How to make sense of partial derivatives w.r.t to polar coordinates?

Question

Ok, so I'm trying to learn multivariable calculus. As I come from a math background, I want to understand it as rigurously as possible. I've seen that differential geometry does the job, but I don't want to go that far yet. So my question is how to make sense of partial derivatives w.r.t to other coordinate systems besides the linear ones? For example, polar coordinates do not behave nicely under linear transformations. Even worse, the vector basis is local, like in physics, so it seems a nightmare to keep track of how to write things like gradients in a vector basis that is always changing with the point. I want a mathematical explanation to keep me sane, physical analogies will not do.

Answer 1

Polar coordinates are a map $\varphi:[-\pi,\pi)\times[0,\infty)\to\mathbb R^2, \varphi(\theta,r)=r(\cos\theta,\sin\theta)$.

Now expressing a function $f:\mathbb R^2\to U$ for some suitable $U$ in polar coordinates just means finding the function $f\circ\varphi$. The partial derivatives with respect to the polar coordinates are now simply the usual partial derivatives of $f\circ\varphi$.

There is some dodginess at the edges of the domain of $\varphi$, but on its interior it works exactly this way.