Polar coordinates: What unit vectors span the $(r,\theta)$ space?
I am thoroughly confused. If in the Cartesian system, the associated orthonormal polar vectors at different points on a circle keep changing, then which ones do we choose to span the corresponding $(r, \theta)$ space?
Suppose you want to describe a 2D vector field, e.g., the electric field due to a line charge. In Cartesian coordinates, I'll claim that (up to scalar factors) this would be $$\vec{E}=\frac{x \hat{x}+y\hat{y}}{x^2+y^2}.$$ An advantage of this form is that we may directly differentiate to verify $\nabla\cdot \vec{E}=0$ is true everywhere except the origin origin, where it is ill-defined; this is what we expect of a line charge.
What is not evident from this form is that it has rotational symmetry. If we instead pass to polar coordinates via $(x,y)=(r\cos\theta,r\sin\theta)$, the electric field is $$\vec{E}=\frac{1}{r}\Big[(\cos\theta) \hat{x}+(\sin\theta) \hat{y}\Big].$$ This form is manifestly rotationally symmetric: the scalar factor $1/r$ does not depend on $\theta$, and the unit-vector factor rotates so as to always point radially away from the origin. We can therefore think of $\vec{E}$ as having magnitude $1/r$ and pointing radially away from the origin.
More generally, suppose we have some vector field $\vec{F}$ and we choose some point $p=(x,y)=(r\cos\theta,r\sin\theta)$. Then there exist vectors $\hat{r}_p,\hat{\theta}_p$ which are defined as such: If I start at $p$ and increase $r$ slightly, then my point moves radially outward (the $\hat{r}_p$ direction); if I instead increase $\theta$ slightly, my point moves counterclockwise (the $\hat{\theta}_p$ direction). In terms of Cartesian coordinates, we have
\begin{align} \hat{r}_p &= (\cos\theta)\hat{x}+(\sin\theta)\hat{y},\\ \hat{\theta}_p &= -(\sin\theta)\hat{x}+(\cos\theta)\hat{y},\\ \end{align}
So long as we avoid the origin (where the choice of $\theta$ becomes ambiguous) these vectors are a perfectly respectable orthonormal basis. Hence we can express $\vec{F}$ at $p$ in this (local!) basis: $$\vec{F}(p)=F_r(p)\hat{r}_p+F_\theta(p) \hat{\theta}_p.$$ In particular, the electric field given earlier may be expressed as $$\vec{E}=\frac{1}{r}\hat{r}_p.$$ This formalizes our earlier description of the electric field. So while we now have to put up with a coordinate-dependent basis, our vector field now takes advantage of rotational symmetry. (Moreover, this form is actually the one I secretly started from; indeed, the derivation of this form using Gauss's law is a standard problem in introductory electrodynamics.)
The bottom line is that $\{\hat{r},\hat{\theta}\}$ is a coordinate-dependent orthonormal basis which we (only) use to express $\vec{F}$ at a particular point $p$. (Or, if you like, there's a whole family of such orthonormal bases which are indexed by the point $p$.) In physics, though, it's conventional to write write these as $(\hat{r},\hat\theta)$ and take the coordinate-dependence as implicit. It is also typical to not distinguish between dependence on the point $p$ vs the Cartesian coordinates $(x,y)$ or polar coordinates $(r,\theta)$. Thus one will typically see the radial component of the above electric field written as $E_r(r,\theta)=1/r$, ignoring the abuse of notation.