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ISBN 10: 0521829259
ISBN 13: 978-0521829250
Author: Duistermaat, Kolk, van Braam Houckgeest
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.
Introduction
Chapter 6 Integration
6.1 Rectangles
6.2 Riemann integrability
6.3 Jordan measurability
6.4 Successive integration
6.5 Examples of successive integration
6.6 Change of Variables Theorem: formulation and examples
6.7 Partitions of unity
6.8 Approximation of Riemann integrable functions
6.9 Proof of Change of Variables Theorem
6.10 Absolute Riemann integrability
6.11 Application of integration: Fourier transformation
6.12 Dominated convergence
6.13 Appendix: two other proofs of Change of Variables Theorem
Chapter 7 Integration over Submanifolds
7.1 Densities and integration with respect to density
7.2 AbsoluteRiemann integrabilitywith respect to density
7.3 Euclidean d-dimensional density
7.4 Examples of Euclidean densities
7.5 Open sets at one side of their boundary
7.6 Integration of a total derivative
7.7 Generalizations of the preceding theorem
7.8 Gauss? Divergence Theorem
7.9 Applications of Gauss? Divergence Theorem
Chapter 8 Oriented Integration
8.1 Line integrals and properties of vector fields
8.2 Antidifferentiation
8.3 Green?s and Cauchy?s Integral Theorems
8.4 Stokes? Integral Theorem
8.5 Applications of Stokes? Integral Theorem
8.6 Apotheosis: differential forms and Stokes? Theorem
8.7 Properties of differential forms
8.8 Applications of differential forms
8.9 Homotopy Lemma
8.10 Poincar??s Lemma
8.11 Degree of mapping
Exercises
Exercises for Chapter 6
Exercises for Chapter 7
Exercises for Chapter 8
Notation
Index
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Tags: Duistermaat, Kolk, van Braam Houckgeest, Multidimensional Real, Analysis II
