Definition:Exterior Derivative

From ProofWiki
Jump to: navigation, search


Let an exact $n$-form $\omega$ be given on an $m$-manifold, with local coordinates $x_1, x_2, \dots, x_m$.

Let a local coordinate expression for $\omega$ be given:

$\omega = f \left({x_1, \ldots, x_m}\right) \, \mathrm d x_{\phi \left({1}\right)} \wedge \mathrm d x_{\phi \left({2}\right)} \wedge \cdots \wedge \mathrm d x_{\phi \left({n}\right)}$


$\phi: \left\{{1, \ldots, n }\right\} \to \left\{{1, \ldots, m}\right\}$ is an injection which determines which coordinate vectors $\omega$ acts on.
$\wedge$ denotes the wedge product.

The exterior derivative $\mathrm d \omega$ is the $\left({n + 1}\right)$-form defined as:

$\displaystyle \mathrm d \omega = \left({\sum_{k \mathop = 1}^m \frac{\partial f}{\partial x_k} \mathrm d x_k}\right) \wedge \mathrm d x_{\phi \left({1}\right)} \wedge \mathrm d x_{\phi \left({2}\right)} \wedge \dots \wedge \mathrm d x_{\phi \left({n}\right)}$

For inexact forms:

$\mathrm d \left({a + b}\right) = \mathrm d a + \mathrm d b$

Also see