Derivation - Revision history

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2008-05-18T19:38:01Z

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Vipul: /* Definition */

2007-08-27T09:37:42Z

Definition

← Older revision		Revision as of 09:37, 27 August 2007
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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> ~~as a <math>R</math>-module~~ is a map <math>D:		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:

Vipul at 09:37, 27 August 2007

2007-08-27T09:37:06Z

New page

==Definition==

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

* <math>D</math> is <math>R</math>-linear (viz, its a map of <math>R</math>-modules)
* <math>D(f) = f(Dg) + (Df)g</math>

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. {{proofat|[[Vector field equals derivation]]}}