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 :\Gamma (E)\to \Gamma (F)} between the associated modules, such that for any elements :