# Definition:Exterior Derivative

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## Definition

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:

- $\displaystyle \d \omega = \paren {\sum_{k \mathop = 1}^m \frac {\partial f} {\partial x_k} \rd x_k} \wedge \d x_{\map \phi 1} \wedge \d x_{\map \phi 2} \wedge \dots \wedge \d x_{\map \phi n}$

For inexact forms:

- $\map \d {a + b} = \d a + \d b$