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| 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:
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| # The whole topological information of a [[smooth manifold]] <math>M</math> is encoded in the algebraic properties of its <math>\mathbb{R}</math>-[[Algebra (ring theory)|algebra]] of smooth functions <math>A=C^\infty (M),</math> as in the [[Banach–Stone theorem]].
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| # [[Vector bundle]]s over <math>M</math> correspond to projective finitely generated [[Module (mathematics)|module]]s over <math>A</math>, via the [[functor]] <math>\Gamma</math> which associates to a vector bundle its module of sections.
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| # [[Vector field]]s on <math>M</math> are naturally identified with [[Derivation (abstract algebra)|derivation]]s of the algebra <math>A</math>.
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| # More generally, a [[linear differential operator]] of order k, sending sections of a vector bundle <math>E\rightarrow M</math> to sections of another bundle <math>F\rightarrow M</math> is seen to be an <math>\mathbb{R}</math>-linear map <math>\Delta: \Gamma (E) \rightarrow \Gamma (F) </math> between the associated modules, such that for any ''k'' + 1 elements <math>f_0,\ldots, f_k\in A</math>:
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| <math>[f_k[f_{k-1}[\cdots[f_0,\Delta]\cdots]]=0</math>
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| where the bracket <math>[f,\Delta]:\Gamma(E)\rightarrow \Gamma(F)</math> is defined as the commutator
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| <math>[f,\Delta](s)=\Delta(f\cdot s)-f\cdot \Delta(s).</math>
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| Denoting the set of ''k''th order linear differential operators from an <math>A</math>-module <math>P</math> to an <math>A</math>-module <math>Q</math> with <math>\mathrm{Diff}_k(P,Q)</math> we obtain a bi-functor with values in the [[category (mathematics)|category]] of <math>A</math>-modules. Other natural concepts of calculus such as [[jet space]]s, [[differential form]]s are then obtained as [[Representable functor|representing object]]s of the functors <math>\mathrm{Diff}_k</math> and related functors.
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| Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects.
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| Replacing the real numbers <math>\mathbb{R}</math> with any [[commutative ring]], and the algebra <math>C^\infty(M)</math> 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 and cohomological physics|secondary calculus]]. Moreover the theory generalizes naturally to the setting of [[supercommutative algebra|graded commutative algebra]], allowing for a natural foundation of calculus on [[supermanifold]]s, [[graded manifold]]s and associated concepts like the [[Berezin integral]].
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| == See also ==
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| * [[Differential algebra]]
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| == References ==
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| *J. Nestruev, ''Smooth Manifolds and Observables'', Graduate Texts in Mathematics '''220''', Springer, 2002.
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| *I. S. Krasil'shchik, "Lectures on Linear Differential Operators over Commutative Algebras". Eprint [http://diffiety.ac.ru/preprint/99/01_99abs.htm DIPS-01/98]
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| *I. S. Krasil'shchik, "Calculus over Commutative Algebras: a concise user's guide", ''Acta Appl. Math.'' '''49''' (1997) 235–248; Eprint [http://www.diffiety.org/preprint/96/01_96abs.htm DIPS-01/99]
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| *I. S. Krasil'shchik, A. M. Verbovetsky, "Homological Methods in Equations of Mathematical Physics", ''Open Ed. and Sciences,'' Opava (Czech Rep.), 1998; Eprint [http://www.diffiety.org/preprint/98/07_98abs.htm DIPS-07/98].
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| *G. Sardanashvily, ''Lectures on Differential Geometry of Modules and Rings'', Lambert Academic Publishing, 2012; Eprint [http://arxiv.org/abs/0910.1515 arXiv:0910.1515] [math-ph] 137 pages.
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| [[Category:Commutative algebra]]
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| [[Category:Differential calculus]]
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Hi there, I am Yoshiko Villareal but I by no means truly favored that title. After being out of my occupation for years I became a production and distribution officer but I plan on changing it. To play croquet is the pastime I will by no means stop doing. Years ago we moved to Arizona but my spouse desires us to transfer.
Here is my website; bjjoutlet.com