From 7a49addafa161ada5dec98b716e083ebf510e3fc Mon Sep 17 00:00:00 2001 From: Juan Marín Noguera Date: Wed, 13 Jan 2021 19:59:57 +0100 Subject: gcs/a3c --- gcs/n3.lyx | 1385 +++++++++++++++++++++++++++++++++++++++++++++--------------- 1 file changed, 1054 insertions(+), 331 deletions(-) (limited to 'gcs') diff --git a/gcs/n3.lyx b/gcs/n3.lyx index ecf63cd..61fab42 100644 --- a/gcs/n3.lyx +++ b/gcs/n3.lyx @@ -5,6 +5,9 @@ \save_transient_properties true \origin unavailable \textclass book +\begin_preamble +\input{../defs} +\end_preamble \use_default_options true \maintain_unincluded_children false \language spanish @@ -77,6 +80,10 @@ \begin_body +\begin_layout Section +Orientación +\end_layout + \begin_layout Standard Dada una superficie regular \begin_inset Formula $S$ @@ -215,213 +222,546 @@ orientable \begin_inset Formula $S$ \end_inset - está orientada cuando se ha escogido una orientación concreta. + está orientada cuando se ha escogido una orientación concreta, en cuyo + caso dicha orientación es su +\series bold +aplicación de Gauss +\series default +. \end_layout \begin_layout Standard -Dos cartas -\begin_inset Formula $(U,X)$ -\end_inset +Ejemplos: +\begin_inset Note Comment +status open - y -\begin_inset Formula $(U',X')$ +\begin_layout Enumerate +La banda de Möbius se puede expresar como la imagen de +\begin_inset Formula $X:\mathbb{R}\times(-1,1)\to\mathbb{R}^{3}$ \end_inset - de -\begin_inset Formula $S$ -\end_inset + dada por +\begin_inset Formula +\[ +X(u,v):=\left((2-v\sin\tfrac{u}{2})\sin u,(2-v\sin\tfrac{u}{2})\cos u,v\cos\tfrac{u}{2}\right). +\] - son -\series bold -compatibles -\series default - si -\begin_inset Formula $V:=X(U)$ \end_inset - y -\begin_inset Formula $V':=X'(U')$ +Esta es una superficie regular no orientable. +\end_layout + +\begin_deeper +\begin_layout Plain Layout +Claramente +\begin_inset Formula $X$ \end_inset - son disjuntos o -\begin_inset Formula $\det(Jh)>0$ + es diferenciable, y es inyectiva en +\begin_inset Formula $U_{1}:=(0,2\pi)\times(-1,1)$ \end_inset -, donde -\begin_inset Formula $h:X^{-1}(V')\to(X')^{-1}(V)$ + y en +\begin_inset Formula $U_{2}:=(-\pi,\pi)\times(-1,1)$ \end_inset - es el cambio de coordenadas de -\begin_inset Formula $V$ +. + Su diferencial es +\begin_inset Formula +\[ +dX(u,v)\equiv\begin{pmatrix}-\frac{v}{2}\cos\frac{u}{2}\sin u+(2-v\sin\frac{u}{2})\cos u & -\sin\frac{u}{2}\sin u\\ +-\frac{v}{2}\cos\frac{u}{2}\cos u-(2-v\sin\frac{u}{2})\sin u & -\sin\frac{u}{2}\cos u\\ +-\frac{v}{2}\sin\frac{u}{2} & \cos\frac{u}{2} +\end{pmatrix}, +\] + \end_inset - a -\begin_inset Formula $V'$ +y el determinante de las dos primeras filas es +\begin_inset Formula +\[ +-\sin\frac{u}{2}\left(-\frac{v}{2}\cos\frac{u}{2}\begin{vmatrix}\sin u & \sin u\\ +\cos u & \cos u +\end{vmatrix}+\left(2-v\sin\frac{u}{2}\right)\begin{vmatrix}\cos u & \sin u\\ +-\sin u & \cos u +\end{vmatrix}\right)=-\sin\frac{u}{2}\left(2-v\sin\frac{u}{2}\right), +\] + \end_inset -. - Un -\series bold -atlas -\series default - para -\begin_inset Formula $S$ +lo que solo se anula cuando +\begin_inset Formula $u\in\{2k\pi\}_{k\in\mathbb{Z}}$ \end_inset - es una familia -\begin_inset Formula $\{(U_{i},X_{i})\}_{i\in I}$ +, pero en tal caso +\begin_inset Formula +\[ +dX(u,v)\equiv\begin{pmatrix}2 & 0\\ +-\frac{v}{2} & 0\\ +0 & 1 +\end{pmatrix} +\] + \end_inset - de cartas tales que -\begin_inset Formula $\bigcup_{i\in I}X_{i}(U_{i})=S$ +y el determinante de la submatriz resultante de quitar la segunda fila es + +\begin_inset Formula $2\neq0$ \end_inset . - Entonces una superficie es orientable si y sólo si existe un atlas cuyas - cartas son compatibles. + Esto prueba que la banda de Möbius es una superficie. \end_layout -\begin_layout Itemize -\begin_inset Argument item:1 -status open - -\begin_layout Plain Layout -\begin_inset Formula $\impliedby]$ +\end_deeper \end_inset \end_layout +\begin_layout Enumerate +El plano +\begin_inset Formula $p_{0}+\langle v\rangle^{\bot}\subseteq\mathbb{R}^{3}$ \end_inset -Sean -\begin_inset Formula ${\cal A}:=\{(U_{i},X_{i})\}_{i\in I}$ + admite la orientación +\begin_inset Formula $N(p):=v/|v|$ \end_inset - un atlas de cartas compatibles en -\begin_inset Formula $S$ +. +\end_layout + +\begin_layout Enumerate +Dados +\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ \end_inset -, -\begin_inset Formula $p\in S$ + +\begin_inset Formula ${\cal C}^{2}$ \end_inset -, -\begin_inset Formula $(U,X)\in{\cal A}(I)$ + y un valor regular +\begin_inset Formula $c$ \end_inset - con -\begin_inset Formula $p\in X(U)$ + de +\begin_inset Formula $f$ \end_inset - y -\begin_inset Formula $N:X(U)\to\mathbb{R}^{3}$ +, la superficie de nivel +\begin_inset Formula $S:=f^{-1}(c)$ \end_inset - dado por + admite la orientación \begin_inset Formula \[ -N(X(u,v)):=N(u,v):=\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}(u,v), +N(p):=\frac{\nabla f(p)}{|\nabla f(p)|}, \] \end_inset - -\begin_inset Formula $N$ + donde +\begin_inset Formula $\nabla f(p):=(\frac{\partial f}{\partial x}(p),\frac{\partial f}{\partial y}(p),\frac{\partial f}{\partial z}(p))$ \end_inset - está bien definido y es diferenciable, normal y unitario. - Sean ahora -\begin_inset Formula $(\overline{U},\overline{X})\in{\cal A}(I)$ + es el +\series bold +gradiente +\series default + de +\begin_inset Formula $f$ \end_inset - con -\begin_inset Formula $p\in\overline{X}(\overline{U})$ + en +\begin_inset Formula $p$ \end_inset -, -\begin_inset Formula $\overline{N}(\overline{X}(u,v)):=\overline{N}(u,v):=\frac{\overline{X}_{u}\cap\overline{X}_{v}}{|\overline{X}_{u}\cap\overline{X}_{v}|}(u,v)$ +. +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $p\in S$ \end_inset - y -\begin_inset Formula $h$ +, +\begin_inset Formula $\alpha:=(x,y,z):I\to S$ \end_inset - el cambio de coordenadas de -\begin_inset Formula $(U,X)$ + una curva diferenciable con +\begin_inset Formula $\alpha(0)=p$ \end_inset - a -\begin_inset Formula $(\overline{U},\overline{X})$ + y +\begin_inset Formula $v:=\alpha'(0)\in T_{p}S$ \end_inset , para -\begin_inset Formula $(u,v)\in X^{-1}(V_{0})$ +\begin_inset Formula $t\in I$ \end_inset -, -\begin_inset Formula -\[ -dX(u,v)=d(\overline{X}\circ h)(u,v)=d\overline{X}(h(u,v))\circ dh(u,v), -\] + es +\begin_inset Formula $f(\alpha(t))=c$ +\end_inset + por ser +\begin_inset Formula $\alpha(t)\in S$ \end_inset - luego -\begin_inset Formula -\[ -N(u,v)=\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}=\frac{\det(Jh(u,v))}{|\det(Jh(u,v))|}\frac{\overline{X}_{u}\wedge\overline{X}_{v}}{|\overline{X}_{u}\wedge\overline{X}_{v}|}(h(u,v))\overset{Jh(u,v)>0}{=}\overline{N}(u,v), -\] +, luego derivando, +\begin_inset Formula $\frac{\partial f}{\partial x}(\alpha(t))x'(t)+\frac{\partial f}{\partial y}(\alpha(t))y'(t)+\frac{\partial f}{\partial z}(\alpha(t))z'(t)=0$ +\end_inset + y +\begin_inset Formula $\nabla f(p)\bot v$ \end_inset - de modo que -\begin_inset Formula $N(p)$ +. + Además, +\begin_inset Formula $\nabla f(p)\neq0$ \end_inset - es diferenciable, normal, unitario y no depende de la carta del atlas escogida. -\end_layout + porque +\begin_inset Formula $p\in S=f^{-1}(c)$ +\end_inset -\begin_layout Itemize -\begin_inset Argument item:1 -status open + y +\begin_inset Formula $c$ +\end_inset -\begin_layout Plain Layout -\begin_inset Formula $\implies]$ + es un valor regular de +\begin_inset Formula $f$ \end_inset +, y claramente +\begin_inset Formula $\nabla f$ +\end_inset + es diferenciable. \end_layout +\end_deeper +\begin_layout Enumerate +\begin_inset Formula $\mathbb{S}^{2}(r)$ \end_inset -Sea -\begin_inset Formula $N$ + admite la orientación +\begin_inset Formula $N(p)=\frac{1}{r}p$ \end_inset - una orientación de -\begin_inset Formula $S$ -\end_inset +. +\end_layout -, para toda carta -\begin_inset Formula $(U,X)$ +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}+z^{2}$ \end_inset - de -\begin_inset Formula $S$ +, +\begin_inset Formula $r^{2}$ \end_inset - es -\begin_inset Formula $N(X(q))=\pm\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}(q)$ + es un valor regular de +\begin_inset Formula $f$ \end_inset - para todo -\begin_inset Formula $q\in U$ + y +\begin_inset Formula $\mathbb{S}^{2}$ \end_inset -. - Entonces, para -\begin_inset Formula $p\in S$ + es la superficie de nivel +\begin_inset Formula $\{p:f(p)=r^{2}\}$ +\end_inset + +, luego admite la orientación +\begin_inset Formula +\[ +N(x,y,z)=\frac{\nabla f(x,y,z)}{|\nabla f(x,y,z)|}=\frac{(2x,2y,2z)}{|(2x,2y,2z)|}=\frac{(x,y,z)}{|(x,y,z)|}=\frac{1}{r}(x,y,z). +\] + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +El