closed differential forms on a simply connected domain

Let D2 be an open set and let ω be a differential formMathworldPlanetmath defined on D.

Theorem 1

If D is simply connected and ω is a closed differential form, then ω is an exact differential form.

The proof of this result is a consequence of the following useful lemmas.

Lemma 1

Let ω be a closed differential form and suppose that γ0 and γ1 are two regularPlanetmathPlanetmathPlanetmathPlanetmath homotopic curves in D (with the same end points). Then

Lemma 2

Let ω be a continuousPlanetmathPlanetmath differential form. If given any two curves γ0, γ1 in D with the same end-points, it holds


then ω is exact.

See the Poincaré Lemma for a generalizationPlanetmathPlanetmath of this result on n-dimensional manifolds.

Title closed differential forms on a simply connected domain
Canonical name ClosedDifferentialFormsOnASimplyConnectedDomain
Date of creation 2013-03-22 13:32:46
Last modified on 2013-03-22 13:32:46
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 14
Author paolini (1187)
Entry type Theorem
Classification msc 53-00
Related topic ClosedCurveTheorem
Related topic PoincareLemma