Math 6510 • Differentiable Manifolds

Let \(S^2=\{(x,y,z)\in\R^3:x^2+y^2+z^2=1\}\) be the unit sphere. Let
\[
  \omega=
  \begin{cases}
    \dfrac{dy\wedge dz}{x}, & \text{if } x\ne0;\\[1ex]
    \dfrac{dz\wedge dx}{y}, & \text{if } y\ne0;\\[1ex]
    \dfrac{dx\wedge dy}{z}, & \text{if } z\ne0.
  \end{cases}
\]
\begin{enumerate}
\item[(a)] Show that \(\omega\) is a well defined two form on \(S^2\)
\item[(b)] Find \(\displaystyle\int_{S^2}\omega\).
\item[(c)] Is \(\omega\) exact? Why?
\end{enumerate}