# 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.

 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