Difference between revisions of "Notes:Differential (manifolds)"
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:* {{MM|1=dF_p\left(\frac{\partial}{\partial x^i}\Big\vert_p\right)=dF_p\left(d(\varphi^{-1})_{\hat{p} }\left(\frac{\partial}{\partial x^i}\Big\vert_{\hat{p} }\right)\right) }}{{MM|1==d(\psi^{-1})_{\hat{F}(\hat{p})}\left(d\hat{F}_{\hat{P} }\left(\frac{\partial}{\partial x^i}\Big\vert_{\hat{p} }\right)\right) }}{{MM|1==d(\psi^{-1})_{\hat{F}(\hat{P})}\left(\frac{\partial \hat{F}^j}{\partial x^i}(\hat{p})\frac{\partial}{\partial}{y^j}\Big\vert_{\hat{F}(\hat{p})}\right)}}{{MM|1==\frac{\partial\hat{F}^j}{\partial x^i}(\hat{P})\frac{\partial}{\partial y^j}\Big\vert_{F(p)} }} and this is apparently a matrix! It's very easy to forget what the operations are, what their elements are, so forth. This notes page is a reminder for me. Example taken from page 62 of [[Books:Introduction to Smooth Manifolds - John M. Lee]] | :* {{MM|1=dF_p\left(\frac{\partial}{\partial x^i}\Big\vert_p\right)=dF_p\left(d(\varphi^{-1})_{\hat{p} }\left(\frac{\partial}{\partial x^i}\Big\vert_{\hat{p} }\right)\right) }}{{MM|1==d(\psi^{-1})_{\hat{F}(\hat{p})}\left(d\hat{F}_{\hat{P} }\left(\frac{\partial}{\partial x^i}\Big\vert_{\hat{p} }\right)\right) }}{{MM|1==d(\psi^{-1})_{\hat{F}(\hat{P})}\left(\frac{\partial \hat{F}^j}{\partial x^i}(\hat{p})\frac{\partial}{\partial}{y^j}\Big\vert_{\hat{F}(\hat{p})}\right)}}{{MM|1==\frac{\partial\hat{F}^j}{\partial x^i}(\hat{P})\frac{\partial}{\partial y^j}\Big\vert_{F(p)} }} and this is apparently a matrix! It's very easy to forget what the operations are, what their elements are, so forth. This notes page is a reminder for me. Example taken from page 62 of [[Books:Introduction to Smooth Manifolds - John M. Lee]] | ||
==Definitions== | ==Definitions== | ||
| + | * '''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.}} | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
Revision as of 18:53, 14 May 2016
- Reason for page: I'm encountering expressions like:
- [math]dF_p\left(\frac{\partial}{\partial x^i}\Big\vert_p\right)=dF_p\left(d(\varphi^{-1})_{\hat{p} }\left(\frac{\partial}{\partial x^i}\Big\vert_{\hat{p} }\right)\right)[/math][math]=d(\psi^{-1})_{\hat{F}(\hat{p})}\left(d\hat{F}_{\hat{P} }\left(\frac{\partial}{\partial x^i}\Big\vert_{\hat{p} }\right)\right)[/math][math]=d(\psi^{-1})_{\hat{F}(\hat{P})}\left(\frac{\partial \hat{F}^j}{\partial x^i}(\hat{p})\frac{\partial}{\partial}{y^j}\Big\vert_{\hat{F}(\hat{p})}\right)[/math][math]=\frac{\partial\hat{F}^j}{\partial x^i}(\hat{P})\frac{\partial}{\partial y^j}\Big\vert_{F(p)}[/math] and this is apparently a matrix! It's very easy to forget what the operations are, what their elements are, so forth. This notes page is a reminder for me. Example taken from page 62 of Books:Introduction to Smooth Manifolds - John M. Lee
Definitions
- 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 1]
- 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]
Notes
- ↑ Lee uses [ilmath]\wedge[/ilmath] (and) where I have written [ilmath]\implies[/ilmath]
