Math 6510 • Differentiable Manifolds

Consider the 1-form on \(\mathbb{R}^2\setminus\{0\}\)
\[
  \omega=\frac{x\,dy-y\,dx}{x^2+y^2}.
\]
\begin{enumerate}
\item Prove that \(\omega\) is closed, i.e. \(d\omega=0\).
\item Prove that \(\omega\) is not exact, i.e. there is no smooth function
\(f:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}\) such that \(\omega=df\).
\end{enumerate}