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. A copy of the license is included in the section entitled GNU Free DocumentationLicense.

2 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. 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.

3 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. 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.

4 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. 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.

5 Vector Algebra .. Dot Product .. Cross Product .. Lines and Planes .. Surfaces .. 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 .. Application: Center of Mass .. Application: Probability and Expected Value .. 1284 Line and Surface Line Integrals .. Properties of Line Integrals .. Green s Theorem.

6 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).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.

7 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). 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 ).

8 Xyz0P(a,b,c)abcFigure 1. VECTORS IN EUCLIDEAN SPACEThe coordinate system shown in Figure is known as aright-handed coordinatesystem, because it is possible, using the right hand, to point the index finger in the positivedirection of thex-axis, the middle finger in the positive direction of they-axis, and the thumbin the positive direction of thez-axis, as in Figure coordinate systemAn equivalent way of defining a right-handed system is if you canpoint your thumb up-wards in the positivez-axis direction while using the remaining four fingers to rotate thex-axis towards they-axis. Doing the same thing with the left hand is what defines aleft-handed coordinate system. Notice that switching thex- andy-axes in a right-handedsystem results in a left-handed system, and that rotating either type of system does notchange its handedness . Throughout the book we will use a right-handed functions of three variables, the graphs exist in 4-dimensional space ( ), whichwe can not see in our 3-dimensional space, let alone simulate in 2-dimensional space.

9 Sowe can only think of 4-dimensional space abstractly. For an entertaining discussion of thissubject, see the book by far, we have discussed thepositionof an object in 2-dimensional or 3-dimensional what about something such as the velocity of the object, orits acceleration? Or thegravitational force acting on the object? These phenomena all seem to involve motion anddirectionin some way. This is where the idea of avectorcomes thing you will learn is why a 4-dimensional creature wouldbe able to reach inside an egg and remove theyolk without cracking the shell! Introduction3 You have already dealt with velocity and acceleration in single-variable Calculus . Forexample, for motion along a straight line, ify=f(t) gives the displacement of an object aftertimet, thendy/dt=f (t) is the velocity of the object at timet. The derivativef (t) is just anumber, which is positive if the object is moving in an agreed-upon positive direction, andnegative if it moves in the opposite of that direction.

10 So you canthink of that number, whichwas called the velocity of the object, as having two components: amagnitude, indicatedby a nonnegative number, preceded by adirection, indicated by a plus or minus symbol(representing motion in the positive direction or the negativedirection, respectively), (t)= afor some numbera 0. Thenais the magnitude of the velocity (normally calledthespeedof the object), and the represents the direction of the velocity (though the+isusually omitted for the positive direction).For motion along a straight line, 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 acurve 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 objectwhich has those features is an arrow, which in elementary geometry is called a directed linesegment.