Exterior derivative extends the concept of the differential[?] of a function
to differential forms of higher degree. Exterior derivative of a differential form of degree k is a differential form of degree k+1.
Exterior differentiation satisfies three important properties:
- <math>d(\omega\wedge\eta) = d\omega\wedge\eta+(-1)^{{\rm deg\,}\omega}(\omega\wedge d\eta)</math>
- and a formula encoding the equality of mixed partial derivatives <math>d(d\omega)=0</math>.
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).
