The trajectory of a particle is given by $\displaystyle \vec r(t)$ and its velocity by $\displaystyle \vec v(t)$.

Calculate $\displaystyle d\vec s = dx \hat i + dy \hat j + dz \hat k$ and $\displaystyle ds^2$ in cylindrical and spherical coordinates.

So far I have converted $\displaystyle \hat i$, $\displaystyle \hat j$ and $\displaystyle \hat k$ 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 $\displaystyle r$, $\displaystyle \varphi$ and z. I realize it will only depend on $\displaystyle r$ and $\displaystyle \varphi$.

I know that $\displaystyle x=r\cos \varphi$. I'm not really sure how to get dx. I don't think I can call "dx" a total derivative.