closed differential forms on a simply connected domain

Let $D\subset\mathbb{R}^{2}$ 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 $\gamma_{0}$ and $\gamma_{1}$ 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 $\gamma_{0}$, $\gamma_{1}$ 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 ClosedDifferentialFormsOnASimplyConnectedDomain 2013-03-22 13:32:46 2013-03-22 13:32:46 paolini (1187) paolini (1187) 14 paolini (1187) Theorem msc 53-00 ClosedCurveTheorem PoincareLemma