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Exterior derivative

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:

  • linearity

<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).



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