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单词 DifferentialField
释义

differential field


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

Additivity

(a+b)=a+b;

Leibniz’ Rule

(ab)=ab+ab.

A derivation is the algebraic abstraction of a derivative from ordinarycalculus. Thus the terms derivation, derivative, anddifferentialMathworldPlanetmath 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 mayalso be denoted as da, δa, a, etc. When there is morethan one derivation i, (F,{i}) is referred to as apartial 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 derivativesand anti-derivatives (indefinite integrals), as well as ordinary and partial differentialequationsMathworldPlanetmath and their solutions. There is an abundance of examples drawnfrom 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 xand y. The field of constants is and thederivations 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. Theyform a ring (or a field if we allow formal inversionMathworldPlanetmathPlanetmath of functionsvanishing in some places). Vector fields on M act naturally asderivations on C(M).

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

Titledifferential field
Canonical nameDifferentialField
Date of creation2013-03-22 14:18:47
Last modified on2013-03-22 14:18:47
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id10
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 13N15
Classificationmsc 12H05
Related topicDifferentialPropositionalCalculus
Definesdifferential ring
Definespartial differential field
Definespartial differential ring
Definesfield of constants
Definesring of constants
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