Derivation

Definition

Let $R$ be a commutative unital ring (resp. sheaf of commutative unital rings). A derivation of $R$ as a $R$-module is a map $D:R\to R$ (resp. a sheaf-theoretic map from $M$ to itself) such that:

• $D$ is $R$-linear (viz, its a map of $R$-modules)
• $D(f)=f(Dg)+(Df)g$

A derivation on a differential manifold is a derivation on its sheaf of differentiable functions, viewed as a sheaf of commutative unital rings.

Facts

Every vector field on a differential manifold gives rise to a derivation, and this gives a correspondence between vector fields and derivations. For full proof, refer: Vector field equals derivation