Michael Corral: Vector Calculus

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iv Preface 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 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
Contents Preface iii 1 Vectors in Euclidean Space 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.7 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.8 Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.9 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2 Functions of Several Variables 65 2.1 Functions of Two or Three Variables . . . . . . . . . . . . . . . . . . . . . 65 2.2 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.3 Tangent Plane to a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.4 Directional Derivatives and the Gradient . . . . . . . . . . . . . . . . . . 78 2.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.6 Unconstrained Optimization: Numerical Methods . . . . . . . . . . . . . 89 2.7 Constrained Optimization: Lagrange Multipliers . . . . . . . . . . . . . . 96 3 Multiple Integrals 101 3.1 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.2 Double Integrals Over a General Region . . . . . . . . . . . . . . . . . . . 105 3.3 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.4 Numerical Approximation of Multiple Integrals . . . . . . . . . . . . . . 113 3.5 Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . 117 3.6 Application: Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.7 Application: Probability and Expected Value . . . . . . . . . . . . . . . . 128 4 Line and Surface Integrals 135 4.1 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2 Properties of Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 v
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188 Bibliography Pogorelov, A.V., A
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190 Appendix A: Answers and Hints t
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Appendix B We will prove the right-
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194 Appendix B: Proof of the Right-
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Appendix C 3D Graphing with Gnuplot
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198 Appendix C: 3D Graphing with Gn
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200 Appendix C: 3D Graphing with Gn
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202 GNU Free Documentation License
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Index Symbols D....................
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212 Index iterated integral .......
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Michael Corral: Vector Calculus

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