cilindro +\begin_inset Formula $\{x^{2}+y^{2}=r^{2}\}$ +\end_inset + + admite la orientación +\begin_inset Formula $N(x,y,z)=(x,y,0)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Es una superficie de nivel y tiene pues orientación +\begin_inset Formula $N(p)=\frac{(2x,2y,0)}{|(2x,2y,0)|}=\frac{(x,y,0)}{|(x,y,0)|}=\frac{1}{r}(x,y,0)$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Dada +\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + + diferenciable en el abierto +\begin_inset Formula $U$ +\end_inset + +, el grafo +\begin_inset Formula $S:=\{(x,y,f(x,y))\}_{x,y\in U}$ +\end_inset + + admite la orientación +\begin_inset Formula +\[ +N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}(u,v). +\] + +\end_inset + +Dada la parametrización +\begin_inset Formula $(U,X)$ +\end_inset + + con +\begin_inset Formula $X(u,v):=(u,v,f(u,v))$ +\end_inset + +, +\begin_inset Formula $X_{u}=(1,0,f_{u})$ +\end_inset + + y +\begin_inset Formula $X_{v}=(0,1,f_{v})$ +\end_inset + +, y +\begin_inset Formula $X_{u}\wedge X_{v}=(-f_{u},-f_{v},1)$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Las superficies orientables tienen exactamente dos orientaciones, una opuesta + de la otra. +\end_layout + +\begin_layout Standard +Dos cartas +\begin_inset Formula $(U,X)$ +\end_inset + + y +\begin_inset Formula $(U',X')$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + son +\series bold +compatibles +\series default + si +\begin_inset Formula $V:=X(U)$ +\end_inset + + y +\begin_inset Formula $V':=X'(U')$ +\end_inset + + son disjuntos o +\begin_inset Formula $\det(Jh)>0$ +\end_inset + +, donde +\begin_inset Formula $h:X^{-1}(V')\to(X')^{-1}(V)$ +\end_inset + + es el cambio de coordenadas de +\begin_inset Formula $V$ +\end_inset + + a +\begin_inset Formula $V'$ +\end_inset + +. + Un +\series bold +atlas +\series default + para +\begin_inset Formula $S$ +\end_inset + + es una familia +\begin_inset Formula $\{(U_{i},X_{i})\}_{i\in I}$ +\end_inset + + de cartas tales que +\begin_inset Formula $\bigcup_{i\in I}X_{i}(U_{i})=S$ +\end_inset + +. + Entonces una superficie es orientable si y sólo si existe un atlas cuyas + cartas son compatibles. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\impliedby]$ +\end_inset + + +\end_layout + +\end_inset + +Sean +\begin_inset Formula ${\cal A}:=\{(U_{i},X_{i})\}_{i\in I}$ +\end_inset + + un atlas de cartas compatibles en +\begin_inset Formula $S$ +\end_inset + +, +\begin_inset Formula $p\in S$ +\end_inset + +, +\begin_inset Formula $(U,X)\in{\cal A}(I)$ +\end_inset + + con +\begin_inset Formula $p\in X(U)$ +\end_inset + + y +\begin_inset Formula $N:X(U)\to\mathbb{R}^{3}$ +\end_inset + + dado por +\begin_inset Formula +\[ +N(X(u,v)):=N(u,v):=\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}(u,v), +\] + +\end_inset + + +\begin_inset Formula $N$ +\end_inset + + está bien definido y es diferenciable, normal y unitario. + Sean ahora +\begin_inset Formula $(\overline{U},\overline{X})\in{\cal A}(I)$ +\end_inset + + con +\begin_inset Formula $p\in\overline{X}(\overline{U})$ +\end_inset + +, +\begin_inset Formula $\overline{N}(\overline{X}(u,v)):=\overline{N}(u,v):=\frac{\overline{X}_{u}\cap\overline{X}_{v}}{|\overline{X}_{u}\cap\overline{X}_{v}|}(u,v)$ +\end_inset + + y +\begin_inset Formula $h$ +\end_inset + + el cambio de coordenadas de +\begin_inset Formula $(U,X)$ +\end_inset + + a +\begin_inset Formula $(\overline{U},\overline{X})$ +\end_inset + +, para +\begin_inset Formula $(u,v)\in X^{-1}(V_{0})$ +\end_inset + +, +\begin_inset Formula +\[ +dX(u,v)=d(\overline{X}\circ h)(u,v)=d\overline{X}(h(u,v))\circ dh(u,v), +\] + +\end_inset + + luego +\begin_inset Formula +\[ +N(u,v)=\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}=\frac{\det(Jh(u,v))}{|\det(Jh(u,v))|}\frac{\overline{X}_{u}\wedge\overline{X}_{v}}{|\overline{X}_{u}\wedge\overline{X}_{v}|}(h(u,v))\overset{Jh(u,v)>0}{=}\overline{N}(u,v), +\] + +\end_inset + + de modo que +\begin_inset Formula $N(p)$ +\end_inset + + es diferenciable, normal, unitario y no depende de la carta del atlas escogida. +\end_layout + +\begin_layout Itemize +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +\begin_inset