Since you are working in cylindrical coordinates r, \(\displaystyle \phi\), and z are independent variables. So by definition of partial derivatives

\(\displaystyle \dfrac{\partial z}{\partial r} = \dfrac{\partial r}{\partial \phi} = 0\)

This is identical in concept to \(\displaystyle \dfrac{\partial y}{\partial x} = 0\) in rectangular coordinates.

-Dan