\begin{definition}[Definition:Exterior Derivative]
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:
:$\omega = \map f {x_1, \ldots, x_m} \rd x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \cdots \wedge \d x_{\map \phi n}$
where:
:$\phi: \set {1, \ldots, n} \to \set {1, \ldots, m}$ is an injection which determines which coordinate vectors $\omega$ acts on.
:$\wedge$ denotes the wedge product.
The '''exterior derivative''' $\d \omega$ is the $\paren {n + 1}$-form defined as:
:$\ds \d \omega = \paren {\sum_{k \mathop = 1}^m \frac {\partial f} {\partial x_k} \rd x_k} \wedge \d x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \dots \wedge \d x_{\map \phi n}$
For inexact forms:
:$\map \d {a + b} = \d a + \d b$
\end{definition}