Formula $\implies]$ +\end_inset + + +\end_layout + +\end_inset + +Sea +\begin_inset Formula $N$ +\end_inset + + una orientación de +\begin_inset Formula $S$ +\end_inset + +, para toda carta +\begin_inset Formula $(U,X)$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + es +\begin_inset Formula $N(X(q))=\pm\frac{X_{u}\wedge X_{v}}{|X_{u}\wedge X_{v}|}(q)$ +\end_inset + + para todo +\begin_inset Formula $q\in U$ +\end_inset + +. + Entonces, para +\begin_inset Formula $p\in S$ \end_inset , podemos tomar una carta @@ -432,257 +772,618 @@ Sea \begin_inset Formula $S$ \end_inset - con -\begin_inset Formula $N(X(q))=\frac{(X_{p})_{u}\wedge(X_{p})_{v}}{|(X_{p})_{u}\wedge(X_{p})_{v}|}(q)$ + con +\begin_inset Formula $N(X(q))=\frac{(X_{p})_{u}\wedge(X_{p})_{v}}{|(X_{p})_{u}\wedge(X_{p})_{v}|}(q)$ +\end_inset + + para +\begin_inset Formula $q\in U$ +\end_inset + +, pues si el normal fuese el opuesto basta cambiar +\begin_inset Formula $X_{p}(u,v)$ +\end_inset + + por +\begin_inset Formula $X_{p}(v,u)$ +\end_inset + + y +\begin_inset Formula $U_{p}$ +\end_inset + + por +\begin_inset Formula $\{(u,v)\}_{(v,u)\in U}$ +\end_inset + +, y el resultado se tiene por la antisimetría del producto vectorial. + Con esto, dados +\begin_inset Formula $a,b\in S$ +\end_inset + + con +\begin_inset Formula $V:=X_{a}(U_{a})\cap X_{b}(U_{b})\neq\emptyset$ +\end_inset + +, queremos ver que el determinante del cambio de coordenadas +\begin_inset Formula $h:X_{a}^{-1}(V)\to X_{b}^{-1}(V)$ +\end_inset + + de +\begin_inset Formula $(U_{a},X_{a})$ +\end_inset + + a +\begin_inset Formula $(U_{b},X_{b})$ +\end_inset + + tiene jacobiano con determinante positivo. + En efecto, +\begin_inset Formula $\det(Jh)$ +\end_inset + + debe ser no nulo, pero si fuera negativo, para un +\begin_inset Formula $p\in V$ +\end_inset + +, sean +\begin_inset Formula $q_{a}:=X_{a}^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $q_{b}:=X_{b}^{-1}(p)$ +\end_inset + +, entonces +\begin_inset Formula +\[ +N(p)=\frac{X_{au}\wedge X_{av}}{|X_{au}\wedge X_{av}|}(q_{a})=\frac{\det(Jh)}{|\det(Jh)|}\frac{X_{bu}\wedge X_{bv}}{|X_{bu}\wedge X_{bv}|}(q_{b})=-N(p), +\] + +\end_inset + +luego +\begin_inset Formula $N(p)=0\#$ +\end_inset + +. + Por tanto +\begin_inset Formula $\det(Jh)>0$ +\end_inset + + y las cartas del atlas +\begin_inset Formula $\{(U_{p},X_{p})\}_{p\in S}$ +\end_inset + + son compatibles. +\end_layout + +\begin_layout Standard +En adelante, cuando consideremos una parametrización +\begin_inset Formula $(U,X)$ +\end_inset + +, escribiremos +\begin_inset Formula $N(u,v):=N(X(u,v))$ +\end_inset + +, +\begin_inset Formula $N_{u}(u,v):=\frac{\partial(N\circ X)}{\partial u}(u,v)$ +\end_inset + + y +\begin_inset Formula $N_{v}(u,v):=\frac{\partial(N\circ X)}{\partial v}(u,v)$ +\end_inset + +. + En general, para +\begin_inset Formula $f:\mathbb{R}^{n}\to\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $f_{x_{i}}:=\frac{\partial f}{\partial x_{i}}$ +\end_inset + +. +\end_layout + +\begin_layout Section +La segunda forma fundamental +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie orientada con aplicación de Gauss +\begin_inset Formula $N:S\to\mathbb{S}^{2}$ +\end_inset + +, llamamos +\series bold +imagen esférica +\series default + de +\begin_inset Formula $S$ +\end_inset + + a +\begin_inset Formula $\text{Im}N\subseteq\mathbb{S}^{2}$ \end_inset - para -\begin_inset Formula $q\in U$ +. + Ejemplos: +\end_layout + +\begin_layout Enumerate +La imagen esférica de un plano es unipuntual. +\end_layout + +\begin_deeper +\begin_layout Standard +Dado el plano +\begin_inset Formula $\Pi:=p_{0}+\langle v\rangle\subseteq\mathbb{R}^{3}$ \end_inset -, pues si el normal fuese el opuesto basta cambiar -\begin_inset Formula $X_{p}(u,v)$ +, donde podemos suponer +\begin_inset Formula $v$ \end_inset - por -\begin_inset Formula $X_{p}(v,u)$ + unitario, la imagen de +\begin_inset Formula $N(p):=v$ \end_inset - y -\begin_inset Formula $U_{p}$ + es +\begin_inset Formula $\{v\}$ \end_inset - por -\begin_inset Formula $\{(u,v)\}_{(v,u)\in U}$ +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La imagen esférica de +\begin_inset Formula $\mathbb{S}^{2}$ \end_inset -, y el resultado se tiene por la antisimetría del producto vectorial. - Con