000 a
999 _c32524
_d32524
008 230901b xxu||||| |||| 00| 0 eng d
020 _a9783030456528
082 _a516.07
_bNES
100 _aNestruev, Jet
245 _aSmooth manifolds and observables
250 _a2nd ed.
260 _bSpringer,
_c2020
_aCham :
300 _axviii, 433 p. ;
_bill.,
_c24 cm
365 _b49.99
_cEUR
_d94.90
490 _aGraduate texts in mathematics
504 _aIncludes bibliographical references and index.
520 _aThis 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.
650 _aAlgebra
650 _aAnalytic geometry
650 _aMathematics Topology
650 _aQuantum physics
942 _2ddc
_cBK