path: root/ggs/n7.lyx

#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_document
\begin_header
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\input{../defs}
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\begin_body

\begin_layout Standard
Una 
\series bold
región
\series default
 de una superficie regular 
\begin_inset Formula $S$
\end_inset

 es un 
\begin_inset Formula $R\subseteq S$
\end_inset

 abierto, conexo y 
\series bold
relativamente compacto
\series default
, es decir, con clausura compacta.
 Si existe una parametrización 
\begin_inset Formula $(U,X)$
\end_inset

 de 
\begin_inset Formula $S$
\end_inset

 con 
\begin_inset Formula $R\subseteq X(U)$
\end_inset

 y 
\begin_inset Formula $f:R\to\mathbb{R}$
\end_inset

 es continua, la 
\series bold
integral
\series default
 de 
\begin_inset Formula $f$
\end_inset

 sobre 
\begin_inset Formula $R$
\end_inset

 es
\begin_inset Formula 
\[
\int_{R}f\,dS=\iint_{X^{-1}(R)}(f\circ X)\left\Vert \frac{\partial X}{\partial u}\wedge\frac{\partial X}{\partial v}\right\Vert =\iint_{X^{-1}(R)}(f\circ X)\sqrt{EG-F^{2}}.
\]

\end_inset

Esta no depende de la parametrización.
 
\series bold
Demostración:
\series default
 Sean 
\begin_inset Formula $(U,X)$
\end_inset

 y 
\begin_inset Formula $(\overline{U},\overline{X})$
\end_inset

 parametrizaciones de 
\begin_inset Formula $S$
\end_inset

 con 
\begin_inset Formula $R\subseteq X(U)\cap\overline{X}(\overline{U})$
\end_inset

, 
\begin_inset Formula $h\coloneqq \overline{X}^{-1}\circ X$
\end_inset

 la reparametrización y 
\begin_inset Formula $h(u,v)=:(\overline{u}(u,v),\overline{v}(u,v))$
\end_inset

, de modo que 
\begin_inset Formula $X=\overline{X}\circ h$
\end_inset

, entonces
\begin_inset Formula 
\begin{align*}
\frac{\partial X}{\partial u} & =\frac{\partial\overline{X}}{\partial\overline{u}}\frac{\partial\overline{u}}{\partial u}+\frac{\partial\overline{X}}{\partial\overline{v}}\frac{\partial\overline{v}}{\partial u}, & \frac{\partial X}{\partial v} & =\frac{\partial\overline{X}}{\partial\overline{u}}\frac{\partial\overline{u}}{\partial v}+\frac{\partial\overline{X}}{\partial\overline{v}}\frac{\partial\overline{v}}{\partial v},
\end{align*}

\end_inset

con las derivadas de 
\begin_inset Formula $\overline{X}$
\end_inset

 evaluadas en 
\begin_inset Formula $h(u,v)$
\end_inset

 y el resto en 
\begin_inset Formula $(u,v)$
\end_inset

, luego
\begin_inset Formula 
\[
\left(\frac{\partial X}{\partial u}\wedge\frac{\partial X}{\partial v}\right)=\frac{\partial\overline{X}}{\partial\overline{u}}\frac{\partial\overline{u}}{\partial u}\wedge\frac{\partial\overline{X}}{\partial\overline{v}}\frac{\partial\overline{v}}{\partial v}+\frac{\partial\overline{X}}{\partial\overline{v}}\frac{\partial\overline{v}}{\partial u}\wedge\frac{\partial\overline{X}}{\partial\overline{u}}\frac{\partial\overline{u}}{\partial v}=\left(\frac{\partial\overline{u}}{\partial u}\frac{\partial\overline{v}}{\partial v}-\frac{\partial\overline{u}}{\partial v}\frac{\partial\overline{v}}{\partial u}\right)\left(\frac{\partial\overline{X}}{\partial\overline{u}}\wedge\frac{\partial\overline{X}}{\partial\overline{v}}\right),
\]

\end_inset

pero 
\begin_inset Formula $\frac{\partial\overline{u}}{\partial u}\frac{\partial\overline{v}}{\partial v}-\frac{\partial\overline{u}}{\partial v}\frac{\partial\overline{v}}{\partial u}=\det(Jh)$
\end_inset