esto, dados -\begin_inset Formula $a,b\in S$ + es +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +La aplicación de Gauss es +\begin_inset Formula $\pm1_{\mathbb{S}^{2}}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La imagen esférica de un grafo +\begin_inset Formula $\{(x,y,f(x,y))\}_{(x,y)\in U}$ \end_inset con -\begin_inset Formula $V:=X_{a}(U_{a})\cap X_{b}(U_{b})\neq\emptyset$ +\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ \end_inset -, queremos ver que el determinante del cambio de coordenadas -\begin_inset Formula $h:X_{a}^{-1}(V)\to X_{b}^{-1}(V)$ + diferenciable está contenida en el hemisferio (estricto) norte o sur. +\end_layout + +\begin_deeper +\begin_layout Standard +Una orientación es +\begin_inset Formula $N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}(u,v)$ +\end_inset + +, y como la coordenada +\begin_inset Formula $z$ \end_inset de -\begin_inset Formula $(U_{a},X_{a})$ +\begin_inset Formula $N$ \end_inset - a -\begin_inset Formula $(U_{b},X_{b})$ + es siempre positiva, +\begin_inset Formula $\text{Im}N$ \end_inset - tiene jacobiano con determinante positivo. - En efecto, -\begin_inset Formula $\det(Jh)$ + está en el hemisferio norte estricto. + Con la orientación opuesta está en el hemisferio sur estricto. +\end_layout + +\end_deeper +\begin_layout Enumerate +La imagen esférica de un cilindro es un circulo máximo de la esfera. +\end_layout + +\begin_deeper +\begin_layout Standard +Los cilindros se obtienen por un movimiento de +\begin_inset Formula $S_{r}:=\{x^{2}+y^{2}=r^{2}\}$ \end_inset - debe ser no nulo, pero si fuera negativo, para un -\begin_inset Formula $p\in V$ + para algún +\begin_inset Formula $r>0$ \end_inset -, sean -\begin_inset Formula $q_{a}:=X_{a}^{-1}(p)$ +, y como su orientación es +\begin_inset Formula $N(x,y,z)=\pm\frac{1}{r}(x,y,0)$ +\end_inset + +, +\begin_inset Formula $N(S_{r})=\{\frac{1}{r}(x,y,0):x^{2}+y^{2}=r^{2}\}=\{(x,y,0):x^{2}+y^{2}=1\}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +El +\series bold +catenoide +\series default +, +\begin_inset Formula $C:=\{x^{2}+y^{2}=\cosh^{2}z\}$ +\end_inset + +, tiene imagen esférica +\begin_inset Formula $\mathbb{S}^{2}\setminus\{\mathsf{N},\mathsf{S}\}$ +\end_inset + +, donde +\begin_inset Formula $\mathsf{N}:=(0,0,1)$ \end_inset + es el +\series bold +polo norte +\series default y -\begin_inset Formula $q_{b}:=X_{b}^{-1}(p)$ +\begin_inset Formula $\mathsf{S}:=(0,0,-1)$ \end_inset -, entonces + es el +\series bold +polo sur +\series default +. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}-\cosh^{2}z$ +\end_inset + +, como +\begin_inset Formula $f_{x}=2x$ +\end_inset + +, +\begin_inset Formula $f_{y}=2y$ +\end_inset + + y +\begin_inset Formula $f_{z}=-2\cosh z\sinh z$ +\end_inset + +, el único punto crítico de +\begin_inset Formula $f$ +\end_inset + + es el origen, con +\begin_inset Formula $f(0)=-1$ +\end_inset + +, de modo que 0 es un valor regular de +\begin_inset Formula $f\in{\cal C}^{\infty}$ +\end_inset + + y +\begin_inset Formula $C=\{f(x,y,z)=0\}$ +\end_inset + + es una superficie de nivel regular y \begin_inset Formula -\[ -N(p)=\frac{X_{au}\wedge X_{av}}{|X_{au}\wedge X_{av}|}(q_{a})=\frac{\det(Jh)}{|\det(Jh)|}\frac{X_{bu}\wedge X_{bv}}{|X_{bu}\wedge X_{bv}|}(q_{b})=-N(p), -\] +\begin{align*} +N(x,y,z) & =\frac{\nabla f(x,y,z)}{\Vert\nabla f(x,y,z)\Vert}=\frac{(2x,2y,-2\cosh z\sinh z)}{2\sqrt{x^{2}+y^{2}+\cosh^{2}z\sinh^{2}z}}\\ + & =\frac{(x,y,-\cosh z\sinh z)}{\sqrt{\cosh^{2}z+\cosh^{2}z\sinh^{2}z}}=\frac{(x,y,-\cosh z\sinh z)}{\cosh^{2}z}. +\end{align*} + +\end_inset + +Como +\begin_inset Formula $N_{1}(p)^{2}+N_{2}(p)^{2}=\frac{x^{2}+y^{2}}{\cosh^{4}z}=\frac{1}{\cosh^{2}z}>0$ +\end_inset + +, no se cubren los polos norte y sur. + Sean ahora +\begin_inset Formula $(\hat{x},\hat{y},\hat{z})\in\mathbb{S}^{2}\setminus\{\mathsf{N},\mathsf{S}\}$ +\end_inset + +, +\begin_inset Formula $z:=\arg\tanh(-\hat{z})$ +\end_inset + + (que existe porque +\begin_inset Formula $\hat{z}\in(-1,1)$ +\end_inset + +), +\begin_inset Formula $x:=\hat{x}\cosh^{2}z$ +\end_inset + + e +\begin_inset Formula $y:=\hat{y}\cosh^{2}z$ +\end_inset + +, es claro que +\begin_inset Formula $N(x,y,z)=(\hat{x},\hat{y},\hat{z})$ +\end_inset + +. + Ahora