# Calculating push forwards of a vector field

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 $x\partial_y - y\partial_x$.

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. :)
• My problem is mainly understanding what is exactly asked for, is the pushforward of $V$ its image under the Jacobian from spherical to cartesian coordinates? Also, I don't quite get what is the second question asking for. – The Bosco Apr 12 '18 at 3:43
• Yes, your answer is correct. I'm suggesting that you write it in cartesian coordinates. Off a small set, $F$ is a diffeomorphism to its image, and $F^{-1}$ maps backward from cartesian to spherical. – Ted Shifrin Apr 12 '18 at 3:47
• For question 2, I know that I have to do it through $J^{-1} v_{cartesian} = v_{spherical}$, but if I solve for $v_{cartesian}$ I end up with the exact same answer and procedure as just getting $F^*(v_{spherical}) = v_{cartesian}$ – The Bosco Apr 12 '18 at 4:04
• In principle, you need the inverse matrix of the Jacobian in terms of $x,y,z$. But did you realize that your answer to question 1 was precisely $-y\partial_x + x\partial_y$? – Ted Shifrin Apr 12 '18 at 5:55