Is there a way to evaluate $\int_{0}^{\phi}\vec{e_{\theta}}d\phi$, where $e_{\theta}$ is the usual unit vector in the theta direction in spherical polars without converting into Cartesian or cylindrical coordinates? If there isn't why, conceptually, not?
Apologies in advance if this question has been answered elsewhere, I couldn't find an answer when looking myself - any help is much appreciated!
For what it's worth, my own attempt at an answer is: Whatever the integral is, we would need to integrate it into a constant unit vector along the line of integration. So one way of doing this is to convert into Cartesian coordinates since the unit vectors then don't depend on position. I'm not sure why the first part of the claim is true, though.