diff options
Diffstat (limited to 'gcs')
| -rw-r--r-- | gcs/n2.lyx | 1821 |
1 files changed, 1582 insertions, 239 deletions
@@ -241,11 +241,90 @@ grafo \end_layout \begin_layout Standard -\begin_inset Note Note +\begin_inset ERT status open \begin_layout Plain Layout -TODO Recordatorio del teorema de la función implícita. + + +\backslash +sremember{FVV3} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +[...][ +\series bold +Teorema de la función implícita: +\series default + Sean +\begin_inset Formula $F:D\subseteq(\mathbb{R}^{n}\times\mathbb{R}^{m})\to\mathbb{R}^{m}$ +\end_inset + + +\begin_inset Formula ${\cal C}^{k}$ +\end_inset + + con +\begin_inset Formula $k\geq1$ +\end_inset + +, +\begin_inset Formula $(x_{0},y_{0})\in D$ +\end_inset + + con +\begin_inset Formula $F(x_{0},y_{0})=0$ +\end_inset + + y +\begin_inset Formula $d(y\mapsto F(x_{0},y))(y_{0}):\mathbb{R}^{m}\to\mathbb{R}^{m}$ +\end_inset + + no singular, existen +\begin_inset Formula ${\cal U}\in{\cal E}(x_{0})$ +\end_inset + + y +\begin_inset Formula ${\cal V}\in{\cal E}(y_{0})$ +\end_inset + + con +\begin_inset Formula ${\cal U}\times{\cal V}\subseteq D$ +\end_inset + + tales que existe una única +\begin_inset Formula $f:{\cal U}\to{\cal V}$ +\end_inset + + con +\begin_inset Formula $F(x,f(x))=0$ +\end_inset + + para todo +\begin_inset Formula $x\in{\cal U}$ +\end_inset + + y esta es +\begin_inset Formula ${\cal C}^{k}$ +\end_inset + +.] +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember \end_layout \end_inset @@ -308,6 +387,14 @@ valor regular \begin_inset Formula $f$ \end_inset + y +\begin_inset Formula $f$ +\end_inset + + es +\begin_inset Formula ${\cal C}^{1}$ +\end_inset + , la \series bold superficie de nivel @@ -395,13 +482,6 @@ Demostración: \end_layout \begin_layout Standard -\begin_inset Newpage newpage -\end_inset - - -\end_layout - -\begin_layout Standard Ejemplos: \end_layout @@ -508,12 +588,7 @@ hiperboloide de dos hojas \begin_inset Formula $H':=\{x^{2}+y^{2}-z^{2}=-1\}$ \end_inset - son superficies regulares -\begin_inset Note Comment -status open - -\begin_layout Plain Layout -. + son superficies regulares. Estas son superficies de revolución resultantes de rotar la hipérbola \begin_inset Formula $\{xy=1\}$ \end_inset @@ -534,12 +609,7 @@ status open \begin_inset Formula $H'$ \end_inset -. -\end_layout - -\end_inset - - +, aplicando antes una transformación lineal. \end_layout \begin_deeper @@ -569,79 +639,282 @@ Sea \end_inset son superficies regulares. -\begin_inset Note Comment -status open +\end_layout -\begin_layout Plain Layout -La recta -\begin_inset Formula $\ell_{1}:=\{y=-x\}$ +\begin_layout Standard +Para ver que los ejes de simetría de la hipérbola son los mencionados, sean + +\begin_inset Formula $v:=(a,b)$ \end_inset - en -\begin_inset Formula $\mathbb{R}^{2}$ + unitario tal que +\begin_inset Formula $\text{span}\{v\}$ +\end_inset + + es un eje de simetría y +\begin_inset Formula $p:=(x,\frac{1}{x})\in\{xy=1\}$ +\end_inset + +, el simétrico de +\begin_inset Formula $p$ \end_inset - tiene un vector ortogonal unitario -\begin_inset Formula $v_{1}:=\frac{1}{\sqrt{2}}(1,1)$ + es +\begin_inset Formula $p-2\langle p,v\rangle v=((1-2a^{2})x-\frac{2ab}{x},\frac{1-2b^{2}}{x}-2abx)$ +\end_inset + +, luego +\begin_inset Formula $((1-2a^{2})x-\frac{2ab}{x})(\frac{1-2b^{2}}{x}-2abx)=1$ \end_inset -, y dado un punto -\begin_inset Formula $p:=(x,\frac{1}{x})$ + y, como esto se cumple para todo +\begin_inset