# differential field

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

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

Leibniz’ Rule

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

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

• Another example is $\mathbb{R}(x,y)$ with two derivations $\partial_{x}$ and $\partial_{y}$. The field of constants is $\mathbb{R}$ 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 $C^{\infty}(M)$ 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 $C^{\infty}(M)$.

• 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