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:

  • 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).