Exterior derivative

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.

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:

$d{\omega }=\sum _{i=1}^{n}{\frac {\partial f_{I}}{\partial x_{i}}}dx_{i}\wedge dx_{I}.$

For general k-forms Σ_I f_I dx_I (where the multi-index I runs over all ordered subsets of {1, ..., n} 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)

$d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{{\rm {deg\,}}\omega }(\omega \wedge d\eta )$

• and d² = 0, a formula encoding the equality of mixed partial derivatives, so that always

$d(d\omega )=0\,\!$

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, …, V_k we have

$d\omega (V_{0},V_{1},...V_{k})=\sum _{i}(-1)^{i}V_{i}\omega (V_{0},...,{\hat {V}}_{i},...,V_{k})$

$+\sum _{i<j}(-1)^{i+j}\omega ([V_{i},V_{j}],V_{0},...,{\hat {V}}_{i},...,{\hat {V}}_{j},...,V_{k})$

where $[V_{i},V_{j}]\,\!$ denotes Lie bracket and $\omega (V_{0},...,{\hat {V}}_{i},...,V_{k})=\omega (V_{0},...,V_{i-1},V_{i+1}...,V_{k}).$

In particular, for 1-forms we have:

$d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]).$

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

$df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,dx_{i}.$

Therefore

$df(V)=\langle {\mbox{grad }}f,V\rangle ,$

where grad f denotes gradient of f and <•, •> is the scalar product.

Curl

For a 1-form $\omega =\sum _{i}f_{i}\,dx_{i}$ on R³,

$d\omega =\sum _{i,j}{\frac {\partial f_{i}}{\partial x_{j}}}dx_{j}\wedge dx_{i},$

which restricted to the three-dimensional case $\omega =u\,dx+v\,dy+w\,dz$ is

$d\omega =\left({\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}\right)dx\wedge dy+\left({\frac {\partial w}{\partial y}}-{\frac {\partial v}{\partial z}}\right)dy\wedge dz+\left({\frac {\partial u}{\partial z}}-{\frac {\partial w}{\partial x}}\right)dz\wedge dx.$

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

(what are U and W here? this assertion needs clarification - Gauge 23:37, 7 Apr 2005 (UTC))

Divergence

For a 2-form $\omega =\sum _{i,j}h_{i,j}\,dx_{i}\,dx_{j},$

$d\omega =\sum _{i,j,k}{\frac {\partial h_{i,j}}{\partial x_{k}}}dx_{k}\wedge dx_{i}\wedge dx_{j}.$

For three dimensions, with $\omega =p\,dy\wedge dz+q\,dz\wedge dx+r\,dx\wedge dy$ we get

$d\omega =\left({\frac {\partial p}{\partial x}}+{\frac {\partial q}{\partial y}}+{\frac {\partial r}{\partial z}}\right)dx\wedge dy\wedge dz={\mbox{div}}Vdx\wedge dy\wedge dz,$

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

$d\sigma =\left({\frac {\partial {v}}{\partial {x}}}-{\frac {\partial {u}}{\partial {y}}}\right)dx\wedge dy$

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

See also

• Exterior covariant derivative
• Green's theorem
• Stoke's theorem