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Derivation
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Derivation

In abstract algebra, a derivation on an algebra A over a field k is a linear map D:A→A that satisfies Leibniz' law:

D(ab) = (Da)b + a(Db).

In addition, if A is unital,

D(1)=0

Examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme.

See also: Kähler differential

If we have a Z2 graded algebra A, D is an antiderivation if

D(ab)=(Da)b+(-1)aa(Db)

The same comment about D(1)=0 if it is unital applies.


Derivation may also be used as a synonym for proof, particularly for formulae.


In music using the twelve tone technique see derived row, where a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator.