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Let R be a commutative unital ring (resp. sheaf of commutative unital rings). A derivation of R 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.


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