Difference between revisions of "Notes:Differential (manifolds)"
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| − | : '' | + | : ''These notes are taken from [[Books:Introduction to Smooth Manifolds - John M. Lee]] unless otherwise noted'' |
| − | + | The [[skeleton (meta)|skeleton]] of what is needed for manifolds is remarkably small. It just looks verbose because of the large amounts of discussion involved. This page is my attempt to "compress" the skeleton of what is needed for understanding the differential into one place. I have applied [[recursive decent (meta)|recursive decent]] to solve this, which means picking something, finding out about what it depends on, and picking them, until a foundation of known facts is reached. | |
==Definitions== | ==Definitions== | ||
| + | * '''Topological n-manifold''' - A [[topological space]], {{Top.|M|J}} that is: | ||
| + | *# [[Hausdorff]] ({{AKA}}: [[T2]]) | ||
| + | *# [[Second countable topological space]] | ||
| + | *# [[Locally Euclidean of dimension n|Locally Euclidean of dimension {{n}}]] - {{M|1=\forall p\in M\exists U\in\mathcal{J}\exists\varphi:U\rightarrow \hat{U}\mathop{\subseteq}_{\text{open} }\mathbb{R}^n[p\in U\implies U\cong_\varphi\hat{U}]}}<ref group="Note">Check this formulation</ref> | ||
| + | * '''[[Chart]]''' - [[Tuple]] {{M|(U,\varphi)}} ([[Mathematicians are lazy]], short for {{M|(U,\varphi:U\in\mathcal{J}\rightarrow \hat{U}\mathop{\subseteq}_\text{open}\mathbb{R}^n)}}, where {{M|\varphi}} is a [[homeomorphism]] between {{M|U}} and {{M|\hat{U} }}) | ||
| + | * '''Smoothness of a map ({{AKA}}: {{M|C^\infty}}''' - a map, {{M|f:U\subseteq\mathbb{R}^n\rightarrow V\subseteq\mathbb{R}^m}} is ''smooth'' if it has continuous partial derivatives of all orders. | ||
| + | * '''[[Smooth map]]''' - Given [[smooth manifold|smooth manifolds]], {{M|M}} and {{M|N}} and a [[map]], {{M|F:M\rightarrow N}}. {{M|F}} is a smooth map if: | ||
| + | ** {{M|\forall p\in M\ \exists (U,\varphi)\in\mathcal{A}_M\ \exists(V,\psi)\in\mathcal{A}_N[p\in U\wedge F(p)\in V\wedge F(U)\subseteq V\implies \psi\circ F\circ \varphi^{-1}:\varphi(U)\rightarrow\psi(V)\text{ is smooth}]}}<ref group="Note">Lee uses {{M|\wedge}} (and) where I have written {{M|\implies}}</ref> | ||
* '''[[Derivation]]''' - a map, {{M|\omega:C^\infty(M)\rightarrow\mathbb{R} }} that is [[linear map|linear]] and satisfies the [[Leibniz rule]]: | * '''[[Derivation]]''' - a map, {{M|\omega:C^\infty(M)\rightarrow\mathbb{R} }} that is [[linear map|linear]] and satisfies the [[Leibniz rule]]: | ||
** {{M|1=\forall f,g\in C^\infty(M)[w(fg)=f(a)w(g)+g(a)w(f)]}} (sometimes called the product rule) | ** {{M|1=\forall f,g\in C^\infty(M)[w(fg)=f(a)w(g)+g(a)w(f)]}} (sometimes called the product rule) | ||
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* '''Differential of {{M|F}} at {{M|p}}'''. For [[smooth manifold|smooth manifolds]], {{M|M}} and {{M|N}} and a [[smooth map]], {{M|F:M\rightarrow N}} we define the ''differential of {{M|F}} as {{M|p\in M}}'' as: | * '''Differential of {{M|F}} at {{M|p}}'''. For [[smooth manifold|smooth manifolds]], {{M|M}} and {{M|N}} and a [[smooth map]], {{M|F:M\rightarrow N}} we define the ''differential of {{M|F}} as {{M|p\in M}}'' as: | ||
