differential field
Let be a field (ring) together with a derivation .The derivation must satisfy two properties:
- Additivity
;
- Leibniz’ Rule
.
A derivation is the algebraic abstraction of a derivative from ordinarycalculus. Thus the terms derivation, derivative, anddifferential are often used interchangeably.
Together, is referred to as a differential field (ring).The subfield (subring) of all elements with vanishing derivative,,is called the field (ring) of constants. Clearly, is -linear.
There are many notations for the derivation symbol, for example mayalso be denoted as , , , etc. When there is morethan one derivation , is referred to as apartial 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 derivativesand anti-derivatives (indefinite integrals), as well as ordinary and partial differentialequations and their solutions. There is an abundance of examples drawnfrom these areas.
- •
The trivial example is a field with for each . Here,nothing new is gained by introducing the derivation.
- •
The most common example is the field of rational functions over an indeterminant satisfying . The field of constants is. This is the setting for ordinary calculus.
- •
Another example is with two derivations and . The field of constants is and thederivations are extended to all elements from the properties ,, and .
- •
Consider the set of smooth functions
on a manifold . Theyform a ring (or a field if we allow formal inversion
of functionsvanishing in some places). Vector fields on act naturally asderivations on .
- •
Let be an algebra and be a one-parameter subgroup ofautomorphisms
of . Here is the infinitesimal generator of theseautomorphisms. From the properties of , must be a linear operatoron that satisfies the Leibniz rule
. So 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 |