Let be a field (ring) together with a derivation . The derivation must satisfy two properties:
- Leibniz’ Rule
There are many notations for the derivation symbol, for example may also be denoted as , , , etc. When there is more than one derivation , 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 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 the derivations are extended to all elements from the properties , , and .
|Date of creation||2013-03-22 14:18:47|
|Last modified on||2013-03-22 14:18:47|
|Last modified by||CWoo (3771)|
|Defines||partial differential field|
|Defines||partial differential ring|
|Defines||field of constants|
|Defines||ring of constants|