bien, +\begin_inset Formula +\begin{multline*} +x^{2}+y^{2}=(\hat{x}^{2}+\hat{y}^{2})\cosh^{4}z=(1-\hat{z}^{2})\cosh^{4}z=\left(1-\tanh^{2}z\right)\cosh^{4}z=\\ +=\frac{\cosh^{2}z-\sinh^{2}z}{\cosh^{2}z}\cosh^{4}z=\frac{\cosh^{4}z}{\cosh^{2}z}=\cosh^{2}z, +\end{multline*} \end_inset luego -\begin_inset Formula $N(p)=0\#$ +\begin_inset Formula $(x,y,z)\in C$ +\end_inset + + y +\begin_inset Formula $N(x,y,z)$ +\end_inset + + cubre +\begin_inset Formula $\mathbb{S}^{2}\setminus\{\mathsf{N},\mathsf{S}\}$ \end_inset . - Por tanto -\begin_inset Formula $\det(Jh)>0$ +\end_layout + +\end_deeper +\begin_layout Standard +Para +\begin_inset Formula $p\in\mathbb{S}^{2}$ \end_inset - y las cartas del atlas -\begin_inset Formula $\{(U_{p},X_{p})\}_{p\in S}$ + es +\begin_inset Formula $T_{N(p)}\mathbb{S}^{2}=T_{p}\mathbb{S}^{2}$ +\end_inset + +, pues +\begin_inset Formula $N(p)=\pm p$ +\end_inset + + y +\begin_inset Formula $T_{-p}\mathbb{S}^{2}=\langle N(-p)\rangle^{\bot}=\langle p\rangle^{\bot}=\langle N(p)\rangle^{\bot}=T_{p}\mathbb{S}^{2}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie regular orientada por +\begin_inset Formula $N$ +\end_inset + +, llamamos +\series bold +operador forma +\series default + o +\series bold +endomorfismo de Weingarten +\series default + en +\begin_inset Formula $p\in S$ +\end_inset + + a +\begin_inset Formula $A_{p}:=-dN_{p}:T_{p}S\to T_{p}S$ +\end_inset + +. + En efecto, como +\begin_inset Formula $N:S\to\mathbb{S}^{2}$ +\end_inset + +, +\begin_inset Formula $dN_{p}:T_{p}S\to T_{N(p)}\mathbb{S}^{2}$ +\end_inset + +, pero como la normal en +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + + es +\begin_inset Formula $1_{\mathbb{S}^{2}}$ +\end_inset + +, +\begin_inset Formula $T_{p'}\mathbb{S}^{2}=\langle p'\rangle^{\bot}$ +\end_inset + + para todo +\begin_inset Formula $p'\in\mathbb{S}^{2}$ +\end_inset + + y en particular +\begin_inset Formula $T_{N(p)}\mathbb{S}^{2}=\langle N(p)\rangle^{\bot}=T_{p}S$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $A_{p}$ +\end_inset + + es +\series bold +autoadjunto +\series default +, es decir, +\begin_inset Formula $\langle A_{p}v,w\rangle=\langle v,A_{p}w\rangle$ +\end_inset + +. + +\series bold +Demostración: +\series default + Por linealidad, basta demostrarlo para una base de +\begin_inset Formula $T_{p}S$ \end_inset - son compatibles. -\end_layout - -\begin_layout Standard -En adelante, cuando consideremos una parametrización +. + Sean \begin_inset Formula $(U,X)$ \end_inset -, escribiremos -\begin_inset Formula $N(u,v):=N(X(u,v))$ + una parametrización de +\begin_inset Formula $S$ \end_inset -, -\begin_inset Formula $N_{u}(u,v):=\frac{\partial(N\circ X)}{\partial u}(u,v)$ + en +\begin_inset Formula $p$ \end_inset y -\begin_inset Formula $N_{v}(u,v):=\frac{\partial(N\circ X)}{\partial v}(u,v)$ +\begin_inset Formula $q:=(u_{0},v_{0}):=X^{-1}(p)$ \end_inset -. -\end_layout - -\begin_layout Standard -Ejemplos: -\begin_inset Note Comment -status open - -\begin_layout Enumerate -La banda de Möbius se puede expresar como la imagen de -\begin_inset Formula $X:\mathbb{R}\times(-1,1)\to\mathbb{R}^{3}$ +, tomamos la base +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ \end_inset - dada por -\begin_inset Formula -\[ -X(u,v):=\left((2-v\sin\tfrac{u}{2})\sin u,(2-v\sin\tfrac{u}{2})\cos u,v\cos\tfrac{u}{2}\right). -\] - + y queremos ver que +\begin_inset Formula $\langle dN_{p}(X_{u}(q)),X_{v}(q)\rangle=\langle X_{u}(q),dN_{p}(X_{v}(q))\rangle$ \end_inset -Esta es una superficie regular no orientable. -\end_layout +. + Sea entonces +\begin_inset Formula $\alpha(u):=X(u_{0}+u,v_{0})$ +\end_inset -\begin_deeper -\begin_layout Plain Layout -Claramente -\begin_inset Formula $X$ +, +\begin_inset Formula $\alpha(0)=p$ \end_inset - es diferenciable, y es inyectiva en -\begin_inset Formula $U_{1}:=(0,2\pi)\times(-1,1)$ + y +\begin_inset Formula $\alpha'(0)=X_{u}(q)$ \end_inset - y en -\begin_inset Formula $U_{2}:=(-\pi,\pi)\times(-1,1)$ +, luego +\begin_inset Formula $dN_{p}(X_{u}(q))=\frac{\partial(N\circ\alpha)}{\partial u}(0)=\frac{\partial(N\circ X)}{\partial u}(u_{0},v_{0})=N_{u}(u_{0},v_{0})$ \end_inset . - Su diferencial es -\begin_inset Formula -\[ -dX(u,v)\equiv\begin{pmatrix}-\frac{v}{2}\cos\frac{u}{2}\sin