Exterior

In mathematics, the exterior derivative operator of differential geometry extends the concept of the differential of a function to differential forms of higher degree. more...

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It is important in the theory of integration on manifolds, and is the differential (coboundary) used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by Élie Cartan.

Definition

The exterior derivative of a differential form of degree k is a differential form of degree k + 1.

Given a multi-index over Rⁿ is defined as

For general k-forms ω=Σ_I f_I dx_I (where the components of the multi-index I run over all the values in {1, ..., n}), the definition of the exterior derivative is extended linearly. Note that whenever i is one of the components of the multi-index I, then

Examples

For a 1-form

Properties

Exterior differentiation satisfies three important properties:

linearity;

the wedge product rule (see antiderivation);

and d² = 0, a formula encoding the equality of mixed partial derivatives, so that always;

Read more at Wikipedia.org