Derivation (differential algebra)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a derivation is a function on an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> that generalizes certain features of the <a href="/facts/Derivative/7Ito70Dh">derivative</a> operator. Specifically, given an algebra A over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> or a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K, a K-derivation is a K-<a href="/facts/Linear_map/Q1PBKNVV">linear map</a> D : A → A that satisfies <a href="/facts/Product_rule/MyKxsP5t">Leibniz's law</a>:

D
        (
        a
        b
        )
        =
        a
        D
        (
        b
        )
        +
        D
        (
        a
        )
        b
        .
      
    
    {\displaystyle D(ab)=aD(b)+D(a)b.}

More generally, if M is an A-<a href="/facts/Bimodule/NskmNGQ1">bimodule</a>, a K-linear map D : A → M that satisfies the Leibniz law is also called a derivation. The collection of all K-derivations of A to itself is denoted by DerK(A). The collection of K-derivations of A into an A-module M is denoted by DerK(A, M).
Derivations occur in many different contexts in diverse areas of mathematics. The <a href="/facts/Partial_derivative/JWHsBkAu">partial derivative</a> with respect to a variable is an R-derivation on the algebra of <a href="/facts/Real-valued/mGYQu2xt">real-valued</a> differentiable functions on Rn. The <a href="/facts/Lie_derivative/1KSJvrzk">Lie derivative</a> with respect to a <a href="/facts/Vector_field/ccFNuPPp">vector field</a> is an R-derivation on the algebra of differentiable functions on a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a>; more generally it is a derivation on the <a href="/facts/Tensor_algebra/H7CKKryf">tensor algebra</a> of a manifold. It follows that the <a href="/facts/Adjoint_representation_of_a_Lie_algebra/bWG4c2su">adjoint representation of a Lie algebra</a> is a derivation on that algebra. The <a href="/facts/Pincherle_derivative/1hvlMmjs">Pincherle derivative</a> is an example of a derivation in <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>. If the algebra A is noncommutative, then the <a href="/facts/Commutator/eYxzFmFY">commutator</a> with respect to an element of the algebra A defines a linear <a href="/facts/Endomorphism/s7NNlSea">endomorphism</a> of A to itself, which is a derivation over K. That is,

[
        F
        G
        ,
        N
        ]
        =
        [
        F
        ,
        N
        ]
        G
        +
        F
        [
        G
        ,
        N
        ]
        ,
      
    
    {\displaystyle [FG,N]=[F,N]G+F[G,N],}

where 
 
 
 
 [
 ⋅
 ,
 N
 ]
 
 
 {\displaystyle [\cdot ,N]}
 
 is the commutator with respect to 
 
 
 
 N
 
 
 {\displaystyle N}
 
. An algebra A equipped with a distinguished derivation d forms a <a href="/facts/Differential_algebra/CXbpIMgk">differential algebra</a>, and is itself a significant object of study in areas such as <a href="/facts/Differential_Galois_theory/neKl6CLC">differential Galois theory</a>.

Derivation (differential algebra) open-in-new

Derivation (differential algebra)