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 antiderivatives (indefinite integrals), as well as ordinary and partial differential equations^{} and their solutions. There is an abundance of examples drawn from these areas.

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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.

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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.

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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 ${\partial}_{x}y={\partial}_{y}x=0$.

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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 ${C}^{\mathrm{\infty}}(M)$.

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Let $A$ be an algebra and ${U}_{t}=\mathrm{exp}(tu)$ be a oneparameter 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  20130322 14:18:47 
Last modified on  20130322 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 