# Manifold

We map each point and a small enough neighbourhood of it to , 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.
Manifolds are cool. Especially differentiable manifolds which we can do calculus on.
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.

## Atlas (topology)

Collection of coordinate charts.
The key element in the definition of a manifold.

## Covariant derivative

A generalized definition of derivative that works on manifolds.
TODO: how does it maintain a single value even across different coordinate charts?

## Differentiable manifold

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...

## Tangent space

TODO what's the point of it.
Bibliography:

## Tangent vector to a manifold

A member of a tangent space.

## One-form

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.