Multidimensional real analysis. 2 Integration

0521829259......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
Acknowledgments......Page 14
Introduction......Page 16
6.1 Rectangles......Page 20
6.2 Riemann integrability......Page 22
6.3 Jordan measurability......Page 26
6.4 Successive integration......Page 32
6.5 Examples of successive integration......Page 36
6.6 Change of Variables Theorem: formulation and examples......Page 41
6.7 Partitions of unity......Page 49
6.8 Approximation of Riemann integrable functions......Page 52
6.9 Proof of Change of Variables Theorem......Page 54
6.10 Absolute Riemann integrability......Page 58
6.11 Application of integration: Fourier transformation......Page 63
6.12 Dominated convergence......Page 68
6.13 Appendix: two other proofs of Change of Variables Theorem......Page 74
7.1 Densities and integration with respect to density......Page 84
7.2 AbsoluteRiemann integrabilitywith respect to density......Page 89
7.3 Euclidean d-dimensional density......Page 92
7.4 Examples of Euclidean densities......Page 95
7.5 Open sets at one side of their boundary......Page 108
7.6 Integration of a total derivative......Page 115
7.7 Generalizations of the preceding theorem......Page 119
7.8 Gauss\' Divergence Theorem......Page 124
7.9 Applications of Gauss\' Divergence Theorem......Page 127
8.1 Line integrals and properties of vector fields......Page 134
8.2 Antidifferentiation......Page 143
8.3 Green\'s and Cauchy\'s Integral Theorems......Page 148
8.4 Stokes\' Integral Theorem......Page 154
8.5 Applications of Stokes\' Integral Theorem......Page 158
8.6 Apotheosis: differential forms and Stokes\' Theorem......Page 164
8.7 Properties of differential forms......Page 173
8.8 Applications of differential forms......Page 178
8.9 Homotopy Lemma......Page 182
8.10 Poincaré\'s Lemma......Page 186
8.11 Degree of mapping......Page 188
Exercises for Chapter 6......Page 196
Exercises for Chapter 7......Page 274
Exercises for Chapter 8......Page 326
Notation......Page 376
Index......Page 380