Let be a commutative unital ring (resp. sheaf of commutative unital rings). A derivation of is a map (resp. a sheaf-theoretic map from to itself) such that:
A derivation on a differential manifold is a derivation on its sheaf of differentiable functions, viewed as a sheaf of commutative unital rings.
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