Differential forms

A differential form $ω$ is a section of the exterior algebra $\overset{k}{⋀} T^{*} M$ of the cotangent bundle of a manifold $M$ . That is,

ω \in Ω^{k} (M) := Γ (M, Λ^{k} T^{*} M) .

It assigns to each point in $M$ an alternating $k$ -linear map on the tangent space.

It can be said that a differential form is, precisely, a geometrical object on a manifold that can be integrated. See integration on Rn#Integration of forms and integration on manifolds.

$L$ -Valued Differential Form

Given a vector bundle $L \to M$ , an $L$ -valued differential form of degree $k$ is a smooth section of the bundle $Λ^{k} T^{*} M \otimes L$ . That is,

ω \in Ω^{k} (M; L) := Γ (M, Λ^{k} T^{*} M \otimes L) .

This means that instead of taking values in real (or complex) numbers, the differential form takes values in the fibers of the bundle $L$ .

They give rise to DeRham cohomology.

Important case: decomposable k-form.

Important: visualization of k-forms; exterior derivative and its visualization

Differential forms

L-Valued Differential Form

Related

$L$ -Valued Differential Form