** {{M|dF_p:T_pM\rightarrow T_{F(p)}M}} given by: {{M|1=dF_p:v\mapsto\left\{\begin{array}{l}:C^\infty(N)\rightarrow \mathbb{R}\\:f\mapsto v(f\circ F)\end{array}\right.}} | ** {{M|dF_p:T_pM\rightarrow T_{F(p)}M}} given by: {{M|1=dF_p:v\mapsto\left\{\begin{array}{l}:C^\infty(N)\rightarrow \mathbb{R}\\:f\mapsto v(f\circ F)\end{array}\right.}} | ||
| + | |||
| + | ==Moving about== | ||
| + | * Changing coordinates - {{MM|1=\frac{\partial}{\partial x_i}\Big\vert_p=\frac{\partial \bar{x}^j}{\partial x^i}(\varphi(p))\frac{\partial}{\partial \bar{x}^j}\Big\vert_p }} - using {{ESC}} | ||
| + | ** Note that this is actually a vector (as there's an implicit sum over {{M|j}}. | ||
| + | |||
| + | The idea is to extend the definitions such that when we are dealing with manifolds that are open chunks of {{M|\mathbb{R}^n}}, we have the Jacobian as usual, and we extend this to a more general case. Much like the extension of continuity from metric to topological spaces. | ||
| + | ==Todo== | ||
| + | * Note that {{M|1=(x^1,\ldots,x^n)=(x^1(x),\ldots,x^n(x)):=\varphi(x)}} for a [[chart]] {{M|(U,\varphi)}} | ||
| + | ** Then expressions like {{MM|\frac{\partial \bar{x}^j}{\partial x^i}(\varphi(p))}} make significantly more sense, as really it is just {{MM|\frac{\partial \bar{x}^j}{\partial [\varphi(x)]^i}(\varphi(p))}} where {{M|[\cdot]^i}} is the {{M|i}}<sup>th</sup> component of a vector in {{M|\mathbb{R}^n}} (recall: {{M|\varphi(U)\subseteq\mathbb{R}^n}}) | ||
| + | *** Expand on this. | ||
| + | * There's another kind of [[derivation]] - although I suspect it is ALSO [[linear map|{{M|K}}-linear]] and satisfies that rule, however it slightly different. | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
Latest revision as of 14:24, 16 May 2016
- These notes are taken from Books:Introduction to Smooth Manifolds - John M. Lee unless otherwise noted
The skeleton of what is needed for manifolds is remarkably small. It just looks verbose because of the large amounts of discussion involved. This page is my attempt to "compress" the skeleton of what is needed for understanding the differential into one place. I have applied recursive decent to solve this, which means picking something, finding out about what it depends on, and picking them, until a foundation of known facts is reached.
Contents
Definitions
- Topological n-manifold - A topological space, [ilmath](M,\mathcal{ J })[/ilmath] that is:
- Hausdorff (AKA: T2)
- Second countable topological space
- Locally Euclidean of dimension [ilmath]n[/ilmath] - [ilmath]\forall p\in M\exists U\in\mathcal{J}\exists\varphi:U\rightarrow \hat{U}\mathop{\subseteq}_{\text{open} }\mathbb{R}^n[p\in U\implies U\cong_\varphi\hat{U}][/ilmath][Note 1]
- Chart - Tuple [ilmath](U,\varphi)[/ilmath] (Mathematicians are lazy, short for [ilmath](U,\varphi:U\in\mathcal{J}\rightarrow \hat{U}\mathop{\subseteq}_\text{open}\mathbb{R}^n)[/ilmath], where [ilmath]\varphi[/ilmath] is a homeomorphism between [ilmath]U[/ilmath] and [ilmath]\hat{U} [/ilmath])
- Smoothness of a map (AKA: [ilmath]C^\infty[/ilmath] - a map, [ilmath]f:U\subseteq\mathbb{R}^n\rightarrow V\subseteq\mathbb{R}^m[/ilmath] is smooth if it has continuous partial derivatives of all orders.
