Exterior derivative
Categories: Differential topology | Multivariate calculus
In mathematics, the exterior derivative operator of differential topology, 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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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:
- <math>d{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x_i} dx_i \wedge dx_I.</math>
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 <math>i = I</math> above then <math>dx_i \wedge dx_I = 0</math> (see wedge product).
Properties
Exterior differentiation satisfies three important properties:
- the wedge product rule (see antiderivation)
- <math>d(\omega \wedge \eta) = d\omega \wedge \eta+(-1)^{{\rm deg\,}\omega}(\omega \wedge d\eta)</math>
- and d2 = 0, a formula encoding the equality of mixed partial derivatives, so that always
- <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).
The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).
Invariant formula
Given a k-form ω and arbitrary smooth vector fields V0,V1, …, Vk we have
- <math>d\omega(V_0,V_1,...V_k)=\sum_i(-1)^i V_i\omega(V_0,...,\hat V_i,...,V_k)</math>
- <math>+\sum_{i<j}(-1)^{i+j}\omega([V_i,V_j],V_0,...,\hat V_i,...,\hat V_j,...,V_k)</math>
where <math>[V_i,V_j]</math> denotes Lie bracket and the hat denotes the omission of that element: <math>\omega(V_0,...,\hat V_i,...,V_k)=\omega(V_0,..., V_{i-1},V_{i+1}...,V_k).</math>
In particular, for 1-forms we have:
- <math>d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]).</math>
More generally, the Lie derivative is defined via the Lie bracket:
- <math>\mathcal{L}_XY=[X,Y]</math>,
and the Lie derivative of a general differential form is closely related to the exterior derivative. The differences are primarily notational; various identities between the two are provided in the article on Lie derivatives.
Connection with vector calculus
The following correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation.
Gradient
For a 0-form, that is a smooth function f: Rn→R, we have
- <math>df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\, dx_i.</math>
Therefore, for vector field <math>V</math>
- <math>df(V) = \langle \mbox{grad }f,V\rangle,</math>
where grad f denotes gradient of f and <•, •> is the scalar product.
Curl
For a 1-form <math>\omega=\sum_{i} f_i\,dx_i</math> on R3,
- <math>d \omega=\sum_{i,j}\frac{\partial f_i}{\partial x_j} dx_j\wedge dx_i,</math>
which restricted to the three-dimensional case <math>\omega= u\,dx+v\,dy+w\,dz </math> is
- <math>d \omega = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) dx \wedge dy
+ \left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) dy \wedge dz + \left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) dz \wedge dx.</math>
Therefore, for vector fields <math>U</math>, <math>V=[u,v,w]</math> and <math>W</math> we have <math>d \omega(U,W)=\langle\mbox{curl}\, V \times U,W\rangle </math> where curl V denotes the curl of V, × is the vector product, and <•, •> is the scalar product.
Divergence
For a 2-form <math> \omega = \sum_{i,j} h_{i,j}\,dx_i\wedge dx_j,</math>
- <math>d \omega = \sum_{i,j,k} \frac{\partial h_{i,j}}{\partial x_k} dx_k \wedge dx_i \wedge dx_j.</math>
For three dimensions, with <math> \omega = p\,dy\wedge dz+q\,dz\wedge dx+r\,dx\wedge dy</math> we get
<math>d \omega\,</math> <math> = \left( \frac{\partial p}{\partial x} + \frac{\partial q}{\partial y} + \frac{\partial r}{\partial z} \right) dx \wedge dy \wedge dz </math> <math>= \mbox{div}V\, dx \wedge dy \wedge dz,</math>
where V is a vector field defined by <math> V = [p,q,r].</math>
Examples
For a 1-form <math>\sigma = u\, dx + v\, dy</math> on R2 we have
- <math>d \sigma = \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) dx \wedge dy</math>
which is exactly the 2-form being integrated in Green's theorem.