In differential geometry, given a manifold with coordinates $q^i$, the tangent basis vectors are defined as $(\frac \partial {\partial q^i})$.
This definition is very general so, even if it not necessary, it should be possible to use it to describe simple cases in Euclidean space.
Can you make an example of use of this definition in a easy Euclidean situation in which it's also possible to use the classical definition $(\frac {\partial \vec P} {\partial x^i})$. I'd like to see how these two definition are equivalent in Euclidean space.