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 itsR -algebra of smooth functionsA = C ∞ ( M ) , as in the Banach–Stone theorem. - Vector bundles over
M correspond to projective finitely generated modules overA , via the functorΓ which associates to a vector bundle its module of sections. - Vector fields on
M are naturally identified with derivations of the algebraA . - More generally, a linear differential operator of order k, sending sections of a vector bundle
E → M to sections of another bundleF → M is seen to be anR -linear mapΔ : Γ ( E ) → Γ ( F ) between the associated modules, such that for any k + 1 elementsf 0 , … , f k ∈ A :
where the bracket
Denoting the set of kth order linear differential operators from an
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