Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] Let \(X\) be a vector field on a smooth manifold \(M\) and let
\(f:M\to N\) be a diffeomorphism. Define \(f_*(X)\) -- the pushforward of \(X\) on \(N\).
\item[(b)] Let \(f:\R^3\to\R\times(-\frac{\pi}{2},\frac{\pi}{2})\times\R\) be
defined by \(f(x,y,z)=(e^x,\tan^{-1}(y),-2z)\). Calculate the bracket
\[
  \left[
    f_*\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial z}\right),
    f_*\left(\frac{\partial}{\partial y}-\frac{\partial}{\partial z}\right)
  \right].
\]
\end{enumerate}