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- Differential_calculus_over_commutative_algebras abstract "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 is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem. Vector bundles over correspond to projective finitely generated modules over , via the functor which associates to a vector bundle its module of sections. Vector fields on are naturally identified with derivations of the algebra . More generally, a linear differential operator of order k, sending sections of a vector bundle to sections of another bundle is seen to be an -linear map between the associated modules, such that for any k + 1 elements where the bracket is defined as the commutatorDenoting the set of kth order linear differential operators from an -module to an -module with we obtain a bi-functor with values in the category of -modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors 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 with any commutative ring, and the algebra 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.".
- Differential_calculus_over_commutative_algebras wikiPageExternalLink 0910.1515.
- Differential_calculus_over_commutative_algebras wikiPageExternalLink 01_96abs.htm.
- Differential_calculus_over_commutative_algebras wikiPageExternalLink 01_99abs.htm.
- Differential_calculus_over_commutative_algebras wikiPageExternalLink 07_98abs.htm.
- Differential_calculus_over_commutative_algebras wikiPageID "7147956".
- Differential_calculus_over_commutative_algebras wikiPageRevisionID "600017888".
- Differential_calculus_over_commutative_algebras hasPhotoCollection Differential_calculus_over_commutative_algebras.
- Differential_calculus_over_commutative_algebras subject Category:Commutative_algebra.
- Differential_calculus_over_commutative_algebras subject Category:Differential_calculus.
- Differential_calculus_over_commutative_algebras comment "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 is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem.".
- Differential_calculus_over_commutative_algebras label "Differential calculus over commutative algebras".
- Differential_calculus_over_commutative_algebras label "Дифференциальное исчисление над коммутативными алгебрами".
- Differential_calculus_over_commutative_algebras sameAs m.025v43_.
- Differential_calculus_over_commutative_algebras sameAs Q5275336.
- Differential_calculus_over_commutative_algebras sameAs Q5275336.
- Differential_calculus_over_commutative_algebras wasDerivedFrom Differential_calculus_over_commutative_algebras?oldid=600017888.
- Differential_calculus_over_commutative_algebras isPrimaryTopicOf Differential_calculus_over_commutative_algebras.