Differential form

Let $(𝑀, 𝒜)$ be a $𝐶 𝛼$ -manifold. A differential $𝑝$ -form is a totally contravariant, totally antisymmetric tensor field. and a general differential form is a direct sum of these.

As a section

The above is equivalent to a $𝐶 𝛼$ -section of the $𝑝$ th exterior power of the cotangent bundle, i.e.

Ω 𝑝 (𝑀) = Γ (⋀ 𝑝 𝑀)

A general (non-homogenous) differential form is a $𝐶 𝛼$ -section of the exterior algebra bundle

Ω ∙ 𝑀 = Γ (⋀ ∙ 𝑀) .

Exterior product

The exterior algebra bundle induces the exterior product of differential forms, so that

(𝜔 \land 𝜇) 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 = (𝑝 + 𝑞)! 𝑝! 𝑞! 𝜔 [𝑎 1 \dots 𝑎 𝑝 𝜇 𝑏 1 \dots 𝑏 𝑞]

which acts on vector fields as

(𝜔 \land 𝜇) (𝑣 1, \dots, 𝑣 𝑝 + 𝑞) = 1 𝑝! 𝑞! \sum 𝜎 \in S 𝑝 + 𝑞 (s g n 𝜎) 𝜔 (𝑣 𝜎 (1), \dots, 𝑣 𝜎 (𝑝)) 𝜇 (𝑣 𝜎 (𝑝 + 1), \dots, 𝑣 𝜎 (𝑝 + 𝑞))

Local coördinates

If $𝑥 : 𝑈 \to ℝ 𝑚$ are a chart then locally a tensor field is given by

𝜔 = 𝜔 𝜇 1 \dots 𝜇 𝑝 d 𝑥 𝜇 1 \otimes \dots \otimes d 𝑥 𝜇 𝑝

and if $𝜔$ is antisymmetric,

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

where

d 𝑥 𝜇 1 \land \dots \land d 𝑥 𝜇 𝑝 = 𝑞! d 𝑥 [𝜇 1 \otimes \dots \otimes d 𝑥 𝜇 𝑝]

and $d : Ω 0 (𝑀) \to Ω 1 (𝑀)$ is the Exterior derivative.

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Differential form

As a section

Exterior product

Local coördinates

Related concepts