Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item Show that \(\Phi:\mathbb{R}^2\to\mathbb{R}^2\) given by
\[
  (p,q)=\Phi(x,y)=(x+y^2+2y,x+y^2-2y)
\]
is a diffeomorphism. Compute \(d\Phi(\partial/\partial y)\) and the pullback
\(\Phi^*(p^2\,dp\wedge dq)\).
\item Let \(X\) be a smooth vector field and \(\omega\) a smooth \(n\)-form on the smooth
compact boundaryless manifold \(M^n\). Let
\(M\times\mathbb{R}\ni(x,t)\mapsto\phi_t(x)\in M\) be the flow generated by \(X\). Let
\[
  A(t)=\int_M\phi_t^*\omega
\]
be the integral of the pulled back \(n\)-form. Find \(dA/dt\).
\end{enumerate}