exterior derivative

In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by lie Cartan.

Definition

The exterior derivative of a differential form of degree k is a differential form of degree k + 1. For a k-form ω = f_I dx_I over Rⁿ, the definition is as follows:

d{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x_i} dx_i \wedge dx_I.

For general k-forms Σ_I f_I dx_I (where the multi-index I runs over all ordered subsets of {1, ..., n} of cardinality k), we just extend linearly. Note that if

i = I

above then

dx_i \wedge dx_I = 0

(see wedge product).

Properties

Exterior differentiation satisfies three important properties:

linearity

the wedge product rule (see antiderivation)

d(\omega \wedge \eta) = d\omega \wedge \eta+(-1)^{\partial x_k} dx_k \wedge dx_i \wedge dx_j.

For three dimensions, with

\omega = p\,dy\wedge dz+q\,dz\wedge dx+r\,dx\wedge dy

we get

d \omega = \left( \frac{\partial p}{\partial x} + \frac{\partial q}{\partial y} + \frac{\partial r}{\partial z} \right) dx \wedge dy \wedge dz = \mbox{div}V dx \wedge dy \wedge dz,

where V is a vector field defined by

V = p,q,r .

Examples

For a 1-form

\sigma = u\, dx + v\, dy

on R² we have

d \sigma = \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) dx \wedge dy

which is exactly the 2-form being integrated in Green's theorem.