Derivation: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math>R</math> be a [[commutative unital ring]] (resp. [[sheaf of commutative unital rings]]). A derivation of <math>R</math> | Let <math>R</math> be a [[commutative unital ring]] (resp. [[sheaf of commutative unital rings]]). A derivation of <math>R</math> is a map <math>D: | ||
R \to R</math> (resp. a [[sheaf-theoretic map]] from <math>M</math> to itself) such that: | R \to R</math> (resp. a [[sheaf-theoretic map]] from <math>M</math> to itself) such that: | ||
Revision as of 09:37, 27 August 2007
Definition
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:
- is -linear (viz, its a map of -modules)
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