We map each point and a small enough neighbourhood of it to $R_{n}$, so we can talk about the manifold points in terms of coordinates.

Does not require any further structure besides a consistent topological map. Notably, does not require metric nor an addition operation to make a vector space.

A notable example of a Non-Euclidean geometry manifold is the space of generalized coordinates of a Lagrangian. For example, in a problem such as the double pendulum, some of those generalized coordinates could be angles, which wrap around and thus are not euclidean.

Collection of coordinate charts.

The key element in the definition of a manifold.

A generalized definition of derivative that works on manifolds.

TODO: how does it maintain a single value even across different coordinate charts?

TODO find a concrete numerical example of doing calculus on a differentiable manifold and visualizing it. Likely start with a boring circle. That would be sweet...

TODO what's the point of it.

Bibliography:

- www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds #6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
- 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.

- jakobschwichtenberg.com/lie-algebra-able-describe-group/ by Jakob Schwichtenberg
- math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on Stack Exchange

A member of a tangent space.

www.youtube.com/watch?v=tq7sb3toTww&list=PLxBAVPVHJPcrNrcEBKbqC_ykiVqfxZgNl&index=19 mentions that it is a bit like a dot product but for a tangent vector to a manifold: it measures how much that vector derives along a given direction.