, luego
\begin_inset Formula 
\begin{multline*}
\iint_{X^{-1}(R)}(f\circ X)\left\Vert \frac{\partial X}{\partial u}\wedge\frac{\partial X}{\partial v}\right\Vert =\iint_{X^{-1}(R)}(f\circ X)|\det(Jh)|\left\Vert \frac{\partial\overline{X}}{\partial\overline{u}}\wedge\frac{\partial\overline{X}}{\partial\overline{v}}\right\Vert =\\
=\iint_{h(X^{-1}(R))=\overline{X}^{-1}(R)}(f\circ X)\left\Vert \frac{\partial\overline{X}}{\partial\overline{u}}\wedge\frac{\partial\overline{X}}{\partial\overline{v}}\right\Vert .
\end{multline*}

\end_inset


\end_layout

\begin_layout Standard
El 
\series bold
área
\series default
 de una región 
\begin_inset Formula $R$
\end_inset

 contenida en la imagen de una parametrización de 
\begin_inset Formula $S$
\end_inset

 es
\begin_inset Formula 
\[
A(R):=\int_{R}dS.
\]

\end_inset


\end_layout

\begin_layout Standard
Si 
\begin_inset Formula $R$
\end_inset

 no está contenida en la imagen de una parametrización, es posible extender
 las definiciones de área y de integral de una función con soporte compacto
 sobre 
\begin_inset Formula $R$
\end_inset

 usando particiones diferenciables de la unidad.
 
\end_layout

\begin_layout Standard
Dada una función 
\begin_inset Formula $\phi:S_{1}\to S_{2}$
\end_inset

 entre superficies regulares, definimos 
\begin_inset Formula $\det(d\phi):S_{1}\to\mathbb{R}$
\end_inset

 como 
\begin_inset Formula $\det(d\phi)(p)\coloneqq \det(J\phi_{p})$
\end_inset

.
 El 
\series bold
soporte
\series default
 de una función 
\begin_inset Formula $f:D\to\mathbb{R}$
\end_inset

 es 
\begin_inset Formula $\text{sop}f\coloneqq \overline{\{x\in D\mid f(x)\neq0\}}$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Teorema del cambio de variable:
\series default
 Si 
\begin_inset Formula $\phi:S_{1}\to S_{2}$
\end_inset

 es un difeomorfismo entre superficies regulares conexas y orientadas y
 
\begin_inset Formula $f:S_{2}\to\mathbb{R}$
\end_inset

 es continua con soporte compacto, entonces
\begin_inset Formula 
\[
\int_{S_{2}}f\,dS_{2}=\int_{S_{1}}(f\circ\phi)|\det(d\phi)|dS_{1}=\pm\int_{S_{1}}(f\circ\phi)\det(d\phi)dS_{1}.
\]

\end_inset


\series bold
Demostración
\series default
 cuando una sola parametrización cubre toda la superficie
\series bold
:
\series default
 Sea 
\begin_inset Formula $(U,X)$
\end_inset

 una parametrización de 
\begin_inset Formula $S_{1}$
\end_inset

 y 
\begin_inset Formula $(U,\overline{X}\coloneqq \phi\circ X)$
\end_inset

 una parametrización de 
\begin_inset Formula $S_{2}$
\end_inset

, entonces
\begin_inset Formula 
\begin{align*}
\frac{\partial\overline{X}}{\partial u} & =d\phi_{X(u,v)}\left(\frac{\partial X}{\partial u}\right), & \frac{\partial\overline{X}}{\partial v} & =d\phi_{X(u,v)}\left(\frac{\partial X}{\partial v}\right),
\end{align*}

\end_inset

luego
\begin_inset Formula 
\[
\left\Vert \frac{\partial\overline{X}}{\partial u}\wedge\frac{\partial\overline{X}}{\partial v}\right\Vert =\left\Vert J\phi_{X(u,v)}\frac{\partial X}{\partial u}\wedge J\phi_{X(u,v)}\frac{\partial X}{\partial v}\right\Vert =|J\theta_{X(u,v)}|\left\Vert \frac{\partial X}{\partial u}\wedge\frac{\partial X}{\partial v}\right\Vert ,
\]

