Exterior derivative

Sponsorship the way you would do it

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 ÃÂlie Cartan.

Table of contents

1 Definition
2 Properties
3 Special cases

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.

Properties

Exterior differentiation satisfies three important properties:

linearity
the wedge product rule

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).

Special cases

This all may not be very enlightening, so let us have some examples. For a 0-form, that is a smooth function f: Rⁿ→R, we have

the familiar gradient of the function. For a 1-form ν = Σ_i g_i dx_i'',

which restricted to the familiar three-dimensional case ν = u dx + v dy + w dz is

the curl of ν viewed as a vector field. For a 2-form μ = Σ_{i, j} h_{i, j} dx_i ∧ dx_j,

For three dimensions, with μ = p dy ∧ dz + q dz ∧ dx + r dx ∧ dy, we get

where we again view μ as a vector field.

These correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation. The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).