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Geometric Integration Theory

Hassler Whitney
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Mihaela Poplicher
, on

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 well-written 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 general problem of integration;
  • Some classical topics (on differentiation and integration and manifolds);
  • Indications of general theory.

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.

  • Part I: Classical Theory contains chapters on Grassmann algebra, Differential forms, Riemann integration theory, Smooth manifolds;
  • Part II: general Theory contains Abstract integration theory, Some relations between chains and functions, General properties of chains and cochains, Chains and cochains in open sets;
  • Part III: Lebesgue Theory contains Flat cochains and differential forms, Lipschitz mappings, Chains and additive set functions.

To make the book self-contained, 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


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