\end_inset

de modo que
\begin_inset Formula 
\begin{align*}
\int_{S_{2}}f\,dS_{2} & =\iint_{\overline{X}^{-1}(S_{2})}(f\circ\overline{X})\left\Vert \frac{\partial\overline{X}}{\partial u}\wedge\frac{\partial\overline{X}}{\partial v}\right\Vert \\
 & =\iint_{X^{-1}(\phi^{-1}(S_{2}))}(f\circ\overline{X})|\det(d\phi_{X})|\left\Vert \frac{\partial X}{\partial u}\wedge\frac{\partial X}{\partial v}\right\Vert \\
 & =\iint_{X^{-1}(S_{1})}(f\circ\phi\circ X)|\det(d\phi_{X})|\left\Vert \frac{\partial X}{\partial u}\wedge\frac{\partial X}{\partial v}\right\Vert =\int_{S_{1}}(f\circ\phi)|\det(d\phi)|dS_{2}.
\end{align*}

\end_inset

Para la última igualdad, como 
\begin_inset Formula $\phi$
\end_inset

 es un difeomorfismo, 
\begin_inset Formula $\det(d\phi_{X(u,v)})$
\end_inset

 no se anula y no cambia de signo.
\end_layout

\begin_layout Standard
Como 
\series bold
teorema
\series default
, si 
\begin_inset Formula $S$
\end_inset

 es una superficie regular orientada por 
\begin_inset Formula $N:S\to\mathbb{S}^{2}$
\end_inset

, 
\begin_inset Formula $p\in S$
\end_inset

 cumple 
\begin_inset Formula $K(p)\neq0$
\end_inset

 y 
\begin_inset Formula $R$
\end_inset

 es una región de 
\begin_inset Formula $S$
\end_inset

 con 
\begin_inset Formula $p\in R$
\end_inset

 tal que 
\begin_inset Formula $N:R\to N(R)$
\end_inset

 es un difeomorfismo, entonces el área de 
\begin_inset Formula $N(R)\subseteq\mathbb{S}^{2}$
\end_inset

 es
\begin_inset Formula 
\[
A(N(R))=\int_{R}|K|dS,
\]

\end_inset

y
\begin_inset Formula 
\[
|K(p)|=\lim_{\varepsilon\to0}\frac{A(N(B(p,\varepsilon)))}{A(B(p,\varepsilon))}.
\]

\end_inset


\series bold
Demostración:
\series default
 Por el teorema del cambio de variable para 
\begin_inset Formula $f(p)\equiv1$
\end_inset

, como 
\begin_inset Formula $\det(dN_{p})=-\det(dA_{p})=-K(p)$
\end_inset

,
\begin_inset Formula 
\[
A(N(R))=\int_{N(R)}d\mathbb{S}^{2}=\int_{R}|\det(dN_{p})|dS=\int_{R}|K|dS.
\]

\end_inset

Ahora bien, por continuidad, 
\begin_inset Formula $K\neq0$
\end_inset

 en un entorno 
\begin_inset Formula $V$
\end_inset

 de 
\begin_inset Formula $p$
\end_inset

, luego 
\begin_inset Formula $\det(dN_{q})\neq0$
\end_inset

 para 
\begin_inset Formula $q\in V$
\end_inset

, 
\begin_inset Formula $N|_{V}$
\end_inset

 es un difeomorfismo y existe un 
\begin_inset Formula $\varepsilon_{0}$
\end_inset

 tal que, para 
\begin_inset Formula $\varepsilon\in(0,\varepsilon_{0}]$
\end_inset

, 
\begin_inset Formula $B(p,\varepsilon)\subseteq V$
\end_inset

 y por tanto
\begin_inset Formula 
\[
A(N(B(p,\varepsilon)))=\int_{B(p,\varepsilon)}|K|dS=|K(p_{\varepsilon})|\int_{B(p,\varepsilon)}dS=|K(p_{\varepsilon})|A(B(p,\varepsilon)),
\]

\end_inset

donde 
\begin_inset Formula $p_{\varepsilon}\in B(p,\varepsilon)$
\end_inset

 se obtiene del teorema del punto medio.
 Despejando 
\begin_inset Formula $|K(p_{\varepsilon})|$
\end_inset

 y tomando límites cuando 
\begin_inset Formula $\varepsilon\to0$
\end_inset

 se obtiene el resultado.
\end_layout

\end_body
\end_document