# Calculus of Several Variables

Springer Science & Business Media, Dec 6, 2012 - Mathematics - 619 pages
The present course on calculus of several variables is meant as a text, either for one semester following A First Course in Calculus, or for a year if the calculus sequence is so structured. For a one-semester course, no matter what, one should cover the first four chapters, up to the law of conservation of energy, which provides a beautiful application of the chain rule in a physical context, and ties up the mathematics of this course with standard material from courses on physics. Then there are roughly two possibilities: One is to cover Chapters V and VI on maxima and minima, quadratic forms, critical points, and Taylor's formula. One can then finish with Chapter IX on double integration to round off the one-term course. The other is to go into curve integrals, double integration, and Green's theorem, that is Chapters VII, VIII, IX, and X, §1. This forms a coherent whole.

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### Contents

 PART 2 4 The Norm of a Vector 17 5 Parametric Lines 32 CHAPTER II 48 2 Length of Curves 62 CHAPTER IV 76 3 Differentiability and Gradient 77 The Chain Rule and the Gradient 87
 CHAPTER X 269 CHAPTER XI 292 3 Center of Mass 313 3 Surface Integrals 333 6 Stokes Theorem 355 2 Multiplication of Matrices 372 CHAPTER XIV 385 2 Linear Mappings 392

 3 Directional Derivative 99 5 The Law of Conservation of Energy 111 CHAPTER V 123 3 Lagrange Multipliers 135 CHAPTER VI 143 2 The Quadratic Term at Critical Points 149 3 Algebraic Study of a Quadratic Form 155 4 Partial Differential Operators 162 5 The General Expression for Taylors Formula 170 CHAPTER 183 3 An Important Special Vector Field 194 5 Proof of the Local Existence Theorem 201 1 Definition and Evaluation of Curve Integrals 207 2 The Reverse Path 217 3 Curve Integrals When the Vector Field Has a Potential Function 220 CHAPTER IX 233 3 Polar Coordinates 252
 3 Geometric Applications 398 4 Composition and Inverse of Mappings 404 CHAPTER XV 412 3 Additional Properties of Determinants 420 4 Independence of Vectors 428 CHAPTER XVI 434 3 The Chain Rule 440 5 Implicit Functions 446 CHAPTER XVII 453 2 Dilations 463 3 Change of Variables Formula in Two Dimensions 469 5 Change of Variables Formula in Three Dimensions 478 APPENDIX 487 2 Computation of Fourier Series 494 Answers 1 Index 108 Copyright