Is it really in "polar coordinates" if there are no $\hat{r}$ and $\hat{\phi}$? Context. Let's say that I am asked to write a $d\vec{r}$ vector along a circular path. I will sometimes hear my classmates say, "We should use polar coordinates," by which they usually mean $$\tag1 d\vec{r} = \langle -r \sin \phi \ d\phi , r \cos \phi \ d\phi \rangle,$$ as opposed to rectangular coordinates, meaning $$\tag2 d\vec{r} =\left\langle dx, \frac{-x \ dx}{\sqrt{r^2 - x^2}} \right\rangle.$$ But I guess I would be tempted to say that (1) is still in rectangular coordinates since $d\vec{r} = \langle -r \sin \phi \ d\phi , r \cos \phi \ d\phi \rangle$ implies $d\vec{r} = (-r \sin \phi \ d\phi) \hat{x} + (r \cos \phi \ d\phi) \hat{y}$, which is still written in the $\{\hat{x}, \hat{y}\}$ basis. And that to truly write this $d\vec{r}$ in polar coordinates would be to write $$\tag3 d\vec{r} = 0 \hat{r} + r d\phi \ \hat{\phi},$$ i.e. written in the $\{\hat{r}, \hat{\phi}\}$ basis. (...or maybe is it now a coordinate frame? I'm not sure.)
Question. I guess my question is: of (1), (2), and (3), what would each representation be called? Which is in "polar coordinates"?
Thanks so much!
 A: To keep things simple, recall the transformation to polar coordinates itself,
$$x=r\cos\phi\tag{1}$$
$$y=r\sin\phi\tag{2}$$
You're then correct to think that having a vector
$$\vec{r}=(x,y)=(r\cos\phi,r\sin\phi)$$
is still written in a cartesian basis of $\hat{x}$ and $\hat{y}$. In most cases for what we do, this is perfectly sufficient.
To then write in a "total polar" form, you would have to transform $\hat{x}$ and $\hat{y}$ as well. That would then give us
$$\mathbf{\hat{x}}=\cos\phi\mathbf{\hat{r}}-\sin\phi\mathbf{\hat{\phi}}\tag{3}$$
$$\mathbf{\hat{y}}=\sin\phi\mathbf{\hat{r}}+\cos\phi\mathbf{\hat{\phi}}\tag{4}$$
As you can see, there is a dependence between the two "coordinate systems." What's really happening is you're parameterizing $x$ and $y$ in terms of $r,\theta$, and SO, there is no fundamental, or pure polar basis. It will always be dependent on a cartesian structure. So is it really in polar coordinates... well... no. There is no defined independent polar basis that has a direct translation to cartesian. $\hat{r}$ and $\hat{\phi}$ are forever dependent.
