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This book was first published by Princeton University Press in 1957. Although the text was completed in May 1956 (50 years ago!), it is so comprehensive and wellwritten that it is still very useful today.
The book begins with an introductory chapter that also gives an overall picture of the content. This includes:
The main body of the book has three parts, and I will list the chapters because I think it is the only way one can realize the complexity of the work and its connections/overlapping with algebraic topology.
To make the book selfcontained, the author introduced appendices on Vector and linear spaces, Geometric and topological preliminaries, as well as Analytical preliminaries.
In short, this is a very good book (otherwise, how could it have been republished after 50 years?) that can be used as a textbook for a course on geometric integration, but mostly as a comprehensive reference for anybody interested in the subject.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
Preface


Introduction  
A. The general problem of integration  
B. Some classical topics  
C. Indications of general theory  
Part I. Classical Theory  
1.  Grassmann algebra  
2.  Differential forms  
3.  Riemann integration theory  
4.  Smooth manifolds  
A. Manifolds in Euclidean space  
B. Triangulation of manifolds  
C. Cohomology in manifolds  
Part II.  General Theory  
5.  Abstract integration theory  
6.  Some relations between chains and functions  
7.  General properties of chains and cochains  
8.  Chains and cochains in open sets  
Part III.  Lebesgue Theory  
9.  Flat cochains and differential forms  
10.  Lipschitz mappings  
11.  Chains and additive set functions  
Appendix I.  Vector and linear spaces  
Appendix II.  Geometric and topological preliminaries  
Appendix III.  Analytical preliminaries  
Index of symbols  
Index of terms 