Sorry for really simple question but I was wondering, if I want to swap between a vector in the polar coordinates, and get to Cartesian coordinates, how do we determine where the radial and angular components actually "go"?
By that I mean, let's say we just have an angular component $r\cos\theta\hat{\mathbf{e}}_\theta$ of the vector. I know that $r\cos\theta=x$ so I can write this as $x\hat{\mathbf{e}}_\theta$ but what about the actual $\hat{\mathbf{e}_\theta}$ itself? Could I do something like $\hat{\mathbf{e}}_\theta\to(-\sin\tan^{-1}\tfrac{y}{x},\cos\tan^{-1}\tfrac{y}{x})=(-\tfrac{y}{\sqrt{x^2+y^2}},\tfrac{1}{\sqrt{x^2+y^2}})$? I just tried this from some trigonometry, but I can't tell if this even makes sense. I would think that it makes more sense to have $\tfrac{1}{\sqrt{x^2+y^2}}(-y,x),$ which is what you get from using $x=r\cos\theta, y=r\sin\theta.$
Thanks.
This is an interesting question. I think the answer is: we can't! That is, there isn't a natural place to send $\hat{e}_\theta$, if we write a polar point $(r_0, \theta_0)_P$ as $r_0\hat{e}_r + \theta_0\hat{e}_\theta$, which is I think misleading notation, as you'll see. We'll go with it for now though.
Of course, to each point $(r, \theta)_P$ written in polar coordinates, we can assign a unique point $(x_1, x_2)_R$. Let's denote this map by $T$, so that $T(r, \theta)_P = (x_1, x_2)_R$; applying $T$ gives us the rectangular coordinates of our point written in polar coordinates.
Now, to see where $(r, \theta)_P = (1, 0)_P$ goes, that's not so bad. That's actually fixed by our map; since the point $1$ unit away from the origin making angle $\theta = 0$ radians is indeed the point $(x_1, x_2)_R = (1, 0)_R$.
The next question is, where does the point $(r, \theta)_P = (0, 1)_P$ get sent? That point is $0$ units away from the origin, so it must be the origin! This makes it incredibly hard to decide where to send $(0, 1)_P$. ; I'm not even going to hazard a suggestion. It seems like a bad idea to send $(1, 0)_P$ to the origin, but that's exactly what it corresponds to.
The issue is that thinking of the set of points $\{(r, \theta)_P : r, \theta \in \Bbb R\}$ as a vector space of polar coordinates isn't compatible with the points $T(r, \theta)_P = (x_1, x_2)_R$. That is, adding $(r_1, \theta_1)_P + (r_2, \theta_2)_P = (r_1 + r_2, \theta_2 + \theta_2)_P\ $ isn't meaningful in polar coordinates; to add vectors written in polar coordinates, we don't just add the magnitudes and angles.
I realize this is a bit of a disjointed answer, but these are the difficulties I personally experienced, trying to figure out where the components go. I wish I had a much more sophisticated reason why such an "addition preserving map" (homomorphism of vector spaces) is extremely unlikely to exist, but I have a feeling it's a combination of reasons. Primarily, we don't add polar components like we do rectangular ones. Secondarily, it seems that the space of polar coordinates should best be viewed as a kind of partial quotient space, where everything happens in the second component "modulo $2\pi$".