- 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 .
- Consider the set of smooth functions on a manifold . They form a ring (or a field if we allow formal inversion of functions vanishing in some places). Vector fields on act naturally as derivations on .
- Let be an algebra and be a one-parameter subgroup of automorphisms of . Here is the infinitesimal generator of these automorphisms. From the properties of , must be a linear operator on that satisfies the Leibniz rule . So can be considered a differential ring.