I have trouble understanding polar coordinates, because they do not seem to be associated with a fixed basis like cartesian coordinates do. One example so you guys can see where I'm struggling.
If we are given the canonical basis $(e_1, e_2)$ of $\mathbb R^2$ and $f:\mathbb R^2 \to \mathbb R$ then we can define $\frac{\partial f}{\partial x}$ as the partial derivative of $f$ with respect to $e_1$ (which can also be written as $f_{e_1}$).
And it works because $e_1$ is the same for all $(x,y)$. That means $\frac{\partial f}{\partial x}(x_1,y_1)$ and $\frac{\partial f}{\partial x}(x_2,y_2)$ are derivative with respect to the same direction, that is $e_1$.
However, how can we define $\frac{\partial f}{\partial r}$ or even $\frac{\partial f}{\partial \theta}$? Indeed, if I write $\frac{\partial f}{\partial r}(r_1, \theta _1)$ and $\frac{\partial f}{\partial r}(r_2, \theta _2)$ how can I know what is direction of the vector we are derivating, since the basis at point $(r_1, \theta _1)$ is not the same as the basis at point $(r_2, \theta _2)$ (since $e_r$ will change with $\theta$). I'm not even sure if I should be talking of basis since polar coordinates clearly don't correspond to the kind of basis we are used to in linear algebra.
I've looked around for a thorough explanation of this kind of stuff without success. The most helpful thing was the article Curvilinear Coordinates on Wikipedia, but wasn't that helpful.
To make a clear question : How is $\frac{\partial f}{\partial r}$ defined? And how should I understand it?