- Smooth map - Given smooth manifolds, [ilmath]M[/ilmath] and [ilmath]N[/ilmath] and a map, [ilmath]F:M\rightarrow N[/ilmath]. [ilmath]F[/ilmath] is a smooth map if:
- [ilmath]\forall p\in M\ \exists (U,\varphi)\in\mathcal{A}_M\ \exists(V,\psi)\in\mathcal{A}_N[p\in U\wedge F(p)\in V\wedge F(U)\subseteq V\implies \psi\circ F\circ \varphi^{-1}:\varphi(U)\rightarrow\psi(V)\text{ is smooth}][/ilmath][Note 2]
- Derivation - a map, [ilmath]\omega:C^\infty(M)\rightarrow\mathbb{R} [/ilmath] that is linear and satisfies the Leibniz rule:
- [ilmath]\forall f,g\in C^\infty(M)[w(fg)=f(a)w(g)+g(a)w(f)][/ilmath] (sometimes called the product rule)
- Tangent space to [ilmath]M[/ilmath] at [ilmath]p[/ilmath] [ilmath]T_pM[/ilmath] is a vector space called the tangent space to [ilmath]M[/ilmath] at [ilmath]p[/ilmath], it's the set of all derivations of [ilmath]C^\infty(M)[/ilmath]
- Differential of [ilmath]F[/ilmath] at [ilmath]p[/ilmath]. For smooth manifolds, [ilmath]M[/ilmath] and [ilmath]N[/ilmath] and a smooth map, [ilmath]F:M\rightarrow N[/ilmath] we define the differential of [ilmath]F[/ilmath] as [ilmath]p\in M[/ilmath] as:
- [ilmath]dF_p:T_pM\rightarrow T_{F(p)}M[/ilmath] given by: [ilmath]dF_p:v\mapsto\left\{\begin{array}{l}:C^\infty(N)\rightarrow \mathbb{R}\\:f\mapsto v(f\circ F)\end{array}\right.[/ilmath]
Moving about
- Changing coordinates - [math]\frac{\partial}{\partial x_i}\Big\vert_p=\frac{\partial \bar{x}^j}{\partial x^i}(\varphi(p))\frac{\partial}{\partial \bar{x}^j}\Big\vert_p[/math] - using Template:ESC
- Note that this is actually a vector (as there's an implicit sum over [ilmath]j[/ilmath].
The idea is to extend the definitions such that when we are dealing with manifolds that are open chunks of [ilmath]\mathbb{R}^n[/ilmath], we have the Jacobian as usual, and we extend this to a more general case. Much like the extension of continuity from metric to topological spaces.
Todo
- Note that [ilmath](x^1,\ldots,x^n)=(x^1(x),\ldots,x^n(x)):=\varphi(x)[/ilmath] for a chart [ilmath](U,\varphi)[/ilmath]
- Then expressions like [math]\frac{\partial \bar{x}^j}{\partial x^i}(\varphi(p))[/math] make significantly more sense, as really it is just [math]\frac{\partial \bar{x}^j}{\partial [\varphi(x)]^i}(\varphi(p))[/math] where [ilmath][\cdot]^i[/ilmath] is the [ilmath]i[/ilmath]th component of a vector in [ilmath]\mathbb{R}^n[/ilmath] (recall: [ilmath]\varphi(U)\subseteq\mathbb{R}^n[/ilmath])
- Expand on this.
- Then expressions like [math]\frac{\partial \bar{x}^j}{\partial x^i}(\varphi(p))[/math] make significantly more sense, as really it is just [math]\frac{\partial \bar{x}^j}{\partial [\varphi(x)]^i}(\varphi(p))[/math] where [ilmath][\cdot]^i[/ilmath] is the [ilmath]i[/ilmath]th component of a vector in [ilmath]\mathbb{R}^n[/ilmath] (recall: [ilmath]\varphi(U)\subseteq\mathbb{R}^n[/ilmath])
- There's another kind of derivation - although I suspect it is ALSO [ilmath]K[/ilmath]-linear and satisfies that rule, however it slightly different.
