Faculty member of the Department ofMathematics atSchoolcraft College. He received a in Mathematics from the University of Californiaat Berkeley, and received an in Mathematics and an in Industrial & OperationsEngineering from the University of text was typeset in LATEX2 with theKOMA-Scriptbundle, using the GNU Emacs texteditor on a Fedora Linux system. The graphics were created using MetaPost, PGF, 2008 Michael is granted to copy, distribute and/or modify this document under the terms of theGNU Free Documentation License, Version or any later version published by the FreeSoftware Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-CoverTexts.

2 A copy of the license is included in the section entitled GNU Free DocumentationLicense .PrefaceThis 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 ( Calculus I andII).I have tried to be somewhat rigorous about proving results. Butwhile it is important forstudents to see full-blown proofs - since that is how mathematics works - too much rigor andemphasis on proofs can impede the flow of learning for the vast majority of the audience atthis level.

3 If I were to rate the level of rigor in the book on a scale of 1 to 10, with 1 beingcompletely informal and 10 being completely rigorous, I would rate it as a are 420 exercises throughout the text, which in my experience are more thanenough 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 exercisesare mostly of a routinecomputational 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.

4 However,many of the B exercises areeasy and not all the C exercises are are a few exercises that require the student to write his or her own computer pro-gram to solve some numerical approximation problems ( the Monte Carlo method forapproximating multiple integrals, in Section ). The code samples in the text are in theJava programming language, hopefully with enough comments so that the reader can figureout what is being done even without knowing Java. Those exercisesdo not mandate the useof Java, so students are free to implement the solutions using the language of their it would have been simple to use a scripting language like Python, and perhaps eveneasier with a functional programming language (such as Haskellor Scheme), Java was cho-sen due to its ubiquity, relatively clear syntax, and easy availability for multiple and hints to most odd-numbered and some even-numbered exercises are pro-vided in Appendix A.

5 Appendix B contains a proof of the right-hand rulefor the cross prod-uct, which seems to have virtually disappeared from Calculus texts over the last few C contains a brief tutorial on Gnuplot for graphing functions of two book is released under the GNU Free Documentation License (GFDL), which allowsothers to not only copy and distribute the book but also to modifyit. For more details, seethe included copy of the GFDL. So that there is no ambiguity on thismatter, anyone canmake as many copies of this book as desired and distribute it as desired, without needingmy permission.

6 The PDF version will always be freely available to the public at no cost(go ). Feel free to contact me questions on this or any other matter involving the book ( comments, suggestions,corrections, etc). I welcome your , I would like to thank my students in Math 240 for being the guinea pigs for theinitial draft of this book, and for finding the numerous errors and typos it 2008 MICHAELCORRALC ontentsPrefaceiii1 Vectors in Euclidean Introduction .. Vector Algebra .. Dot Product .. Cross Product .. Lines and Planes .. Surfaces.

7 Curvilinear Coordinates .. Vector -Valued Functions .. Arc Length .. 592 Functions of Several Functions of Two or Three Variables .. Partial Derivatives .. Tangent Plane to a Surface .. Directional Derivatives and the Gradient .. Maxima and Minima .. Unconstrained Optimization: Numerical Methods .. Constrained Optimization: Lagrange Multipliers .. 963 Multiple Double Integrals .. Double Integrals Over a General Region .. Triple Integrals .. Numerical Approximation of Multiple Integrals .. Change of Variables in Multiple Integrals.

8 Application: Center of Mass .. Application: Probability and Expected Value .. 1284 Line and Surface Line Integrals .. Properties of Line Integrals .. Green s Theorem .. Surface Integrals and the Divergence Theorem .. Stokes Theorem .. Gradient, Divergence, Curl and Laplacian .. 177 Bibliography187 Appendix A: Answers and Hints to Selected Exercises189 Appendix B: Proof of the Right-Hand Rule for the Cross Product192 Appendix C: 3D Graphing with Gnuplot196 GNU Free Documentation License201 History209 Index2101 Vectors in Euclidean IntroductionIn single-variable Calculus , the functions that one encounters are functions of a variable(usuallyxort) that varies over some subset of the real number line (which we denote byR).

9 For such a function, say,y=f(x), thegraphof the functionfconsists of the points (x,y)=(x,f(x)). These points lie in theEuclidean plane, which, in theCartesianorrectangularcoordinate 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. Wedenote the Euclidean plane byR2; the 2 represents the number ofdimensionsof the Euclidean plane has two perpendicularcoordinate axes: thex-axis and Vector (or multivariable) Calculus , we will deal with functions of two or three variables(usuallyx,yorx,y,z, respectively).

10 The graph of a function of two variables, say,z=f(x,y),lies inEuclidean space, which in the Cartesian coordinate system consists of all orderedtriples of real numbers (a,b,c). Since Euclidean space is 3-dimensional, we denote it graph offconsists of the points (x,y,z)=(x,y,f(x,y)). The 3-dimensional coordinatesystem 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 space has three mutually perpendicular coordinate axes(x,yandz), and threemutually perpendicular coordinate planes: thexy-plane,yz-plane andxz-plane (see ).