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 [ilmath]\mathbb{R}^n[/ilmath] - 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 [ilmath]U[/ilmath] to some open subset of [ilmath]\mathbb{R}^n[/ilmath], for example:
[math]\phi:U\rightarrow\mathbb{R}^n[/math]
This is often thought of as: [math]\phi(p)=(\phi_1(p),\cdots,\phi_n(p))[/math] where the [math]\phi_i:U\rightarrow\mathbb{R}[/math] 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 [ilmath]n[/ilmath] if all charts have dimension [ilmath]n[/ilmath]
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 [ilmath]\mathbb{R}^n[/ilmath] or even [ilmath]\mathbb{R}^3[/ilmath] 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:
