Exterior derivative

The exterior derivative generalizes the concept of differential to general differential forms on a $𝐶 𝛼$ -manifold $(𝑀, 𝒜)$ . For $𝑝 \geq 0$ , we have a map

d : Ω 𝑝 (𝑀) \to Ω 𝑝 + 1 (𝑀) .

Note that to any 0-form, i.e. continuous function $𝑓 \in 𝐶 𝛼 (𝑀)$ , we can naturally associate a 1-form by the

(d 𝑓) 𝑎 𝑣 𝑎 = \nabla 𝑣 𝑓 .

The general exterior derivative is then the unique extension of this operation to a graded derivation $Ω ∙ (𝑀) \to Ω ∙ (𝑀)$ such that $d 2 = 0$ , i.e. if $𝜔 \in Ω 𝑝 (𝑀)$ and $𝜇 \in Ω 𝑞 (𝑀)$ we have

d (𝜔 \land 𝜇) = d 𝜔 \land 𝜇 + (- 1) 𝑝 𝜔 \land d 𝜇 .

Proof of existence and uniqueness

proof

Local coördinates

Let $𝑥 : 𝑈 \to ℝ 𝑚$ be a chart. Then for

𝜔 = 1 𝑝! 𝜔 𝜇 1 \dots 𝜇 𝑝 d 𝑥 𝜇 1 \land \dots \land d 𝑥 𝜇 𝑝

we have

d 𝜔 = 1 𝑝! 𝜕 𝜇 𝑝 + 1 𝜔 𝜇 1 \dots 𝜇 𝑝 d 𝑥 𝜇 𝑝 + 1 \land d 𝑥 𝜇 1 \land \dots \land d 𝑥 𝜇 𝑝

This can be seen as a special case of From a covariant derivative.

From a covariant derivative

Let $\nabla$ denote a torsion-free affine connexion on $𝑀$ . Then for $𝜔 𝑎 1 \dots 𝑎 𝑝 \in Ω 𝑝 (𝑀)$ , the covariant derivative gives

(d 𝜔) 𝑏 𝑎 1 \dots 𝑎 𝑝 = (𝑝 + 1) \nabla [𝑏 𝜔 𝑎 1 \dots 𝑎 𝑝]

which is independent of the choice of $\nabla$ .¹

Proof

proof

Table of Contents

Exterior derivative

Local coördinates

From a covariant derivative

See also

Graph View

Backlinks

Table of Contents

Exterior derivative

Local coördinates

From a covariant derivative

See also

Footnotes

Graph View

Backlinks