VC5IntSup

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

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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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Page i Contents Preface iii 2 DiFerentiation 1 2.7 Some Technical Diferentiation Theorems ........... 1 3 Higher-Order Derivatives and Extrema 17 3.4A Second Derivative Test: Constrained Extrema ....... 17 3.4B ProoF oF the Implicit ±unction Theorem .......... 21 4Vector Valued ±unctions 25 4.1A Equilibria in Mechanics ................... 25 4.1B Rotations and the Sunshine ±ormula ............ 30 4.1C The Principle oF Least Action ................ 42 4.4 ±lows and the Geometry oF the Divergence ......... 58 5 Double and Triple Integrals 65 5.2 Alternative De²nition oF the Integral 65 5.6 Technical Integration Theorems ............... 71 6I ntegrals 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 Diferentials ....................... 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

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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 diFerent 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: http://www.whfreeman.com/MarsdenVC5e Please send any new corrections you may ±nd 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. ²or spherical geometry in ²igure 8.2.13 of the main text, see http://torus.math.uiuc.edu/jms/java/dragsphere/ which gives a nice JAVA applet for parallel transport on the sphere. 2. ²or further information on the Sunshine formula (see § 4.1C of this internet supplement), see http://www.math.niu.edu/~rusin/uses-math/position.sun/
Page iv 3. For the Genesis Orbit shown in Figure 4.1.11, see http://genesismission.jpl.nasa.gov/ 4. For more on Newton, see for instance, http://scienceworld.wolfram.com/biography/Newton.html and for Feynman, see http://www.feynman.com/ 5. For surface integrals using Mathematica Notebooks, see http://www.math.umd.edu/~jmr/241/surfint.htm Practice Final Examination. At the end of this Internet Supplement, students will ±nd 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.

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

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