Let's say I have: $$x=r\sin\theta\cos\phi$$ $$y=r\sin\theta\sin\phi$$ $$z=r\cos\theta$$ I would like to get an expression for: $$dx^2+dy^2+dz^2$$ in terms of $r$, $\theta$, $\phi$. For example: $$dx=\sin\theta\cos\phi dr+r\sin\theta(-\sin\phi)d\phi+r\cos\theta\cos\phi$$ How can I achieve this with Mathematica? I know I can start with:
x[_r,_theta,_phi]=r*Sin[theta]*cos[Phi]
and so on, but then I don't see a straightforward line forward.
