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.
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[edit] 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 (see wedge product).
[edit] Properties
Exterior differentiation satisfies three important properties:
- the wedge product rule (see antiderivation)
- 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).
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.
[edit] Invariant formula
Given a k-form ω and arbitrary smooth vector fields V0,V1, …, Vk we have
where [Vi,Vj] denotes Lie bracket and the hat denotes the omission of that element:
In particular, for 1-forms we have:
- dω(X,Y) = X(ω(Y)) − Y(ω(X)) − ω([X,Y]).
[edit] The exterior derivative in calculus
The following correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation.
[edit] Gradient
For a 0-form, that is a smooth function f: Rn→R, we have
Therefore, for vector field V
where grad f denotes gradient of f and 〈•, •〉 is the scalar product.
[edit] Curl
For a 1-form on R3,
or
which restricted to the three-dimensional case is
Therefore, for vector fields U, V = [u,v,w] and W we have where curl V denotes the curl of V, × is the vector product, and 〈•, •〉 is the scalar product.
[edit] Divergence
For a 2-form
For three dimensions, with we get
where V is a vector field defined by V = [p,q,r].
[edit] Examples
For a 1-form on R2 we have
which is exactly the 2-form being integrated in Green's theorem.