Definition:Exterior Derivative

Definition

Given an exact $n$-form $\omega$ on an $m$-manifold, local coordinates $x_1, x_2, \dots, x_m$, and a local coordinate expression for $\omega$:

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

where $\phi:\left\{{1, \dots, n }\right\} \to \left\{{1, \dots, m}\right\}$ is an injection which determines which coordinate vectors $\omega$ acts on, the exterior derivative $\mathrm d \omega$ is the $(n+1)$-form defined as:

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

For inexact forms, $\mathrm d (a+b) = \mathrm d a + \mathrm d b$

Also see

• Manipulation of Exterior Derivative

• Wedge Product