Smooth manifolds and observables / Jet Nestruev.

Includes bibliographical references (pages [217]-218) and index.

1. Introduction -- 2. Cutoff and Other Special Smooth Functions on R[superscript n] -- 3. Algebras and Points -- 4. Smooth Manifolds (Algebraic Definition) -- 5. Charts and Atlases -- 6. Smooth Maps -- 7. Equivalence of Coordinate and Algebraic Definitions -- 8. Spectra and Ghosts -- 9. The Differential Calculus as a Part of Commutative Algebra -- 10. Smooth Bundles -- 11. Vector Bundles and Projective Modules -- App. Observability Principle, Set Theory and the "Foundations of Mathematics" / A.M. Vinogradov.

"Completely new approach to the subject. This book is a self-contained introduction to fiber spaces and differential operators on smooth manifolds that is accessible to graduate students specializing in mathematics and physics. Over the last 20 years the authors developed an algebraic approach to the subject and they explain in this book why differential calculus on manifolds can be considered as an aspect of commutative algebra. This new approach is based on the fundamential notion of "observable" which is used by physicists and it will further the understanding of the mathematics underlying quantum field theory. The prerequisites for this book are a standard advanced calculus course as well as courses in linear algebra and algebraic structures."--Publisher's website.