Change of variables in a differential operator.

Question

I would like to know how could I change the coordinates to cilindrical coordinates of the following differential operator.

$y\frac{\partial f}{\partial x} + xy^2z^5\frac{\partial f}{\partial y} + x^3yz^2\frac{\partial f }{\partial z}$

EDIT: cilindrical rather than polar coordinates

Answer 1

As you say, $$\left(\begin{matrix}\frac{\partial f}{\partial r}\\ \frac{\partial f}{\partial \theta}\end{matrix}\right) = \left(\begin{matrix}\cos\theta & \sin\theta\\ -r\sin\theta & r\cos\theta\end{matrix}\right) \left(\begin{matrix}\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\end{matrix}\right)$$ implying that (where $r\neq 0$) $$\left(\begin{matrix}\frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y}\end{matrix}\right) = \left(\begin{matrix}\cos\theta & -\frac{\sin\theta}{r}\\ \sin\theta & \frac{\cos\theta}{r}\end{matrix}\right) \left(\begin{matrix}\frac{\partial f}{\partial r}\\ \frac{\partial f}{\partial \theta}\end{matrix}\right)$$