Smooth Manifolds and Observables. 2nd ed. 2020

種類:

電子ブック

責任表示:

by Jet Nestruev

出版情報:

Cham : Springer International Publishing : Imprint: Springer, 2020

著者名:

• Nestruev, Jet.
• SpringerLink (Online service)

シリーズ名:

Graduate Texts in Mathematics ; 220

ISBN:

9783030456504 [3030456501]    

注記:

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.
This textbook demonstrates how differential calculus, smooth manifolds, and commutative algebra constitute a unified whole, despite having arisen at different times and under different circumstances. Motivating this synthesis is the mathematical formalization of the process of observation from classical physics. A broad audience will appreciate this unique approach for the insight it gives into the underlying connections between geometry, physics, and commutative algebra. 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 a

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