 Tangent space

www.youtube.com/watch?v=oxB4aH8h5j4 actually gives a more concrete example. Basically, the vectors are defined by saying "we are doing the Directional derivative of any function along this direction".

One thing to remember is that of course, the most convenient way to define a function $f$ and to specify a direction, is by using one of the coordinate charts.

We can then just switch between charts by change of basis.