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 formand 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.