Motivation for smooth manifolds

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 $\mathbb{R}^2$ 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

$$\mathbb{R}_{++}^2=\{(x,y)\in\mathbb{R}^2|x>0\wedge y>0\}$$ .

Already you're thinking of this as a part of the plane, and clearly with coordinate $(x,y)$ - 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 $\mathbb{R}^n$ as in this case) - it's worth having a look now.

Our charts

Symbol	Name	Chart definition	Map (WRT standard)
$\alpha$	standard coordinates	$(\mathbb{R}_{++}^2,\alpha:\mathbb{R}^2_{++}\longrightarrow\mathbb{R}^2_{++})$	$\alpha(x,y)\mapsto(x,y)$
$\beta$	Polar coordinates	$(\mathbb{R}_{++}^2,\beta:\mathbb{R}^2_{++}\longrightarrow\mathbb{R}_+\times(0,\tfrac{\pi}{2}))$	$\beta(x,y)\mapsto\left(\sqrt{x^2+y^2},\arctan(\tfrac{y}{x})\right)$
$\gamma$	Made up for example	$(\mathbb{R}^2_{++},\gamma:\mathbb{R}^2_{++}\longrightarrow\mathbb{R}^2_{++})$	$\gamma(x,y)\mapsto(x^3,y^3)$

We will then look at $\beta$ and $\gamma$ which are two "alien" structures on $\mathbb{R}^2_{++}$

Are these charts in the same smooth structure?

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TODO: Manifold example page

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