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)