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 was invented, in its current form, by Élie Cartan.

The exterior derivative of a differential form of degree k is a differential form of degree k+1. Exterior differentiation satisfies three important properties:

• linearity

• the wedge product rule

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

.

It can be shown that exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions).

Special cases of exterior differentiation correspond to familiar differential operators of vector calculus along the same lines as the differential corresponds to the gradient. For example, in 3 dimensional Euclidean space, exterior derivative of a 1-form corresponds to curl and exterior derivative of a 2-form corresponds to divergence.

This correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation. The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).