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Derivation (differential algebra)
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{{Short description|Algebraic generalization of the derivative}} In [[mathematics]], a '''derivation''' is a function on an [[algebra over a field|algebra]] that generalizes certain features of the [[derivative]] operator. Specifically, given an algebra ''A'' over a [[ring (mathematics)|ring]] or a [[field (mathematics)|field]] ''K'', a ''K''-derivation is a ''K''-[[linear map]] {{nowrap|''D'' : ''A'' β ''A''}} that satisfies [[Product rule|Leibniz's law]]: :<math> D(ab) = a D(b) + D(a) b.</math> More generally, if ''M'' is an ''A''-[[bimodule]], a ''K''-linear map {{nowrap|''D'' : ''A'' β ''M''}} that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der<sub>''K''</sub>(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by {{nowrap|Der<sub>''K''</sub>(''A'', ''M'')}}. Derivations occur in many different contexts in diverse areas of mathematics. The [[partial derivative]] with respect to a variable is an '''R'''-derivation on the algebra of [[real-valued]] differentiable functions on '''R'''<sup>''n''</sup>. The [[Lie derivative]] with respect to a [[vector field]] is an '''R'''-derivation on the algebra of differentiable functions on a [[differentiable manifold]]; more generally it is a derivation on the [[tensor algebra]] of a manifold. It follows that the [[adjoint representation of a Lie algebra]] is a derivation on that algebra. The [[Pincherle derivative]] is an example of a derivation in [[abstract algebra]]. If the algebra ''A'' is noncommutative, then the [[commutator]] with respect to an element of the algebra ''A'' defines a linear [[endomorphism]] of ''A'' to itself, which is a derivation over ''K''. That is, :<math>[FG,N]=[F,N]G+F[G,N],</math> where <math>[\cdot,N]</math> is the commutator with respect to <math>N</math>. An algebra ''A'' equipped with a distinguished derivation ''d'' forms a [[differential algebra]], and is itself a significant object of study in areas such as [[differential Galois theory]].
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