Difference between revisions of "Motivation for smooth manifolds"
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| − | We will chart <math>\mathbb{R}_{++}^2=\{(x,y)\in\mathbb{R}^2|x>0\wedge y>0</math>. | + | 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 {{M|(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 {{M|\mathbb{R}^n}} as in this case) - it's worth having a look now. | Already you're thinking of this as a part of the plane, and clearly with coordinate {{M|(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 {{M|\mathbb{R}^n}} as in this case) - it's worth having a look now. | ||
Latest revision as of 18:08, 10 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 R2 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 R2++={(x,y)∈R2|x>0∧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 Rn as in this case) - it's worth having a look now.
Our charts
| Symbol | Name | Chart definition | Map (WRT standard) |
|---|---|---|---|
| α | standard coordinates | (R2++,α:R2++⟶R2++) | α(x,y)↦(x,y) |
| β | Polar coordinates | (R2++,β:R2++⟶R+×(0,π2)) | β(x,y)↦(√x2+y2,arctan(yx)) |
| γ | Made up for example | (R2++,γ:R2++⟶R2++) | γ(x,y)↦(x3,y3) |
We will then look at β and γ which are two "alien" structures on R2++
Are these charts in the same smooth structure?
TODO: Manifold example page
