Multidimensional real analysis. 2 Integration

Volume 2 provides a comprehensive review of integral analysis in multidimensional Euclidean space. 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 Part one of the authors'comprehensive and innovative work on multidimensional real analysis. This book is based on extensive teaching experience at Utrecht University and gives a thorough account of differential analysis in multidimensional Euclidean space. It is an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects this book presents an original treatment of the subject and it contains many results and exercises that cannot be found elsewhere. The numerous exercises illustrate a variety of applications in mathematics and physics. This combined with the exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study. Part two of the authors'comprehensive and innovative work on multidimensional real analysis. This book is based on extensive teaching experience at Utrecht University and gives a thorough account of integral analysis in multidimensional Euclidean space. It is an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects this book presents an original treatment of the subject and it contains many results and exercises that cannot be found elsewhere. The numerous exercises illustrate a variety of applications in mathematics and physics. This combined with the exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study. Part one of the authors' comprehensive and innovative work on multidimensional real analysis. This book is based on extensive teaching experience at Utrecht University and gives a thorough account of differential analysis in multidimensional Euclidean space. It is an ideal preparation for students who wish to go on to more advanced study. The notation is carefully organized and all proofs are clean, complete and rigorous. The authors have taken care to pay proper attention to all aspects of the theory. In many respects this book presents an original treatment of the subject and it contains man Part two of the authors' comprehensive and innovative work on multidimensional real analysis. Numerous illustrative exercises combined with an exhaustive and transparent treatment of subject matter make the book ideal as either the text for a course, a source of problems for a seminar or for self study. 1. Differentiation -- 2. Integration. J.j. Duistermaat, J.a.c. Kolk ; Translated From Dutch By J.p. Van Braam Houckgeest. Includes Indexes.