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 algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles. *

• Smooth Manifolds and Observables is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics. This second edition adds ten new chapters to further develop the notion of differential calculus over commutative algebras, showing it to be a generalization of the differential calculus on smooth manifolds. Applications to diverse areas, such as symplectic manifolds, de Rham cohomology, and Poisson brackets are explored. Additional examples of the basic functors of the theory are presented alongside numerous new exercises, providing readers with many more opportunities to practice these concepts.

Motivates the algebraic study of smooth manifolds by looking at them from the point of view of physics, in particular by using the observability principle Bridges between ideas from physics, geometry, and algebra, offering new perspectives to students and researchers in each field Incorporates a wide array of exercises and detailed illustrations, allowing readers to better familiarize themselves with the material Includes ten new chapters in the second edition, as well as additional exercises, examples, and illustrations Includes supplementary material: sn.pub/extras

Autorentext

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).

Inhalt
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.