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| -rw-r--r-- | gcs/n3.lyx | 1001 |
1 files changed, 862 insertions, 139 deletions
@@ -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 $S$ @@ -215,7 +222,340 @@ 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 +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 $\mathbb{S}^{2}(r)$ +\end_inset + + admite la orientación +\begin_inset Formula $N(p)=\frac{1}{r}p$ +\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 + +, +\begin_inset Formula $r^{2}$ +\end_inset + + 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 $\{p:f(p)=r^{2}\}$ +\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 @@ -536,227 +876,514 @@ En adelante, cuando consideremos una parametrización \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 -Ejemplos: -\begin_inset Note Comment -status open +Sea +\begin_inset Formula $S$ +\end_inset -\begin_layout Enumerate -La banda de Möbius se puede expresar como la imagen de -\begin_inset Formula $X:\mathbb{R}\times(-1,1)\to\mathbb{R}^{3}$ + una superficie orientada con aplicación de Gauss +\begin_inset Formula $N:S\to\mathbb{S}^{2}$ \end_inset - dada por -\begin_inset Formula -\[ -X(u,v):=\left((2-v\sin\tfrac{u}{2})\sin u,(2-v\sin\tfrac{u}{2})\cos u,v\cos\tfrac{u}{2}\right). -\] +, 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 -Esta es una superficie regular no orientable. +. + Ejemplos: +\end_layout + +\begin_layout Enumerate +La imagen esférica de un plano es unipuntual. \end_layout \begin_deeper -\begin_layout Plain Layout -Claramente -\begin_inset Formula $X$ +\begin_layout Standard +Dado el plano +\begin_inset Formula $\Pi:=p_{0}+\langle v\rangle\subseteq\mathbb{R}^{3}$ \end_inset - es diferenciable, y es inyectiva en -\begin_inset Formula $U_{1}:=(0,2\pi)\times(-1,1)$ +, donde podemos suponer +\begin_inset Formula $v$ \end_inset - y en -\begin_inset Formula $U_{2}:=(-\pi,\pi)\times(-1,1)$ + unitario, la imagen de +\begin_inset Formula $N(p):=v$ +\end_inset + + es +\begin_inset Formula $\{v\}$ \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_layout +\end_deeper +\begin_layout Enumerate +La imagen esférica de +\begin_inset Formula $\mathbb{S}^{2}$ \end_inset -y el determinante de las dos primeras filas es -\begin_inset Formula -\[ --\sin\frac{u}{2}\left(-\frac{v}{2}\cos\frac{u}{2}\begin{vmatrix}\sin u & \sin u\\ -\cos u & \cos u -\end{vmatrix}+\left(2-v\sin\frac{u}{2}\right)\begin{vmatrix}\cos u & \sin u\\ --\sin u & \cos u -\end{vmatrix}\right)=-\sin\frac{u}{2}\left(2-v\sin\frac{u}{2}\right), -\] + 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 -lo que solo se anula cuando -\begin_inset Formula $u\in\{2k\pi\}_{k\in\mathbb{Z}}$ +. +\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 -, pero en tal caso -\begin_inset Formula -\[ -dX(u,v)\equiv\begin{pmatrix}2 & 0\\ --\frac{v}{2} & 0\\ -0 & 1 -\end{pmatrix} -\] + con +\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ +\end_inset + + 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 el determinante de la submatriz resultante de quitar la segunda fila es - -\begin_inset Formula $2\neq0$ +, y como la coordenada +\begin_inset Formula $z$ \end_inset -. - Esto prueba que la banda de Möbius es una superficie. - + de +\begin_inset Formula $N$ +\end_inset + + es siempre positiva, +\begin_inset Formula $\text{Im}N$ +\end_inset + + 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 + + para algún +\begin_inset Formula $r>0$ \end_inset +, 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 plano -\begin_inset Formula $p_{0}+\langle v\rangle^{\bot}\subseteq\mathbb{R}^{3}$ +El +\series bold +catenoide +\series default +, +\begin_inset Formula $C:=\{x^{2}+y^{2}=\cosh^{2}z\}$ \end_inset - admite la orientación -\begin_inset Formula $N(p):=v/|v|$ +, 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 $\mathsf{S}:=(0,0,-1)$ \end_inset + es el +\series bold +polo sur +\series default . \end_layout -\begin_layout Enumerate -Dados -\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$ +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}-\cosh^{2}z$ \end_inset - -\begin_inset Formula ${\cal C}^{2}$ +, como +\begin_inset Formula $f_{x}=2x$ \end_inset - y un valor regular -\begin_inset Formula $c$ +, +\begin_inset Formula $f_{y}=2y$ \end_inset - de + y +\begin_inset Formula $f_{z}=-2\cosh z\sinh z$ +\end_inset + +, el único punto crítico de \begin_inset Formula $f$ \end_inset -, la superficie de nivel -\begin_inset Formula $S:=f^{-1}(c)$ + es el origen, con +\begin_inset Formula $f(0)=-1$ \end_inset - admite la orientación +, 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 -\[ -N(p):=\frac{\nabla f(p)}{|\nabla f(p)|}, -\] +\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 - donde -\begin_inset Formula $\nabla f(p):=(\frac{\partial f}{\partial x}(p),\frac{\partial f}{\partial y}(p),\frac{\partial f}{\partial z}(p))$ +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 - es el -\series bold -gradiente -\series default - de -\begin_inset Formula $f$ +, 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 - en -\begin_inset Formula $p$ +, +\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 $(x,y,z)\in C$ +\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 . \end_layout -\begin_deeper +\end_deeper \begin_layout Standard -Sean +Para +\begin_inset Formula $p\in\mathbb{S}^{2}$ +\end_inset + + 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 $\alpha:=(x,y,z):I\to S$ +\begin_inset Formula $dN_{p}:T_{p}S\to T_{N(p)}\mathbb{S}^{2}$ \end_inset - una curva diferenciable con -\begin_inset Formula $\alpha(0)=p$ +, pero como la normal en +\begin_inset Formula $\mathbb{S}^{2}$ \end_inset - y -\begin_inset Formula $v:=\alpha'(0)\in T_{p}S$ + es +\begin_inset Formula $1_{\mathbb{S}^{2}}$ \end_inset -, para -\begin_inset Formula $t\in I$ +, +\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 -\begin_inset Formula $f(\alpha(t))=c$ +\series bold +autoadjunto +\series default +, es decir, +\begin_inset Formula $\langle A_{p}v,w\rangle=\langle v,A_{p}w\rangle$ \end_inset - por ser -\begin_inset Formula $\alpha(t)\in S$ +. + +\series bold +Demostración: +\series default + Por linealidad, basta demostrarlo para una base de +\begin_inset Formula $T_{p}S$ \end_inset -, luego derivando, -\begin_inset Formula $\frac{\partial f}{\partial x}(\alpha(t))x'(t)+\frac{\partial f}{\partial y}(\alpha(t))y'(t)+\frac{\partial f}{\partial z}(\alpha(t))z'(t)=0$ +. + Sean +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ \end_inset y -\begin_inset Formula $\nabla f(p)\bot v$ +\begin_inset Formula $q:=(u_{0},v_{0}):=X^{-1}(p)$ +\end_inset + +, tomamos la base +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ +\end_inset + + 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 . - Además, -\begin_inset Formula $\nabla f(p)\neq0$ + Sea entonces +\begin_inset Formula $\alpha(u):=X(u_{0}+u,v_{0})$ \end_inset - porque -\begin_inset Formula $p\in S=f^{-1}(c)$ +, +\begin_inset Formula $\alpha(0)=p$ \end_inset y -\begin_inset Formula $c$ +\begin_inset Formula $\alpha'(0)=X_{u}(q)$ \end_inset - es un valor regular de -\begin_inset Formula $f$ +, 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 -, y claramente -\begin_inset Formula $\nabla f$ +. + Análogamente +\begin_inset Formula $dN_{p}(X_{v}(q))=N_{v}(u_{0},v_{0})$ \end_inset - es diferenciable. +, 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 + +. + Sabemos que +\begin_inset Formula $\langle N,X_{u}\rangle=\langle N,X_{v}\rangle=0$ +\end_inset + +, 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 + +, pero +\begin_inset Formula $X_{uv}=X_{vu}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Ejemplos: +\end_layout + +\begin_layout Enumerate +Para un plano, +\begin_inset Formula $A_{p}\equiv0$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $N$ +\end_inset + + es