翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク differential calculus over commutative algebras ： ウィキペディア英語版 differential calculus over commutative algebras In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: # The whole topological information of a smooth manifold $M$ is encoded in the algebraic properties of its $\backslash mathbb$-algebra of smooth functions $A=C^\backslash infty\; (M),$ as in the Banach–Stone theorem. # Vector bundles over $M$ correspond to projective finitely generated modules over $A$, via the functor $\backslash Gamma$ which associates to a vector bundle its module of sections. # Vector fields on $M$ are naturally identified with derivations of the algebra $A$. # More generally, a linear differential operator of order k, sending sections of a vector bundle $E\backslash rightarrow\; M$ to sections of another bundle $F\backslash rightarrow\; M$ is seen to be an $\backslash mathbb$-linear map $\backslash Delta:\; \backslash Gamma\; (E)\; \backslash rightarrow\; \backslash Gamma\; (F)$ between the associated modules, such that for any ''k'' + 1 elements $f\_0,\backslash ldots,\; f\_k\backslash in\; A$: $]=0$ where the bracket $():\backslash Gamma(E)\backslash rightarrow\; \backslash Gamma(F)$ is defined as the commutator $()(s)=\backslash Delta(f\backslash cdot\; s)-f\backslash cdot\; \backslash Delta(s).$ Denoting the set of ''k''th order linear differential operators from an $A$-module $P$ to an $A$-module $Q$ with $\backslash mathrm\_k(P,Q)$ we obtain a bi-functor with values in the category of $A$-modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors $\backslash mathrm\_k$ and related functors. Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects. Replacing the real numbers $\backslash mathbb$ with any commutative ring, and the algebra $C^\backslash infty(M)$ with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral. == See also == * Differential algebra 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「differential calculus over commutative algebras」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.