Math 6510 • Differentiable Manifolds

Let \(\omega\) be the 1-form on \(\mathbb{R}^2-\{0\}\) given by
\[
  \omega(x,y)=\frac{-y\,dx}{x^2+y^2}+\frac{x\,dy}{x^2+y^2}.
\]
\begin{enumerate}
\item Calculate \(\int_C\omega\) for any circle \(C\) centered at the origin.
\item Prove that in the half-plane \(x>0\) there is a smooth function \(f\) such that
\(\omega=df\).
\item Prove that there is no smooth function \(f\) defined on \(\mathbb{R}^2-\{0\}\) such
that \(\omega=df\).
\end{enumerate}