Exterior derivative - Definition


Exterior derivative - Definition

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 �lie Cartan.

Contents

1 Definition

2 Properties

3 Invariant formula

4 Connection with vector calculus

4.1 Gradient
4.2 Curl
4.3 Divergence

5 Examples

6 See also

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:

<math>d{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x_i} dx_i \wedge dx_I.<math>

For general k-forms Σ_I f_I dx_I (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:

linearity

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 <math>k<math>-form <math>\omega\,\!<math> and arbitrary smooth vector fields <math>V_0,V_1,...V_k<math> 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

where <math>[V_i,V_j]\,\!<math> denotes Lie bracket and <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>

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_i}{\partial x_i}\, dx_j.<math>

Therefore

<math>df(V) = \langle \mbox{grad }f,V\rangle,<math>

where <math>\mbox{grad}f<math> denotes gradient of <math>f<math> and <math>\langle *,*\rangle<math> is scalar product.

Curl

For a 1-form <math>\omega=\sum_{i} f_i\,dx_i<math> on R^3,

<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 field <math>V=[u,v,w]<math> we have <math>d \omega(U,W)=\langle\mbox{rot}V\times U,W\rangle <math> the where <math>\mbox{rot}V<math> denotes curl of <math>V<math>, <math>\times<math> is the vector product and <math>\langle *,*\rangle<math> is scalar product.

Divergence

For a 2-form <math> \omega = \sum_{i,j} h_{i,j}\,dx_i\,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 = \left( \frac{\partial p}{\partial x} + \frac{\partial q}{\partial y} + \frac{\partial r}{\partial z} \right) dx \wedge dy \wedge dz = \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.