smooth in polar but not in rectangular

Question

Can anyone please give an example or two of functions on $\mathbb{R}^2$ which are smooth in the polar coordinates $(r,\theta)$ but not smooth in the Cartesian coordinates $(x,y)$?

Thank you!

Answer 1

Maybe this is cheap, but one example is the function $\theta$ itself. Certainly this function is smooth in polar coordinates, where it's just the projection onto the second coordinate.

However, in $x$ and $y$ coordinates it is (up to adding $\pi$) just $\arctan(y/x)$ which cannot be extended in a smooth, or even continuous, way to the origin.