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In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Leibniz's law: : More generally, if ''M'' is an ''A''-bimodule, a ''K''-linear map that satisfies the Leibniz law is also called a derivation. The collection of all ''K''-derivations of ''A'' to itself is denoted by Der''K''(''A''). The collection of ''K''-derivations of ''A'' into an ''A''-module ''M'' is denoted by . 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''n''. 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. 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''. 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. ==Properties== The Leibniz law itself has a number of immediate consequences. Firstly, if , then it follows by mathematical induction that : (the last equality holds if, for all commutes with ). In particular, if ''A'' is commutative and , then this formula simplifies to the familiar power rule . Secondly, if ''A'' has a unit element 1, then since . Moreover, because ''D'' is ''K''-linear, it follows that "the derivative of any constant function is zero"; more precisely, for any , . If is a subring, and ''A'' is a ''k''-algebra, then there is an inclusion : since any ''K''-derivation is ''a fortiori'' a ''k''-derivation. The set of ''k''-derivations from ''A'' to ''M'', is a module over ''k''. Furthermore, the ''k''-module Der''k''(''A'') forms a Lie algebra with Lie bracket defined by the commutator: : It is readily verified that the Lie bracket of two derivations is again a derivation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Derivation (differential algebra)」の詳細全文を読む スポンサード リンク
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