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Exterior
In mathematics, the exterior derivative operator of differential geometry extends the concept of the differential of a function to differential forms of higher degree. more...
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It is important in the theory of integration on manifolds, and is the differential (coboundary) 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 = fI dxI over Rn, the definition is as follows:
-
 For general k-forms ΣI fI dxI (where the multi-index I runs over all ordered subsets of {1, ..., n} of cardinality k), we just extend linearly. Note that if i = I above then .</span></p>
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Properties
Exterior differentiation satisfies three important properties:
linearity;
the wedge product rule (see antiderivation);
-
-
and d2 = 0, a formula encoding the equality of mixed partial derivatives, so that always;
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- It can be shown that the exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions).
The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).
The exterior derivative is natural. If f: M → N is a smooth map and Ωk and Ωk+1 are the contravariant smooth functors that assign correspondingly to each manifold the space of k- and k+1-forms on the manifold, then the following diagram commutes
so d(f*ω) = f*dω, where f* denotes the pullback of f. Thus d is a natural transformation from Ωk to Ωk+1.
Invariant formula
Given a k-form ω and arbitrary smooth vector fields V0,V1, …, Vk we have
Read more at Wikipedia.org
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