# Curvilinear coordinates on tangent spaces

1. Dec 7, 2013

### mnb96

Hello,

if we consider a diffeomorphism f:M-->N between two manifolds, we can easily obtain a basis for the tangent space of N at p from the differential of f.

I was wondering, why should we always express tangent vectors as linear combinations of tangent basis vectors?

Could it be useful in some circumstances to express tangent vectors in some convenient system of curvilinear coordinates?

If so, then could we interpret the introduction of local curvilinear coordinates as a diffeomorphism of the tangent space?