In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a derivation is a function on an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
which generalizes certain features of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
operator. Specifically, given an algebra ''A'' over a
ring or a
field ''K'', a ''K''-derivation is a ''K''-
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
that satisfies
Leibniz's law:
:
More generally, if ''M'' is an ''A''-
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
, 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
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
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
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
; 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
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
. If the algebra ''A'' is noncommutative, then the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
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,
: