Find infinitesimal displacement in any coordinate system

[Msilva]

I am wondering how can I find the infinitesimal displacement in any coordinate system. For example, in spherical coordinates we have the folow relations:
$x = \, \rho sin\theta cos\phi$
$y = \, \rho sin\theta sin\phi$
$z = \, \rho cos\theta$

And we have that $$d\vec l = dr\hat r +rd\theta\hat \theta + r sin\theta d\phi \hat \phi$$
How do I found this for any system? What book has this explanation? I am not finding.

 

[kontejnjer]

In general the infinitesimal displacement given in some coordinates $q_i$ is $d\vec r=\frac{\partial \vec r}{\partial q_j}dq_j$ (sum over j), so if you have a vector given in cartesian coordinates, and know how to transform those to the new coordinates, you can use the above formula. For more info, I'd suggest Riley & Hobson, chapters 10 and 26.

Note: this works fine in Euclidean space $\mathbb R^n$, but for an arbitrary Riemannian manifold I'm not so sure.