u+(2-v\sin\frac{u}{2})\cos u & -\sin\frac{u}{2}\sin u\\ --\frac{v}{2}\cos\frac{u}{2}\cos u-(2-v\sin\frac{u}{2})\sin u & -\sin\frac{u}{2}\cos u\\ --\frac{v}{2}\sin\frac{u}{2} & \cos\frac{u}{2} -\end{pmatrix}, -\] - + Análogamente +\begin_inset Formula $dN_{p}(X_{v}(q))=N_{v}(u_{0},v_{0})$ \end_inset -y el determinante de las dos primeras filas es -\begin_inset Formula -\[ --\sin\frac{u}{2}\left(-\frac{v}{2}\cos\frac{u}{2}\begin{vmatrix}\sin u & \sin u\\ -\cos u & \cos u -\end{vmatrix}+\left(2-v\sin\frac{u}{2}\right)\begin{vmatrix}\cos u & \sin u\\ --\sin u & \cos u -\end{vmatrix}\right)=-\sin\frac{u}{2}\left(2-v\sin\frac{u}{2}\right), -\] - +, por lo que queda ver que +\begin_inset Formula $\langle N_{u},X_{v}\rangle(q)=\langle N_{v},X_{u}\rangle(q)$ \end_inset -lo que solo se anula cuando -\begin_inset Formula $u\in\{2k\pi\}_{k\in\mathbb{Z}}$ +. + Sabemos que +\begin_inset Formula $\langle N,X_{u}\rangle=\langle N,X_{v}\rangle=0$ \end_inset -, pero en tal caso -\begin_inset Formula -\[ -dX(u,v)\equiv\begin{pmatrix}2 & 0\\ --\frac{v}{2} & 0\\ -0 & 1 -\end{pmatrix} -\] - +, y derivando, +\begin_inset Formula $\langle N_{v},X_{u}\rangle+\langle N,X_{uv}\rangle=\langle N_{u},X_{v}\rangle+\langle N,X_{vu}\rangle=0$ \end_inset -y el determinante de la submatriz resultante de quitar la segunda fila es - -\begin_inset Formula $2\neq0$ +, pero +\begin_inset Formula $X_{uv}=X_{vu}$ \end_inset . - Esto prueba que la banda de Möbius es una superficie. - \end_layout -\end_deeper -\end_inset +\begin_layout Standard +Ejemplos: +\end_layout +\begin_layout Enumerate +Para un plano, +\begin_inset Formula $A_{p}\equiv0$ +\end_inset +. \end_layout -\begin_layout Enumerate -El plano -\begin_inset Formula $p_{0}+\langle v\rangle^{\bot}\subseteq\mathbb{R}^{3}$ +\begin_deeper +\begin_layout Standard +\begin_inset Formula $N$ \end_inset - admite la orientación -\begin_inset Formula $N(p):=v/|v|$ + es fijo, luego +\begin_inset Formula $-dN_{p}\equiv0$ \end_inset . \end_layout +\end_deeper \begin_layout Enumerate -Dados -\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ +Para +\begin_inset Formula $\mathbb{S}^{2}(r)$ \end_inset - -\begin_inset Formula ${\cal C}^{2}$ + orientada con +\begin_inset Formula $N(p)=\pm\frac{1}{r}p$ \end_inset - y un valor regular -\begin_inset Formula $c$ +, +\begin_inset Formula $A_{p}=\mp\frac{1}{r}1_{T_{p}\mathbb{S}^{2}(r)}$ \end_inset - de -\begin_inset Formula $f$ -\end_inset +. +\end_layout -, la superficie de nivel -\begin_inset Formula $S:=f^{-1}(c)$ +\begin_layout Enumerate +Para el cilindro +\begin_inset Formula $X(\mathbb{R}^{2})$ \end_inset - admite la orientación -\begin_inset Formula -\[ -N(p):=\frac{\nabla f(p)}{|\nabla f(p)|}, -\] + con +\begin_inset Formula $X(u,v):=(r\cos u,r\sin u,v)$ +\end_inset +, si +\begin_inset Formula $p\in C$ \end_inset - donde -\begin_inset Formula $\nabla f(p):=(\frac{\partial f}{\partial x}(p),\frac{\partial f}{\partial y}(p),\frac{\partial f}{\partial z}(p))$ + y +\begin_inset Formula $q\in X^{-1}(p)$ \end_inset - es el -\series bold -gradiente -\series default - de -\begin_inset Formula $f$ +, +\begin_inset Formula $A_{p}=\text{diag}(-\frac{1}{r},0)$ \end_inset - en -\begin_inset Formula $p$ + respecto a la base +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ \end_inset . @@ -690,153 +1391,175 @@ gradiente \begin_deeper \begin_layout Standard -Sean -\begin_inset Formula $p\in S$ -\end_inset - -, -\begin_inset Formula $\alpha:=(x,y,z):I\to S$ -\end_inset - - una curva diferenciable con -\begin_inset Formula $\alpha(0)=p$ +Si +\begin_inset Formula $p=:(x,y,z)$ \end_inset y -\begin_inset Formula $v:=\alpha'(0)\in T_{p}S$ +\begin_inset Formula $q=:(u,v)$ \end_inset -, para -\begin_inset Formula $t\in I$ +, +\begin_inset Formula $X_{u}(q)=(-r\sin u,r\cos u,0)$ \end_inset - es -\begin_inset Formula $f(\alpha(t))=c$ +, +\begin_inset Formula $X_{v}(q)=(0,0,1)$ \end_inset - por ser -\begin_inset Formula $\alpha(t)\in S$ + y, como +\begin_inset Formula $N(x,y,z)=\frac{1}{r}(x,y,0)=(\cos u,\sin u,0)$ \end_inset -, luego derivando, -\begin_inset Formula $\frac{\partial f}{\partial x}(\alpha(t))x'(t)+\frac{\partial f}{\partial y}(\alpha(t))y'(t)+\frac{\partial f}{\partial z}(\alpha(t))z'(t)=0$ +, +\begin_inset