fijo, luego +\begin_inset Formula $-dN_{p}\equiv0$ +\end_inset + +. \end_layout \end_deeper \begin_layout Enumerate -\begin_inset Formula $\mathbb{S}^{2}$ +Para +\begin_inset Formula $\mathbb{S}^{2}(r)$ \end_inset - admite la orientación -\begin_inset Formula $N(p):=p$ + orientada con +\begin_inset Formula $N(p)=\pm\frac{1}{r}p$ +\end_inset + +, +\begin_inset Formula $A_{p}=\mp\frac{1}{r}1_{T_{p}\mathbb{S}^{2}(r)}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Para el cilindro +\begin_inset Formula $X(\mathbb{R}^{2})$ +\end_inset + + 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 + + y +\begin_inset Formula $q\in X^{-1}(p)$ +\end_inset + +, +\begin_inset Formula $A_{p}=\text{diag}(-\frac{1}{r},0)$ +\end_inset + + respecto a la base +\begin_inset Formula $(X_{u}(q),X_{v}(q))$ \end_inset . @@ -764,79 +1391,175 @@ Sean \begin_deeper \begin_layout Standard -Sea -\begin_inset Formula $f(x,y,z):=x^{2}+y^{2}+z^{2}$ +Si +\begin_inset Formula $p=:(x,y,z)$ \end_inset -, 1 es un valor regular de -\begin_inset Formula $f$ + y +\begin_inset Formula $q=:(u,v)$ \end_inset - y -\begin_inset Formula $\mathbb{S}^{2}$ +, +\begin_inset Formula $X_{u}(q)=(-r\sin u,r\cos u,0)$ \end_inset - es la superficie de nivel -\begin_inset Formula $\{p:f(p)=1\}$ +, +\begin_inset Formula $X_{v}(q)=(0,0,1)$ \end_inset -, luego admite la orientación -\begin_inset Formula -\[ -N(x,y,z)=\frac{\nabla f(x,y,z)}{|\nabla f(x,y,z)|}=\frac{(2x,2y,2z)}{|(2x,2y,2z)|}=\frac{(x,y,z)}{|(x,y,z)|}=(x,y,z). -\] + y, como +\begin_inset Formula $N(x,y,z)=\frac{1}{r}(x,y,0)=(\cos u,\sin u,0)$ +\end_inset +, +\begin_inset Formula $N_{u}(q)=(-\sin u,\cos u,0)=-\frac{1}{r}X_{u}$ \end_inset + y +\begin_inset Formula $N_{v}(q)=0$ +\end_inset +. \end_layout \end_deeper \begin_layout Enumerate -Dada -\begin_inset Formula $f:U\subseteq\mathbb{R}^{2}\to\mathbb{R}$ +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 - diferenciable en el abierto -\begin_inset Formula $U$ +, +\begin_inset Formula $A_{p}(0)\equiv\text{diag}(-2,2)$ \end_inset -, el grafo -\begin_inset Formula $S:=\{(x,y,f(x,y))\}_{x,y\in U}$ + respecto a la base +\begin_inset Formula $(X_{u}(0),X_{v}(0))$ \end_inset - admite la orientación +. +\end_layout + +\begin_deeper +\begin_layout Standard +\begin_inset Formula $S$ +\end_inset + + 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 + +. + Entonces \begin_inset Formula \[ -N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}(u,v). +N(u,v)=\frac{(-f_{u},-f_{v},1)}{\sqrt{1+f_{u}^{2}+f_{v}^{2}}}=\frac{(2u,-2v,1)}{\sqrt{1+4u^{2}+4v^{2}}}, \] \end_inset -Dada la parametrización -\begin_inset Formula $(U,X)$ +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 - con -\begin_inset Formula $X(u,v):=(u,v,f(u,v))$ +y en particular +\begin_inset Formula $N_{u}(0)=(2,0,0)$ \end_inset -, -\begin_inset Formula $X_{u}=(1,0,f_{u})$ + y +\begin_inset Formula $N_{v}(0)=(0,-2,0)$ +\end_inset + +, pero +\begin_inset Formula $X_{u}(0)=(1,0,0)$ \end_inset y -\begin_inset Formula $X_{v}=(0,1,f_{v})$ +\begin_inset Formula $X_{v}(0)=(0,1,0)$ \end_inset -, y -\begin_inset Formula $X_{u}\wedge X_{v}=(-f_{u},-f_{v},1)$ +, 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 Standard +El operador forma +\begin_inset Formula $A_{p}$ +\end_inset + + lleva asociada unívocamente una forma bilineal simétrica +\begin_inset Formula $\sigma_{p}:T_{p}S\times T_{p}S\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula $\sigma_{p}(v,w):=\langle A_{p}v,w\rangle$ +\end_inset + +, así como una forma cuadrática +\begin_inset Formula ${\cal II}_{p}:T_{p}S\to\mathbb{R}$ +\end_inset + + dada por +\begin_inset Formula ${\cal II}_{p}(v):=\sigma_{p}(v,v)=\langle A_{p}v,v\rangle$ +\end_inset + +. + +\begin_inset Formula ${\cal II}_{p}$ +\end_inset + + es la +\series bold +segunda forma fundamental +\series default + de +\begin_inset Formula $S$ +\end_inset + + 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 |