Preface. PART I BACKGROUND MATERIAL. 1 Sets and Functions. 1.1 Sets in General. 1.2 Sets of Numbers. 1.3 Functions. 2 Real Numbers. 2.1 Review of the Order Relations. 2.2 Completeness of Real Numbers. 2.3 Sequences of Real Numbers. 2.4 Subsequences. 2.5 Series of Real Numbers. 2.6 Intervals and Connected Sets. 3 Vector Functions. 3.1 Vector Spaces: The Basics. 3.2 Bilinear Functions. 3.3 Multilinear Functions. 3.4 Inner Products. 3.5 Orthogonal Projections. 3.6 Spectral Theorem. PART II DIFFERENTIATION. 4 Normed Vector Spaces. 4.1 Preliminaries. 4.2 Convergence in Normed Spaces. 4.3 Norms of Linear and Multilinear Transformations. 4.4 Continuity in Normed Spaces. 4.5 Topology of Normed Spaces. 5 Derivatives. 5.1 Functions of a Real Variable. 5.2 Differentiable Functions. 5.3 Existence of Derivatives. 5.4 Partial Derivatives. 5.5 Rules of Differentiation. 5.6 Differentiation of Products. 6 Diffeomorphisms and Manifolds. 6.1 The Inverse Function Theorem. 6.2 Graphs. 6.3 Manifolds in Parametric Representations. 6.4 Manifolds in Implicit Representations. 6.5 Differentiation on Manifolds. 7 Higher-Order Derivatives. 7.1 Definitions. 7.2 Change of Order in Differentiation. 7.3 Sequences of Polynomials. 7.4 Local Extremal Values. PART III INTEGRATION. 8 Multiple Integrals. 8.1 Jordan Sets and Volume. 8.2 Integrals. 8.3 Images of Jordan Sets. 8.4 Change of Variables. 9 Integration on Manifolds. 9.1 Euclidean Volumes. 9.2 Integration on Manifolds. 9.3 Oriented Manifolds. 9.4 Integrals of Vector Fields. 9.5 Integrals of Tensor Fields. 9.6 Integration on Graphs. 10 Stokes' Theorem. 10.1 Basic Stokes' Theorem. 10.2 Flows. 10.3 Flux and Change of Volume in a Flow. 10.4 Exterior Derivatives. 10.5 Regular and Almost Regular Sets. 10.6 Stokes' Theorem on Manifolds. PART IV APPENDICES. Appendix A: Construction of the Real Numbers. A.1 Field and Order Axioms in Q. A.2 Equivalence Classes of Cauchy Sequences in Q. A.3 Completeness of R. Appendix B: Dimension of a Vector Space. B.1 Bases and Linearly Independent Subsets. Appendix C: Determinants. C.1 Permutations. C.2 Determinants of Square Matrices. C.3 Determinant Functions. C.4 Determinant of a Linear Transformation. C.5 Determinants on Cartesian Products. C.6 Determinants in Euclidean Spaces. C.7 Trace of an Operator. Appendix D: Partitions of Unity. D.1 Partitions of Unity. Index
Author(s): Mustafa A. Akcoglu, Paul F.A. Bartha, Dzung Minh Ha
Publisher: Wiley
Year: 2009
Language: English
Pages: 479
Tags: Математика;Математический анализ;
Cover......Page 1
Analysis in Vector Spaces: A Course in Advanced Calculus......Page 5
CONTENTS......Page 7
Preface......Page 11
PART I BACKGROUND MATERIAL......Page 15
1.1 Sets in General......Page 17
1.2 Sets of Numbers......Page 24
1.3 Functions......Page 31
2 Real Numbers......Page 45
2.1 Review of the Order Relations......Page 46
2.2 Completeness of Real Numbers......Page 50
2.3 Sequences of Real Numbers......Page 54
2.4 Subsequences......Page 59
2.5 Series of Real Numbers......Page 64
2.6 Intervals and Connected Sets......Page 68
3 Vector Functions......Page 75
3.1 Vector Spaces: The Basics......Page 76
3.2 Bilinear Functions......Page 96
3.3 Multilinear Functions......Page 102
3.4 Inner Products......Page 109
3.5 Orthogonal Projections......Page 117
3.6 Spectral Theorem......Page 123
PART II DIFFERENTIATION......Page 135
4 Normed Vector Spaces......Page 137
4.1 Preliminaries......Page 138
4.2 Convergence in Normed Spaces......Page 142
4.3 Norms of Linear and Multilinear Transformations......Page 149
4.4 Continuity in Normed Spaces......Page 156
4.5 Topology of Normed Spaces......Page 170
5 Derivatives......Page 189
5.1 Functions of a Real Variable......Page 190
5.2 Differentiable Functions......Page 204
5.3 Existence of Derivatives......Page 215
5.4 Partial Derivatives......Page 219
5.5 Rules of Differentiation......Page 225
5.6 Differentiation of Products......Page 232
6 Diffeomorphisms and Manifolds......Page 239
6.1 The Inverse Function Theorem......Page 240
6.2 Graphs......Page 252
6.3 Manifolds in Parametric Representations......Page 257
6.4 Manifolds in Implicit Representations......Page 266
6.5 Differentiation on Manifolds......Page 274
7.1 Definitions......Page 281
7.2 Change of Order in Differentiation......Page 284
7.3 Sequences of Polynomials......Page 287
7.4 Local Extremal Values......Page 296
PART III INTEGRATION......Page 299
8 Multiple Integrals......Page 301
8.1 Jordan Sets and Volume......Page 303
8.2 Integrals......Page 317
8.3 Images of Jordan Sets......Page 335
8.4 Change of Variables......Page 342
9 Integration on Manifolds......Page 353
9.1 Euclidean Volumes......Page 354
9.2 Integration on Manifolds......Page 359
9.3 Oriented Manifolds......Page 367
9.4 Integrals of Vector Fields......Page 375
9.5 Integrals of Tensor Fields......Page 380
9.6 Integration on Graphs......Page 385
10 Stokes' Theorem......Page 395
10.1 Basic Stokes' Theorem......Page 396
10.2 Flows......Page 400
10.3 Flux and Change of Volume in a Flow......Page 404
10.4 Exterior Derivatives......Page 410
10.5 Regular and Almost Regular Sets......Page 415
10.6 Stokes' theorem on Manifolds......Page 426
PART IV APPENDICES......Page 431
Appendix A: Construction of the real numbers......Page 433
A.1 Field and Order Axioms in Q......Page 434
A.2 Equivalence Classes of Cauchy Sequences in Q......Page 435
A.3 Completeness of R......Page 441
Appendix B: Dimension of a vector space......Page 445
B.1 Bases and linearly independent subsets......Page 446
C.1 Permutations......Page 449
C.2 Determinants of Square Matrices......Page 451
C.3 Determinant Functions......Page 453
C.4 Determinant of a Linear Transformation......Page 457
C.5 Determinants on Cartesian Products......Page 458
C.6 Determinants in Euclidean Spaces......Page 459
C.7 Trace of an Operator......Page 462
Appendix D: Partitions of unity......Page 465
D.1 Partitions of Unity......Page 466
Index......Page 469
