1. ## Prove the theorem.

Let $U=\mathbb{R} \setminus \{(0,0)\}$ ; let
$\omega \dfrac{-y dx + x dy}{x^2 + y^2}$
be a $1-$ from in $U$. Then $d\omega =0$ ,but there is no $0-$ form $g$ on $U$ such that $dg = \omega$

(***) Possible way of proving: (see the attached picture)

2. ## Re: Prove the theorem.

If such a form existed, than it should be at least $C^2$ away from the origin.
By Schwartz's theorem, $g_{xy}=g_{yx}$. See how this checks out.