Differential algebra

Derivation (differential algebra)

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 D : A → A that satisfies Leibniz's law: More generally, if M is an A-bimodule, 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 partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on Rn. 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, where is the commutator with respect to . 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. (Wikipedia).

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Derivative of the exponential map | Commutator | Differential algebra | Differential form | Exterior derivative | Formal power series | Product rule | Subring | Tangent space | Derivative | Superalgebra | Wirtinger derivatives | Bimodule | Algebra over a field | Partial derivative | Hasse derivative | Pincherle derivative | Hasse–Schmidt derivation | Interior product | Differentiable manifold | P-derivation | Tensor algebra | Linear map | Differential Galois theory | Mathematics | Field (mathematics) | Lie derivative | Kähler differential | Lie algebra | Mathematical induction | Ring (mathematics) | Abstract algebra | Differential geometry | Endomorphism | Vector field | Module (mathematics)