closed differential forms on a simply connected domain

Let $D\subset {ℝ}^2$ be an open set and let $\omega$ be a differential form defined on $D$.

Theorem 1

If $D$ is simply connected and $\omega$ is a closed differential form, then $\omega$ is an exact differential form.

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

Lemma 1

Let $\omega$ be a closed differential form and suppose that ${\gamma }_{\mathrm{0}}$ and ${\gamma }_{\mathrm{1}}$ are two regular homotopic curves in $D$ (with the same end points). Then

$${\int }_{{\gamma }_0}\omega ={\int }_{{\gamma }_1}\omega .$$

Lemma 2

Let $\omega$ be a continuous differential form. If given any two curves ${\gamma }_{\mathrm{0}}$, ${\gamma }_{\mathrm{1}}$ in $D$ with the same end-points, it holds

$${\int }_{{\gamma }_0}\omega ={\int }_{{\gamma }_1}\omega ,$$

then $\omega$ is exact.

See the Poincaré Lemma for a generalization 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