As far as I am aware, converting unit vectors from Cartesian to polar coordinates works as follows
- Express the transformation rules
$$ x = r \cos \theta \\ y = r \sin \theta $$
- Express the position vector in Cartesian coordinates
$$ \vec{r} = x \vec{e}_x + y \vec{e}_y $$
- Write the components of the position vector in polar coordinates
$$ \vec{r} = r \cos \theta \vec{e}_x + r \sin \theta \vec{e}_y $$
- The unit vectors in polar coordinates will be
$$ \vec{e}_r = \frac{ \frac{\partial \vec{r}}{\partial r} }{\left| \frac{\partial \vec{r}}{\partial r} \right| } = \cos \theta \vec{e}_x + \sin \theta \vec{e}_y; \qquad \vec{e}_\theta = \frac{ \frac{\partial \vec{r}}{\partial \theta} }{\left| \frac{\partial \vec{r}}{\partial \theta} \right| } = -\sin \theta \vec{e}_x + \cos \theta \vec{e}_y $$
I'd say I'm quite confident about this transformation rule as it agrees with all the sources I can find on this topic. This makes me believe that the steps in the derivation are good and consistent.
I am trying to do the same thing backwards, eg. start from polar coordinates and transform the unit vectors back to Cartesian. I've attempted the following steps, quite analogous to the ones above
- Express the transformation rules (with being aware of the limitation of $\tan^{-1}$)
$$ r = \sqrt{x^2 + y^2} \\ \theta = \tan^{-1} (y/x) $$
- Express the position vector in polar coordinates
$$ \vec{r} = r \vec{e}_r $$
- Write the components of the position vector in polar coordinates
$$ \vec{r} = \sqrt{x^2+y^2} \vec{e}_r $$
- The unit vectors in Cartesian coordinates will be
$$ \vec{e}_x = \frac{ \frac{\partial \vec{r}}{\partial x} }{\left| \frac{\partial \vec{r}}{\partial x} \right| } = \vec{e}_r ; \qquad \vec{e}_y = \frac{ \frac{\partial \vec{r}}{\partial y} }{\left| \frac{\partial \vec{r}}{\partial y} \right| } = \vec{e}_r $$
Here I am pretty sure that my result is incorrect, seeing that I am getting the same exact vector for both $x$ and $y$ unit vectors, which makes no sense to me. However, I think I am faithfully following the necessary steps, and I don't think I've made a miscalculation anywhere.
This makes me believe that there's a conceptual mistake somewhere. Could someone kindly point out to me what I did wrong, and how to do it correctly?
Best Answer
The issue is that $\hat{r}$ implicitly relies on $\theta$, so you have to use the chain rule:
$$\vec{r}(r, \theta) = r\vec{e}_r$$ $$\frac{\partial \vec{r}}{\partial x} = \frac{\partial \vec{r}}{\partial r}\frac{\partial r}{\partial x} + \frac{\partial \vec{r}}{\partial \theta}\frac{\partial \theta}{\partial x}$$ $$ = \vec{e}_r\frac{x}{\sqrt{x^2 + y^2}}+\sqrt{x^2 + y^2}\vec{e}_{\theta}\frac{1}{1 + (\frac{y}{x})^2}\frac{-y}{x^2}$$ $$= \cos\theta \vec{e}_r - \sin\theta \vec{e}_{\theta}$$
which brings us to the correct set of conversions.