Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] Define \(f:\R\to\R^2\) by \(f(x)=(x^2,x^3)\). Find \(f^*dx_1\).
\item[(b)] Assume \(g:\R^2\to\R^2\) is a smooth map and for each
\((x_1,x_2)\in\R^2\) the derivative is given by the matrix
\[
  \begin{pmatrix}
    2x_1+x_2 & x_1+3\\
    \dfrac{x_1}{(x_1^2+x_2^2)^2} &
    \dfrac{x_2}{(x_1^2+x_2^2)^2}
  \end{pmatrix}.
\]
Find \(g^*dx_1\).
\end{enumerate}