Difference between revisions of "Motivation for smooth manifolds"
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Revision as of 22:44, 9 April 2015
Here I will build up an example of a Smooth manifold
Intro
You already deal with manifolds (to a limited extent) - when you deal with polar coordinates you're indeed charting [ilmath]\mathbb{R}^2[/ilmath] just in a different way. In Basis and coordinates I explain how coordinates with a basis are the same point, this is much the same, just more general.
Example
We will chart [math]\mathbb{R}_{++}^2=\{(x,y)\in\mathbb{R}^2|x>0\wedge y>0[/math].
Already you're thinking of this as a part of the plane, and clearly with coordinate [ilmath](x,y)[/ilmath] - coordinates in "the standard basis" to use linear algebra terminology. These are the Standard coordinates in manifolds terminology and are well defined on subsets of Euclidean spaces (like subsets of [ilmath]\mathbb{R}^n[/ilmath] as in this case) - it's worth having a look now.
Our charts
| Symbol | Name | Chart definition | Map (WRT standard) |
|---|---|---|---|
| [ilmath]\alpha[/ilmath] | standard coordinates | [ilmath](\mathbb{R}_{++}^2,\alpha:\mathbb{R}^2_{++}\longrightarrow\mathbb{R}^2_{++})[/ilmath] | [ilmath]\alpha(x,y)\mapsto(x,y)[/ilmath] |
| [ilmath]\beta[/ilmath] | Polar coordinates | [ilmath](\mathbb{R}_{++}^2,\beta:\mathbb{R}^2_{++}\longrightarrow\mathbb{R}_+\times(0,\tfrac{\pi}{2}))[/ilmath] | [ilmath]\beta(x,y)\mapsto\left(\sqrt{x^2+y^2},\arctan(\tfrac{y}{x})\right)[/ilmath] |
| [ilmath]\gamma[/ilmath] | Made up for example | [ilmath](\mathbb{R}^2_{++},\gamma:\mathbb{R}^2_{++}\longrightarrow\mathbb{R}^2_{++})[/ilmath] | [ilmath]\gamma(x,y)\mapsto(x^3,y^3)[/ilmath] |
We will then look at [ilmath]\beta[/ilmath] and [ilmath]\gamma[/ilmath] which are two "alien" structures on [ilmath]\mathbb{R}^2_{++} [/ilmath]
Are these charts in the same smooth structure?
TODO: Manifold example page
