differential field

Let F be a field (ring) together with a derivation ():FF. The derivation must satisfy two properties:



Leibniz’ Rule


A derivation is the algebraic abstraction of a derivative from ordinary calculus. Thus the terms derivation, derivative, and differentialMathworldPlanetmath are often used interchangeably.

Together, (F,) is referred to as a differential field (ring). The subfieldMathworldPlanetmath (subring) of all elements with vanishing derivative, K={aFa=0}, is called the field (ring) of constants. Clearly, () is K-linear.

There are many notations for the derivation symbol, for example a may also be denoted as da, δa, a, etc. When there is more than one derivation i, (F,{i}) is referred to as a partial differential field (ring).

1 Examples

Differential fields and rings (together under the name of differential algebra) are a natural setting for the study of algebraic properties of derivatives and anti-derivatives (indefinite integrals), as well as ordinary and partial differential equationsMathworldPlanetmath and their solutions. There is an abundance of examples drawn from these areas.

  • The trivial example is a field F with a=0 for each aF. Here, nothing new is gained by introducing the derivation.

  • The most common example is the field of rational functions (z) over an indeterminant satisfying z=1. The field of constants is . This is the setting for ordinary calculus.

  • Another example is (x,y) with two derivations x and y. The field of constants is and the derivations are extended to all elements from the properties xx=1, yy=1, and xy=yx=0.

  • Consider the set of smooth functionsMathworldPlanetmath C(M) on a manifold M. They form a ring (or a field if we allow formal inversionMathworldPlanetmathPlanetmath of functions vanishing in some places). Vector fields on M act naturally as derivations on C(M).

  • Let A be an algebra and Ut=exp(tu) be a one-parameter subgroup of automorphismsMathworldPlanetmathPlanetmathPlanetmath of A. Here u is the infinitesimal generator of these automorphisms. From the properties of Ut, u must be a linear operator on A that satisfies the Leibniz rulePlanetmathPlanetmath u(ab)=u(a)b+au(b). So (A,u) can be considered a differential ring.

Title differential field
Canonical name DifferentialField
Date of creation 2013-03-22 14:18:47
Last modified on 2013-03-22 14:18:47
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 13N15
Classification msc 12H05
Related topic DifferentialPropositionalCalculus
Defines differential ring
Defines partial differential field
Defines partial differential ring
Defines field of constants
Defines ring of constants