Formula $N_{u}(q)=(-\sin u,\cos u,0)=-\frac{1}{r}X_{u}$ \end_inset y -\begin_inset Formula $\nabla f(p)\bot v$ +\begin_inset Formula $N_{v}(q)=0$ \end_inset . - Además, -\begin_inset Formula $\nabla f(p)\neq0$ -\end_inset - - porque -\begin_inset Formula $p\in S=f^{-1}(c)$ -\end_inset +\end_layout - y -\begin_inset Formula $c$ +\end_deeper +\begin_layout Enumerate +Para el +\series bold +paraboloide hiperbólico +\series default + o +\series bold +silla de montar +\series default +, +\begin_inset Formula $S:=\{y^{2}-x^{2}=z\}=\{(u,v,v^{2}-u^{2})\}_{(u,v)\in\mathbb{R}^{2}}$ \end_inset - es un valor regular de -\begin_inset Formula $f$ +, +\begin_inset Formula $A_{p}(0)\equiv\text{diag}(-2,2)$ \end_inset -, y claramente -\begin_inset Formula $\nabla f$ + respecto a la base +\begin_inset Formula $(X_{u}(0),X_{v}(0))$ \end_inset - es diferenciable. +. \end_layout -\end_deeper -\begin_layout Enumerate -\begin_inset Formula $\mathbb{S}^{2}$ +\begin_deeper +\begin_layout Standard +\begin_inset Formula $S$ \end_inset - admite la orientación -\begin_inset Formula $N(p):=p$ + es una superficie porque es el grafo de +\begin_inset Formula $f:\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula $f(u,v):=v^{2}-u^{2}$ \end_inset . -\end_layout + Entonces +\begin_inset Formula +\[ +N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}=\frac{(2u,-2v,1)}{\sqrt{1+4u^{2}+4v^{2}}}, +\] -\begin_deeper -\begin_layout Standard -Sea -\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}+z^{2}$ \end_inset -, 1 es un valor regular de -\begin_inset Formula $f$ +luego +\begin_inset Formula +\begin{align*} +N_{u}(u,v) & =\frac{(2(1+4u^{2}+4v^{2})-8u^{2},8uv,-4u)}{(1+4u^{2}+4v^{2})^{3/2}}=\frac{(2(1+4v^{2}),8uv,-4u)}{(1+4u^{2}+4v^{2})^{3/2}},\\ +N_{v}(u,v) & =\frac{(-8uv,-2(1+4u^{2}+4v^{2})+8v^{2},-4v)}{(1+4u^{2}+4v^{2})^{3/2}}=\frac{(-8uv,-2(1+4u^{2}),-4v)}{(1+4u^{2}+4v^{2})^{3/2}}, +\end{align*} + +\end_inset + +y en particular +\begin_inset Formula $N_{u}(0)=(2,0,0)$ \end_inset y -\begin_inset Formula $\mathbb{S}^{2}$ +\begin_inset Formula $N_{v}(0)=(0,-2,0)$ \end_inset - es la superficie de nivel -\begin_inset Formula $\{p:f(p)=1\}$ +, pero +\begin_inset Formula $X_{u}(0)=(1,0,0)$ \end_inset -, luego admite la orientación -\begin_inset Formula -\[ -N(x,y,z)=\frac{\nabla f(x,y,z)}{|\nabla f(x,y,z)|}=\frac{(2x,2y,2z)}{|(2x,2y,2z)|}=\frac{(x,y,z)}{|(x,y,z)|}=(x,y,z). -\] + y +\begin_inset Formula $X_{v}(0)=(0,1,0)$ +\end_inset +, luego +\begin_inset Formula $N_{u}(0)=2X_{u}(0)$ \end_inset + y +\begin_inset Formula $N_{v}(0)=2X_{v}(0)$ +\end_inset +. \end_layout \end_deeper -\begin_layout Enumerate -Dada -\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ -\end_inset - - diferenciable en el abierto -\begin_inset Formula $U$ +\begin_layout Standard +El operador forma +\begin_inset Formula $A_{p}$ \end_inset -, el grafo -\begin_inset Formula $S:=\{(x,y,f(x,y))\}_{x,y\in U}$ + lleva asociada unívocamente una forma bilineal simétrica +\begin_inset Formula $\sigma_{p}:T_{p}S\times T_{p}S\to\mathbb{R}$ \end_inset - admite la orientación -\begin_inset Formula -\[ -N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}(u,v). -\] - + dada por +\begin_inset Formula $\sigma_{p}(v,w):=\langle A_{p}v,w\rangle$ \end_inset -Dada la parametrización -\begin_inset Formula $(U,X)$ +, así como una forma cuadrática +\begin_inset Formula ${\cal II}_{p}:T_{p}S\to\mathbb{R}$ \end_inset - con -\begin_inset Formula $X(u,v):=(u,v,f(u,v))$ + dada por +\begin_inset Formula ${\cal II}_{p}(v):=\sigma_{p}(v,v)=\langle A_{p}v,v\rangle$ \end_inset -, -\begin_inset Formula $X_{u}=(1,0,f_{u})$ +. + +\begin_inset Formula ${\cal II}_{p}$ \end_inset - y -\begin_inset Formula $X_{v}=(0,1,f_{v})$ + es la +\series bold +segunda forma fundamental +\series default + de +\begin_inset Formula $S$ \end_inset -, y -\begin_inset Formula $X_{u}\wedge X_{v}=(-f_{u},-f_{v},1)$ + en +\begin_inset Formula $p$ \end_inset . \end_layout \begin_layout Standard +Las tres formas dan la misma información usando la +\series bold +identidad de polarización: +\series default + +\begin_inset Formula +\[ +\sigma_{p}(v,w)=\frac{1}{2}\left({\cal II}_{p}(v+w)-{\cal II}_{p}(v)-{\cal II}_{p}(w)\right). +\] + +\end_inset + \end_layout -- cgit v1.2.3