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Prove the theorem.
Let $\displaystyle U=\mathbb{R} \setminus \{(0,0)\} $ ; let
$\displaystyle \omega \dfrac{y dx + x dy}{x^2 + y^2}$
be a $\displaystyle 1$ from in $\displaystyle U$. Then $\displaystyle d\omega =0$ ,but there is no $\displaystyle 0$ form $\displaystyle g$ on $\displaystyle U$ such that $\displaystyle dg = \omega $
Please help!
(***) Possible way of proving: (see the attached picture)

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