Advanced Calculus of Several Variables

Title......Page 1
Contents......Page 4
Preface......Page 6
1 The Vector Space Rn......Page 9
2 Subspaces of Rn......Page 13
3 Inner Products and Orthogonality......Page 18
4 Linear Mappings and Matrices......Page 28
5 The Kernel and Image of a Linear Mapping......Page 37
6 Determinants......Page 42
7 Limits and Continuity......Page 49
8 Elementary Topology of Rn......Page 57
II Multivariable Differential Calculus......Page 64
1 Curves in Rm......Page 65
2 Directional Derivatives and the Differential......Page 71
3 The Chain Rule......Page 84
4 Lagrange Multipliers and the Classification of Critical Points for Functions of Two Variables......Page 98
5 Maxima and Minima, Manifolds, and Lagrange Multipliers......Page 109
6 Taylor\'s Formula for Single-Variable Functions......Page 125
7 Taylor\'s Formula in Several Variables......Page 137
8 The Classification of Critical Points......Page 150
1 Newton\'s Method and Contraction Mappings......Page 168
2 The Multivariable Mean Value Theorem......Page 180
3 The Inverse and Implicit Mapping Theorems......Page 189
4 Manifolds in Rn......Page 204
5 Higher Derivatives......Page 209
IV Multiple Integrals......Page 211
1 Area and the 1-Dimensional Integral......Page 212
2 Volume and the n-Dimensional Integral......Page 222
3 Step Functions and Riemann Sums......Page 231
4 Iterated Integrals and Fubini\'s Theorem......Page 243
5 Change of Variables......Page 252
6 Improper Integrals and Absolutely Integrable Functions......Page 276
V Line and Surface Integrals; Differential Forms and Stokes\' Theorem......Page 294
1 Pathlength and Line Integrals......Page 295
2 Green\'s Theorem......Page 312
3 Multilinear Functions and the Area of a Parallelepiped......Page 330
4 Surface Area......Page 338
5 Differential Forms......Page 353
6 Stokes\' Theorem......Page 371
7 The Classical Theorems of Vector Analysis......Page 388
8 Closed and Exact Forms......Page 403
VI The Calculus of Variations......Page 410
1 Normed Vector Spaces and Uniform Convergence......Page 412
2 Continuous Linear Mappings and Differentials......Page 419
3 The Simplest Variational Problem......Page 426
4 The Isoperimetric Problem......Page 437
5 Multiple Integral Problems......Page 446
Appendix: The Completeness of R......Page 453
Suggested Reading......Page 457
Subject Index......Page 461