differential field

Let $F$ be a field (ring) together with a derivation ${(\cdot )}^{\prime }:F\to F$. The derivation must satisfy two properties:

Additivity

${(a+b)}^{\prime }=a^{\prime }+b^{\prime }$;

Leibniz’ Rule

${(ab)}^{\prime }=a^{\prime }b+a{b}^{\prime }$.

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

Together, $(F,{}^{\prime })$ is referred to as a differential field (ring). The subfield (subring) of all elements with vanishing derivative, $K=\{a\in F\mid a^{\prime }=0\}$, is called the field (ring) of constants. Clearly, ${(\cdot )}^{\prime }$ is $K$-linear.

There are many notations for the derivation symbol, for example $a^{\prime }$ may also be denoted as $da$, $\delta a$, $\partial a$, etc. When there is more than one derivation ${\partial }_i$, $(F,\{{\partial }_i\})$ 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 equations and their solutions. There is an abundance of examples drawn from these areas.

• The trivial example is a field $F$ with $a^{\prime }=0$ for each $a\in F$. Here, nothing new is gained by introducing the derivation.

• The most common example is the field of rational functions $ℝ(z)$ over an indeterminant satisfying $z^{\prime }=1$. The field of constants is $ℝ$. This is the setting for ordinary calculus.

• Another example is $ℝ(x,y)$ with two derivations ${\partial }_x$ and ${\partial }_y$. The field of constants is $ℝ$ and the derivations are extended to all elements from the properties ${\partial }_{x}x=1$, ${\partial }_{y}y=1$, and ${\partial }_{x}y={\partial }_{y}x=0$.

• Consider the set of smooth functions $C^{\mathrm{\infty }}(M)$ on a manifold $M$. They form a ring (or a field if we allow formal inversion of functions vanishing in some places). Vector fields on $M$ act naturally as derivations on $C^{\mathrm{\infty }}(M)$.

• Let $A$ be an algebra and $U_t=\exp (tu)$ be a one-parameter subgroup of automorphisms of $A$. Here $u$ is the infinitesimal generator of these automorphisms. From the properties of $U_t$, $u$ must be a linear operator on $A$ that satisfies the Leibniz rule $u(ab)=u(a)b+au(b)$. 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