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Diffstat (limited to 'gcs')
| -rw-r--r-- | gcs/n3.lyx | 759 |
1 files changed, 759 insertions, 0 deletions
@@ -78,6 +78,765 @@ \begin_body \begin_layout Standard +Dada una superficie regular +\begin_inset Formula $S$ +\end_inset + +, un +\series bold +campo de vectores +\series default + sobre +\begin_inset Formula $S$ +\end_inset + + es una función +\begin_inset Formula $\xi:S\to\mathbb{R}^{3}$ +\end_inset + +, y es +\series bold +tangente +\series default + si +\begin_inset Formula $\xi(p)\in T_{p}S$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + +, +\series bold +normal +\series default + si +\begin_inset Formula $\xi(p)\in(T_{p}S)^{\bot}$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + + y +\series bold +unitario +\series default + si +\begin_inset Formula $|\xi(p)|=1$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + +. + Llamamos +\begin_inset Formula $\mathfrak{X}(S)$ +\end_inset + + al conjunto de campos de vectores tangentes sobre +\begin_inset Formula $S$ +\end_inset + + y +\begin_inset Formula $\mathfrak{X}(S)^{\bot}$ +\end_inset + + al conjunto de campos de vectores normales sobre +\begin_inset Formula $S$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Una +\series bold +orientación +\series default + de una superficie regular +\begin_inset Formula $S$ +\end_inset + + es un campo de vectores diferenciable, normal y unitario sobre +\begin_inset Formula $S$ +\end_inset + +. + +\begin_inset Formula $S$ +\end_inset + + es +\series bold +orientable +\series default + si admite una orientación, si y sólo si existe un campo +\begin_inset Formula $\xi$ +\end_inset + + normal y diferenciable sobre +\begin_inset Formula $S$ +\end_inset + + que no se anula en ningún punto, pues las orientaciones son de esta forma + y, dado +\begin_inset Formula $\xi$ +\end_inset + +, basta tomar la orientación +\begin_inset Formula $N(p):=\xi(p)/|\xi(p)|$ +\end_inset + +. + Una orientación +\begin_inset Formula $N$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + da a cada +\begin_inset Formula $p\in S$ +\end_inset + + un sentido de giro para +\begin_inset Formula $T_{p}S$ +\end_inset + + dado por el producto vectorial en +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + +. + +\begin_inset Formula $S$ +\end_inset + + está orientada cuando se ha escogido una orientación concreta. +\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 +\begin_inset Formula $(U_{p},X_{p})$ +\end_inset + + de +\begin_inset Formula $S$ +\end_inset + + con +\begin_inset Formula $N(X(q))=\frac{(X_{p})_{u}\wedge(X_{p})_{v}}{|(X_{p})_{u}\wedge(X_{p})_{v}|}(q)$ +\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 + +. +\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 $X:\mathbb{R}\times(-1,1)\to\mathbb{R}^{3}$ +\end_inset + + dada por +\begin_inset Formula +\[ +X(u,v):=\left((2-v\sin\tfrac{u}{2})\sin u,(2-v\sin\tfrac{u}{2})\cos u,v\cos\tfrac{u}{2}\right). +\] + +\end_inset + +Esta es una superficie regular no orientable. +\end_layout + +\begin_deeper +\begin_layout Plain Layout +Claramente +\begin_inset Formula $X$ +\end_inset + + es diferenciable, y es inyectiva en +\begin_inset Formula $U_{1}:=(0,2\pi)\times(-1,1)$ +\end_inset + + y en +\begin_inset Formula $U_{2}:=(-\pi,\pi)\times(-1,1)$ +\end_inset + +. + Su diferencial es +\begin_inset Formula +\[ +dX(u,v)\equiv\begin{pmatrix}-\frac{v}{2}\cos\frac{u}{2}\sin u+(2-v\sin\frac{u}{2})\cos u & -\sin\frac{u}{2}\sin u\\ +-\frac{v}{2}\cos\frac{u}{2}\cos u-(2-v\sin\frac{u}{2})\sin u & -\sin\frac{u}{2}\cos u\\ +-\frac{v}{2}\sin\frac{u}{2} & \cos\frac{u}{2} +\end{pmatrix}, +\] + +\end_inset + +y el determinante de las dos primeras filas es +\begin_inset Formula +\[ +-\sin\frac{u}{2}\left(-\frac{v}{2}\cos\frac{u}{2}\begin{vmatrix}\sin u & \sin u\\ +\cos u & \cos u +\end{vmatrix}+\left(2-v\sin\frac{u}{2}\right)\begin{vmatrix}\cos u & \sin u\\ +-\sin u & \cos u +\end{vmatrix}\right)=-\sin\frac{u}{2}\left(2-v\sin\frac{u}{2}\right), +\] + +\end_inset + +lo que solo se anula cuando +\begin_inset Formula $u\in\{2k\pi\}_{k\in\mathbb{Z}}$ +\end_inset + +, pero en tal caso +\begin_inset Formula +\[ +dX(u,v)\equiv\begin{pmatrix}2 & 0\\ +-\frac{v}{2} & 0\\ +0 & 1 +\end{pmatrix} +\] + +\end_inset + +y el determinante de la submatriz resultante de quitar la segunda fila es + +\begin_inset Formula $2\neq0$ +\end_inset + +. + Esto prueba que la banda de Möbius es una superficie. + +\end_layout + +\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 + + admite la orientación +\begin_inset Formula $N(p):=v/|v|$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Dados +\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ +\end_inset + + +\begin_inset Formula ${\cal C}^{2}$ +\end_inset + + y un valor regular +\begin_inset Formula $c$ +\end_inset + + de +\begin_inset Formula $f$ +\end_inset + +, la superficie de nivel +\begin_inset Formula $S:=f^{-1}(c)$ +\end_inset + + admite la orientación +\begin_inset Formula +\[ +N(p):=\frac{\nabla f(p)}{|\nabla f(p)|}, +\] + +\end_inset + + donde +\begin_inset Formula $\nabla f(p):=(\frac{\partial f}{\partial x}(p),\frac{\partial f}{\partial y}(p),\frac{\partial f}{\partial z}(p))$ +\end_inset + + es el +\series bold +gradiente +\series default + de +\begin_inset Formula $f$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +. +\end_layout + +\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 $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $v:=\alpha'(0)\in T_{p}S$ +\end_inset + +, para +\begin_inset Formula $t\in I$ +\end_inset + + es +\begin_inset Formula $f(\alpha(t))=c$ +\end_inset + + por ser +\begin_inset Formula $\alpha(t)\in S$ +\end_inset + +, luego derivando, +\begin_inset Formula $\frac{\partial f}{\partial x}(\alpha(t))x'(t)+\frac{\partial f}{\partial y}(\alpha(t))y'(t)+\frac{\partial f}{\partial z}(\alpha(t))z'(t)=0$ +\end_inset + + y +\begin_inset Formula $\nabla f(p)\bot v$ +\end_inset + +. + Además, +\begin_inset Formula $\nabla f(p)\neq0$ +\end_inset + + porque +\begin_inset Formula $p\in S=f^{-1}(c)$ +\end_inset + + y +\begin_inset Formula $c$ +\end_inset + + 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 $\mathbb{S}^{2}$ +\end_inset + + admite la orientación +\begin_inset Formula $N(p):=p$ +\end_inset + +. +\end_layout + +\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 $f$ +\end_inset + + y +\begin_inset Formula $\mathbb{S}^{2}$ +\end_inset + + es la superficie de nivel +\begin_inset Formula $\{p:f(p)=1\}$ +\end_inset + +, luego admite la orientación +\begin_inset Formula +\[ +N(x,y,z)=\frac{\nabla f(x,y,z)}{|\nabla f(x,y,z)|}=\frac{(2x,2y,2z)}{|(2x,2y,2z)|}=\frac{(x,y,z)}{|(x,y,z)|}=(x,y,z). +\] + +\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 \end_layout |