Smooth Manifolds and Observables

By: Jet Nestruev

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Hardcover | 4 October 2002

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244 Pages

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24.13 x 16.51 x 1.27

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Smooth Manifolds and Observables is about the differential calculus, smooth manifolds, and commutative algebra. While these theories arose at different times and under completely different circumstances, this book demonstrates how they constitute a unified whole. The motivation behind this synthesis is the mathematical formalization of the process of observation in classical physics. The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles. This unique textbook contains a large number of exercises and is intended for advanced undergraduates, graduate students, and research mathematicians and physicists.

Industry Reviews

From the reviews:

"Main themes of the book are manifolds, fibre bundles and differential operators acting on sections of vector bundles. ... A classical treatment of these topics starts with a coordinate description of a manifold M ... . The present book is based on an alternative point of view, where calculus on manifolds is treated as a part of commutative algebra. ... The book contains quite a few exercises and many useful illustrations." (EMS, September, 2004)

"The book provides a self-contained introduction to the theory of smooth manifolds and fibre bundles, oriented towards graduate students in mathematics and physics. The approach followed here, however, substantially differs from most textbooks on manifold theory. ... This book is certainly quite interesting and may appeal even to people who merely want to study algebraic geometry, in the sense that they will gain extra insight here by the attention which is paid to making certain constructions in algebraic geometry physically or intuitively acceptable." (Willy Sarlet, Zentralblatt MATH, Vol. 1021, 2003)

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Preface to English Edition
Preface
Introduction
Cutoff and Other Special Smooth Functions on R^n
Algebras and Points
Smooth Manifolds (Algebraic Definition)
Charts and Atlases
Smooth Maps
Equivalence of Coordinate and Algebraic Definitions
Spectra and Ghosts
The Differential Calculus as a Part of Commutative Algebra
Smooth Bundles
Vector Bundles and Projective Modules
Afterword
Appendix
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

Smooth Manifolds and Observables

At a Glance

Hardcover

Industry Reviews

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