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| -rw-r--r-- | ggs/n.lyx | 14 | ||||
| -rw-r--r-- | ggs/n7.lyx | 503 |
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@@ -231,5 +231,19 @@ filename "n6.lyx" \end_layout +\begin_layout Chapter +Integración en superficies +\end_layout + +\begin_layout Standard +\begin_inset CommandInset include +LatexCommand input +filename "n7.lyx" + +\end_inset + + +\end_layout + \end_body \end_document diff --git a/ggs/n7.lyx b/ggs/n7.lyx new file mode 100644 index 0000000..d73c097 --- /dev/null +++ b/ggs/n7.lyx @@ -0,0 +1,503 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass book +\begin_preamble +\input{../defs} +\end_preamble +\use_default_options true +\maintain_unincluded_children false +\language spanish +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize default +\use_geometry false +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style french +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\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 $h:=\overline{X}^{-1}\circ X$ +\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 $\det(d\phi)(p):=\det(J\phi_{p})$ +\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 $\text{sop}f:=\{x\in D:f(x)\neq0\}$ +\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 $(U,\overline{X}:=\phi\circ X)$ +\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 |