VC5IntSup - Page i Internet Supplement for Vector Calculus...

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Chapter 10 / Exercise 45
Applied Calculus
Berresford/Rockett
Expert Verified
Page i Internet Supplement for Vector Calculus Fifth Edition Version: October, 2003 Jerrold E. Marsden California Institute of Technology Anthony Tromba University of California, Santa Cruz W.H. Freeman and Co. New York
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Chapter 10 / Exercise 45
Applied Calculus
Berresford/Rockett
Expert Verified
Page i Contents Preface iii 2 Differentiation 1 2.7 Some Technical Differentiation Theorems . . . . . . . . . . . 1 3 Higher-Order Derivatives and Extrema 17 3.4A Second Derivative Test: Constrained Extrema . . . . . . . 17 3.4B Proof of the Implicit Function Theorem . . . . . . . . . . 21 4 Vector Valued Functions 25 4.1A Equilibria in Mechanics . . . . . . . . . . . . . . . . . . . 25 4.1B Rotations and the Sunshine Formula . . . . . . . . . . . . 30 4.1C The Principle of Least Action . . . . . . . . . . . . . . . . 42 4.4 Flows and the Geometry of the Divergence . . . . . . . . . 58 5 Double and Triple Integrals 65 5.2 Alternative Definition of the Integral . . . . . . . . . . . . 65 5.6 Technical Integration Theorems . . . . . . . . . . . . . . . 71 6 Integrals over Curves and Surfaces 83 6.2A A Challenging Example . . . . . . . . . . . . . . . . . . . 84 6.2A The Gaussian Integral . . . . . . . . . . . . . . . . . . . . 88 8 The Integral Theorems of Vector Analysis 91 8.3 Exact Differentials . . . . . . . . . . . . . . . . . . . . . . . 91
Page ii 8.5 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . 94 Selected Answers for the Internet Supplement 107 Practice Final Examination 123 Practice Final Examination Solutions 129
Page iii Preface The Structure of this Supplement. This Internet Supplement is in- tended to be used with the 5th Edition of our text Vector Calculus. It contains supplementary material that gives further information on various topics in Vector Calculus, including different applications and also technical proofs that were omitted from the main text. The supplement is intended for students who wish to gain a deeper un- derstanding, usually by self study, of the material—both for the theory as well as the applications. Corrections and Website. A list of corrections and suggestions con- cerning the text and instructors guide are available available on the book’s website: Please send any new corrections you may find to one of us. More Websites. There is of course a huge number of websites that con- tain a wealth of information. Here are a few sample sites that are relevant for the book: 1. For spherical geometry in Figure 8.2.13 of the main text, see which gives a nice JAVA applet for parallel transport on the sphere. 2. For further information on the Sunshine formula (see § 4.1C of this internet supplement), see
Page iv 3. For the Genesis Orbit shown in Figure 4.1.11, see 4. For more on Newton, see for instance, and for Feynman, see 5. For surface integrals using Mathematica Notebooks, see Practice Final Examination. At the end of this Internet Supplement, students will find a Practice Final Examination that covers topics in the whole book, complete with solutions. We recommend, if you wish to prac- tice your skills, that you allow yourself 3 hours to take the exam and then self-mark it, keeping in mind that there is often more than one way to approach a problem.