Notes:Differential (manifolds)

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]
• 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].