diferensial field

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

$ (a+b)' = a' + b'$;
Leibniz’ Rule
$ (ab)' = a'b+ab'$.

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

There are many notations for the derivation symbol, for example $ a'$ 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).


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'=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'=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.

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