Manifolds

Consider the Earth, the planet you are on right now. Locally you can go forward and back, left and right - you may move around the surface as if it were a plane. Of course it is not actually a plane.

Manifolds are much the same. They are "blobs" for lack of a better word that we can pretend - locally - look like $\mathbb{R}^n$ - such a way of looking at them is called a chart (sometimes a "coordinate chart"), it is a map, from an open set of the manifold $U$ to some open subset of $\mathbb{R}^n$, for example:

$$\phi:U\rightarrow\mathbb{R}^n$$

This is often thought of as:

$$\phi(p)=(\phi_1(p),\cdots,\phi_n(p))$$ where the $$\phi_i:U\rightarrow\mathbb{R}$$ are called "local coordinate functions" - or often just "local coordinates"

A collection of charts such that every point belongs to at least one chart in the collection is called an "atlas" of the manifold.

A manifold has dimension $n$ if all charts have dimension $n$

A rather large amount of work is required to study manifolds because all the calculus the reader has likely done so far involved things (surfaces) that could be easily put in $\mathbb{R}^n$ or even $\mathbb{R}^3$ in all likelihood. Integrating these is easy enough! The tangent vector is an intuitive idea, however in a manifold we do not have an "ambient space" we can have a tangent in.

Studying

First steps:

• Topological manifold
• Chart
• Atlas
• Tangent space
• Derivation
• Smooth