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:

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 (see wedge product).

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

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

where denotes Lie bracket and

In particular, for 1-forms we have:

dω(X,Y) = X(ω(Y)) - Y(ω(X)) - ω([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

Therefore

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

Curl

For a 1-form on R³,

which restricted to the three-dimensional case is

Therefore, for vector field V=[u,v,w] we have 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

For three dimensions, with we get

where V is a vector field defined by V = [p,q,r].

Examples

For a 1-form on R² we have

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

See also

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