In mathematics, the exterior derivative operator of differential geometry 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:

$d\{\backslash omega\}\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; f\_I\}\{\backslash partial\; x\_i\}\; dx\_i\; \backslash wedge\; dx\_I.$

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 $dx\_i\; \backslash wedge\; dx\_I\; =\; 0$ (see wedge product).

Properties

Exterior differentiation satisfies three important properties:

• linearity

• the wedge product rule (see antiderivation)

$d(\backslash omega\; \backslash wedge\; \backslash eta)\; =\; d\backslash omega\; \backslash wedge\; \backslash eta+(-1)^\{\backslash partial\; x\_k\}\; dx\_k\; \backslash wedge\; dx\_i\; \backslash wedge\; dx\_j.$

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

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

where V is a vector field defined by $V\; =*.$

Examples

For a 1-form $\backslash sigma\; =\; u\backslash ,\; dx\; +\; v\backslash ,\; dy$ on R² we have

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

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

See also

• Exterior covariant derivative
• Green's theorem
• Lie derivative
• Stokes theorem

Differential forms

Derivada exterior | Dérivée extérieure | 外微分