= Tangent space
{wiki}
TODO what's the point of it.
Bibliography:
* https://www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds \#6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
* https://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.
* http://jakobschwichtenberg.com/lie-algebra-able-describe-group/ by <Jakob Schwichtenberg>
* https://math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on <Stack Exchange>