There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via the velocity of curves is intuitively the simplest, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.
If the tangent space is defined via differentiable curves, then this map is defined by
If, instead, the tangent space is defined via derivations, then this map is defined by
This is a generalization of the inverse function theorem to maps between manifolds.