The trajectory of a particle is given by \vec r(t) and its velocity by \vec v(t).
Calculate d\vec s = dx \hat i + dy \hat j + dz \hat k and ds^2 in cylindrical and spherical coordinates.
So far I have converted \hat i, \hat j and \hat k into a combination of unit vectors in cylindrical coordinates. I also done the opposite.
I just don't know how to convert dx into a differential depending on r, \varphi and z. I realize it will only depend on r and \varphi.
I know that x=r\cos \varphi. I'm not really sure how to get dx. I don't think I can call "dx" a total derivative.