Math 6510 • Differentiable Manifolds

Let
\[
  \omega=(x^2+y^2+z^2)^\alpha
  (x\,dy\wedge dz+y\,dz\wedge dx+z\,dx\wedge dy)
\]
be a 2-form in \(\mathbb{R}^3\setminus\{0\}\).
\begin{enumerate}
\item Find \(\alpha\in\mathbb{R}\) such that \(\omega\) is closed.
\item Show that \(\omega\) is not exact.
\end{enumerate}