Exterior derivative

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In mathematics, the exterior derivative operator of differential geometry 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 Élie 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 = \sum_^n \frac 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 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)^ 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 + \frac + \frac \right) dx \wedge dy \wedge dz ]
	[= \mboxV\, 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}} - \frac}}\right) dx \wedge dy]

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

See also

• Exterior covariant derivative
• Green's theorem
• Lie derivative
• Stokes theorem

 

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