Jet Nestruev

Smooth Manifolds and Observables (eBook, PDF)

• Format: PDF

Produktbeschreibung

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. The authors offer an algebraic approach which is based on the fundamential notion of "observable" used by physicists, and which 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.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 222
• Erscheinungstermin: 6. April 2006
• Englisch
• ISBN-13: 9780387227399
• Artikelnr.: 43998832

Autorenporträt

Jet Nestruev is a collective of authors, who originally convened for a seminar run by Alexandre Vinogradov at the Mechanics and Mathematics Department of Moscow State University in 1969. In the present edition, Jet Nestruev consists of Alexander Astashov (Senior Researcher at the State Research Institute of Aviation Systems), Alexandre Vinogradov (Professor of Mathematics at Salerno University), Mikhail Vinogradov (Diffiety Institute), and Alexey Sossinsky (Professor at the Independent University of Moscow).

Inhaltsangabe

Foreword.- Preface.- 1. Introduction.- 2. Cutoff and Other Special Smooth Functions on R^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. Points, Spectra and Ghosts.- 9. The Differential Calculus as Part of Commutative Algebra.- 10. Symbols and the Hamiltonian Formalism.- 11. Smooth Bundles.- 12. Vector Bundles and Projective Modules.- 13. Localization.- 14. Differential 1-forms and Jets.- 15. Functors of the differential calculus and their representations.- 16. Cosymbols, Tensors, and Smoothness.- 17. Spencer Complexes and Differential Forms.- 18. The (co)chain complexes that come from the Spencer Sequence.- 19. Differential forms: classical and algebraic approach.- 20. Cohomology.- 21. Differential operators over graded algebras.- Afterword.- Appendix.- References.- Index.

Rezensionen

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)