Calculus of Several Variables

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 619 pages
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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
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