Advanced calculus of several variables

✍ Scribed by Edwards C.H.

• Publisher: AP
• Year: 1973
• Tongue: English
• Leaves: 465
• Category: Library

✦ Synopsis

Modern conceptual treatment of multivariable calculus, emphasizing the interplay of geometry and analysis via linear algebra and the approximation of nonlinear mappings by linear ones. At the same time, ample attention is paid to the classical applications and computational methods. Hundreds of examples, problems and figures. 1973 edition.

✦ Table of Contents

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