Calculating push forwards of a vector field

Question

For the transformation from spherical coordinates to cartesian

$$F(r,\theta,\phi) = (r\cos\theta\sin\phi,r\sin\theta\sin\phi,r\cos\phi)$$

Calculate the push forward of the vector field $V = \frac{\partial}{\partial \theta}$

Is this just computing the Jacobian of $F$ and multiplying it by $V$?

In that case, is the answer just

$$\begin{pmatrix} \cos\theta\sin\phi & -r\sin\theta\sin\phi & r\cos\theta\cos\phi \\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi \\ \cos\phi & 0 & -r\sin\phi \end{pmatrix} \begin{pmatrix}0 \\ 1 \\ 0\end{pmatrix} = -r\sin\theta\sin\phi\partial_x + r\cos\theta\sin\phi \partial_y$$

The next part confuses me more regarding the directions of the transformation

Question 2:

Find the pushforward via $F^{-1}$ of $x\partial_y - y\partial_x$.

Answer 1

I would write the image just in terms of $x,y,z$ and $\partial_x, \partial_y, \partial_z$. That will help you do the second question. :)