Exterior Derivative

to Differential Form s of higher degree. It is important in the theory of Integration on Manifold s, and is the Differential (coboundary) 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 R^''n'', the definition is as follows:

::

d{\omega} = \sum_{i=1}^n rac{\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 Linear ly. 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 )

For three dimensions, with

\omega = p\,dy\wedge dz+q\,dz\wedge dx+r\,dx\wedge dy

we get

where ''V'' is a Vector Field defined by

V =

{Link without Title} .

EXAMPLES

For a 1-form

\sigma = u\, dx + v\, dy

on R^''2'' we have

:

d \sigma = \left(rac{\partial{v}}{\partial{x}} - rac{\partial{u}}{\partial{y}}
ight) dx \wedge dy

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

The Vector Calculus Identities
:

abla 	imes ( 
abla f )  = 0

and
:

abla \cdot ( 
abla 	imes \mathbf{F} ) = 0

are special cases of the third property of the exterior derivative,

d^2=0

.

SEE ALSO

Exterior Covariant Derivative

Green's Theorem

Lie Derivative

Stokes Theorem

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Exterior Derivative