Formula $x\in\mathbb{R}^{*}$ \end_inset - de la hipérbola, su simétrico por -\begin_inset Formula $\ell_{1}$ +, multiplicando todo por +\begin_inset Formula $x^{2}$ +\end_inset + + y haciendo +\begin_inset Formula $x$ +\end_inset + + tender a +\begin_inset Formula $+\infty$ \end_inset , -\begin_inset Formula $p_{1}:=p-2\langle p,v_{1}\rangle v_{1}=(x,\frac{1}{x})-(x+\frac{1}{x})(1,1)=(-\frac{1}{x},-x)$ +\begin_inset Formula $(-2ab)(1-2b^{2})=0$ +\end_inset + +, por lo que bien +\begin_inset Formula $-2ab=0$ \end_inset -, también está en la hipérbola, y -\begin_inset Formula $\ell_{1}$ + y +\begin_inset Formula $v\in\{(\pm1,0),(0,\pm1)\}$ \end_inset - es efectivamente un eje de simetría. - Análogamente, -\begin_inset Formula $v_{2}:=\frac{1}{\sqrt{2}}(1,-1)$ +, para lo que cualquier punto de la hipérbola sirve de contraejemplo, bien + +\begin_inset Formula $1-2b^{2}=0$ \end_inset - es unitario y ortogonal a -\begin_inset Formula $\ell_{2}:=\{y=x\}$ + y entonces +\begin_inset Formula $v\in\{\frac{1}{\sqrt{2}}(\pm1,\pm1)\}$ \end_inset -, y el simétrico de -\begin_inset Formula $p$ +, lo que nos da las rectas +\begin_inset Formula $\{y=-x\}$ \end_inset - por -\begin_inset Formula $\ell_{2}$ + e +\begin_inset Formula $\{y=x\}$ \end_inset - es -\begin_inset Formula $p_{2}:=p-2\langle p,v_{2}\rangle v_{2}=(x,\frac{1}{x})-(x-\frac{1}{x})(1,-1)=(\frac{1}{x},x)$ +. + Si +\begin_inset Formula $v=\frac{1}{\sqrt{2}}(1,1)$ \end_inset -, que también está en la hipérbola. -\end_layout +, +\begin_inset Formula $p-2\langle p,v\rangle v=(-\frac{1}{x},-x)$ +\end_inset -\begin_layout Plain Layout -Finalmente, -\begin_inset Note Note -status open +, y efectivamente +\begin_inset Formula $(-\frac{1}{x})(-x)=1$ +\end_inset -\begin_layout Plain Layout -TODO Ver que efectivamente salen las superficies buscadas. +, y si +\begin_inset Formula $v=\frac{1}{\sqrt{2}}(1,-1)$ +\end_inset + +, +\begin_inset Formula $p-2\langle p,v\rangle v=(\frac{1}{x},x)$ +\end_inset + +, y efectivamente +\begin_inset Formula $\frac{1}{x}x=1$ +\end_inset + +. \end_layout +\begin_layout Standard +Queda ver que las figuras de revolución son efectivamente las mencionadas. + Para +\begin_inset Formula $H$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +\frac{1}{2}\begin{pmatrix}1 & 1\\ +-1 & 1 +\end{pmatrix}\begin{pmatrix}u\\ +\frac{1}{u} +\end{pmatrix} & =\frac{1}{2}\begin{pmatrix}u+\frac{1}{u}\\ +-u+\frac{1}{u} +\end{pmatrix}, & \frac{1}{2}\begin{pmatrix}1 & 1\\ +-1 & 1 +\end{pmatrix}\begin{pmatrix}-1\\ +1 +\end{pmatrix} & =\begin{pmatrix}0\\ +1 +\end{pmatrix}, +\end{align*} + +\end_inset + +y los puntos son de la forma +\begin_inset Formula $(x,y,z):=((u+\frac{1}{u})\cos v,(u+\frac{1}{u})\sin v,-u+\frac{1}{u})/2$ +\end_inset + +, de modo que +\begin_inset Formula $x^{2}+y^{2}-z^{2}=\frac{1}{4}((u+\frac{1}{u})^{2}-(u-\frac{1}{u})^{2})=\frac{1}{4}\cdot4=1$ +\end_inset + +. + Recíprocamente, dado +\begin_inset Formula $(x,y,z)$ +\end_inset + + con +\begin_inset Formula $x^{2}+y^{2}-z^{2}=1$ +\end_inset + +, como +\begin_inset Formula $u\mapsto(u+\frac{1}{u})/2$ +\end_inset + + tiene como imagen +\begin_inset Formula $\mathbb{R}\setminus(-1,1)$ +\end_inset + + y +\begin_inset Formula $x^{2}+y^{2}=z^{2}+1\geq1$ +\end_inset + +, existe +\begin_inset Formula $u$ +\end_inset + + con +\begin_inset Formula $(u+\frac{1}{u})/2=\sqrt{x^{2}+y^{2}}$ +\end_inset + +, y existe +\begin_inset Formula $v$ +\end_inset + + tal que +\begin_inset Formula $(u+\frac{1}{u})/2\cdot(\cos v,\sin v)=(x,y)$ +\end_inset + +. + Finalmente, +\begin_inset Formula $z^{2}=x^{2}+y^{2}-1=\frac{1}{4}(u+\frac{1}{u})^{2}-1=\frac{1}{4}(u-\frac{1}{u})^{2}$ \end_inset +, luego +\begin_inset Formula $z\in\{\pm\frac{1}{2}(-u+\frac{1}{u})\}$ +\end_inset +, y si los signos son opuestos basta cambiar +\begin_inset Formula $u$ +\end_inset + + por +\begin_inset Formula $\frac{1}{u}$ +\end_inset + +. \end_layout +\begin_layout Standard +Para +\begin_inset Formula $H'$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +\frac{1}{2}\begin{pmatrix}1 & -1\\ +1 & 1 +\end{pmatrix}\begin{pmatrix}u\\ +\frac{1}{u} +\end{pmatrix} & =\frac{1}{2}\begin{pmatrix}u-\frac{1}{u}\\ +u+\frac{1}{u} +\end{pmatrix}, & \frac{1}{2}\begin{pmatrix}1 & -1\\ +1 & 1 +\end{pmatrix}\begin{pmatrix}1\\ +1 +\end{pmatrix} & =\begin{pmatrix}0\\ +1 +\end{pmatrix}, +\end{align*} + +\end_inset + +y los puntos son de la forma +\begin_inset Formula $(x,y,z):=((u-\frac{1}{u})\cos v,(u-\frac{1}{u})\sin v,u+\frac{1}{u})/2$ \end_inset +, de modo que +\begin_inset Formula $x^{2}+y^{2}-z^{2}=\frac{1}{4}((u-\frac{1}{u})^{2}-(u+\frac{1}{u})^{2})=-1$ +\end_inset + +. + Recíprocamente, dado +\begin_inset Formula $(x,y,z)$ +\end_inset + + con +\begin_inset Formula $x^{2}+y^{2}-z^{2}=-1$ +\end_inset + +, como +\begin_inset Formula $u\mapsto(u-\frac{1}{u})/2$ +\end_inset + + tiene como imagen todo +\begin_inset Formula $\mathbb{R}$ +\end_inset + +, existe +\begin_inset Formula $u$ +\end_inset + + con +\begin_inset Formula $(u-\frac{1}{u})/2=\sqrt{x^{2}+y^{2}}$ +\end_inset + +, y existe +\begin_inset Formula $v$ +\end_inset + tal que +\begin_inset Formula $(u-\frac{1}{u})/2\cdot(\cos v,\sin v)=(x,y)$ +\end_inset + +. + Finalmente, +\begin_inset Formula $z^{2}=x^{2}+y^{2}+1=\frac{1}{4}(u-\frac{1}{u})^{2}+1=\frac{1}{4}(u+\frac{1}{u})$ +\end_inset + +, luego +\begin_inset Formula $z\in\{\pm\frac{1}{2}(u+\frac{1}{u})\}$ +\end_inset + + y, si los signos son opuestos, basta cambiar +\begin_inset Formula $u$ +\end_inset + + por +\begin_inset Formula $-\frac{1}{u}$ +\end_inset + +. \end_layout \end_deeper @@ -1725,22 +1998,6 @@ Demostración: \end_layout \begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -begin{samepage} -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Standard Una función \begin_inset Formula $f:S\to\mathbb{R}^{m}$ \end_inset @@ -1881,22 +2138,6 @@ Sea . \end_layout -\begin_layout Standard -\begin_inset ERT -status open - -\begin_layout Plain Layout - - -\backslash -end{samepage} -\end_layout - -\end_inset - - -\end_layout - \begin_layout Subsection Propiedades \end_layout @@ -2189,152 +2430,6 @@ Dada una parametrización \end_layout \end_deeper -\begin_layout Standard -Ejemplos: -\end_layout - -\begin_layout Enumerate -Sean -\begin_inset Formula $S$ -\end_inset - - una superficie regular y -\begin_inset Formula $p_{0}\in\mathbb{R}^{3}$ -\end_inset - -, -\begin_inset Formula $f(p):=|p-p_{0}|^{2}$ -\end_inset - - es diferenciable en -\begin_inset Formula $S$ -\end_inset - -, pero la función distancia, -\begin_inset Formula $g(p):=|p-p_{0}|$ -\end_inset - -, es diferenciable en -\begin_inset Formula $S$ -\end_inset - - si y sólo si -\begin_inset Formula $p_{0}\notin S$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -Como -\begin_inset Formula $f$ -\end_inset - - es polinómica, es diferenciable en todo -\begin_inset Formula $\mathbb{R}^{3}$ -\end_inset - - y por tanto en -\begin_inset Formula $S$ -\end_inset - -. - Entonces -\begin_inset Formula $g$ -\end_inset - - es diferenciable en todo -\begin_inset Formula $\mathbb{R}^{3}\setminus\{p_{0}\}$ -\end_inset - -, luego lo es en -\begin_inset Formula $S$ -\end_inset - - si -\begin_inset Formula $p_{0}\notin S$ -\end_inset - -, pero si -\begin_inset Formula $p_{0}\in S$ -\end_inset - -, dada una parametrización -\begin_inset Formula $(U,X)$ -\end_inset - - de -\begin_inset Formula $S$ -\end_inset - - con -\begin_inset Formula $p_{0}\in X(U)$ -\end_inset - -, -\begin_inset Formula $(g\circ X)(q)=|X(q)-p_{0}|=|X(q)-X(q_{0})|$ -\end_inset - - para algún -\begin_inset Formula $q_{0}\in U$ -\end_inset - -. - Entonces -\begin_inset Formula $\frac{\partial(g\circ X)}{\partial q}=\frac{\sum_{i=1}^{3}(X_{i}(q)-X_{i}(q_{0}))X'_{i}(q)}{|X(q)-X(q_{0})|}$ -\end_inset - -, que no está definida en -\begin_inset Formula $q_{0}$ -\end_inset - -. -\end_layout - -\end_deeper -\begin_layout Enumerate -Dado un vector -\begin_inset Formula $v$ -\end_inset - - unitario, la -\series bold -función altura -\series default -, -\begin_inset Formula $h(p):=\langle p,v\rangle$ -\end_inset - -, representa la distancia de -\begin_inset Formula $p$ -\end_inset - - al plano ortogonal a -\begin_inset Formula $v$ -\end_inset - - por el origen y es diferenciable en toda superficie regular -\begin_inset Formula $S$ -\end_inset - -. -\end_layout - -\begin_deeper -\begin_layout Standard -Por ser -\begin_inset Formula $h$ -\end_inset - - polinómica y por tanto diferenciable en todo -\begin_inset Formula $\mathbb{R}^{3}$ -\end_inset - -. -\end_layout - -\end_deeper \begin_layout Subsection Funciones diferenciables entre dos superficies \end_layout @@ -3178,6 +3273,1202 @@ Tomando signo positivo, la base \end_layout \begin_layout Section +Diferencial de funciones en superficies +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $S$ +\end_inset + + una superficie regular, +\begin_inset Formula $f:S\to\mathbb{R}^{n}$ +\end_inset + + diferenciable y +\begin_inset Formula $p\in S$ +\end_inset + +, llamamos +\series bold +diferencial +\series default + de +\begin_inset Formula $f$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + a +\begin_inset Formula $df_{p}:T_{p}S\to\mathbb{R}^{n}$ +\end_inset + + dada por +\begin_inset Formula $df_{p}(v):=(f\circ\alpha)'(0)$ +\end_inset + +, siendo +\begin_inset Formula $\alpha:(-\varepsilon,\varepsilon)\to S$ +\end_inset + + una curva regular con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + +. + La función +\begin_inset Formula $df_{p}$ +\end_inset + + está bien definida y es lineal, y si +\begin_inset Formula $(U,X)$ +\end_inset + + es una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $q:=X^{-1}(p)$ +\end_inset + +, entonces +\begin_inset Formula $df_{p}(v_{1}X_{u}(q)+v_{2}X_{v}(q))=d(f\circ X)_{q}(v_{1},v_{2})$ +\end_inset + +. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula $a\in T_{p}S$ +\end_inset + +, +\begin_inset Formula $q:=X^{-1}(p)$ +\end_inset + +, +\begin_inset Formula $\alpha:I\to S$ +\end_inset + + una curva diferenciable con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=a$ +\end_inset + + y +\begin_inset Formula $\tilde{\alpha}:=X^{-1}\circ\alpha:\alpha^{-1}(X(U))\to U$ +\end_inset + +, entonces +\begin_inset Formula $\tilde{\alpha}(0)=q$ +\end_inset + +, y como +\begin_inset Formula $f\circ\alpha=(f\circ X)\circ(X^{-1}\circ\alpha)=:\tilde{f}\circ\tilde{\alpha}$ +\end_inset + +, si +\begin_inset Formula $\tilde{\alpha}(t)=:(u(t),v(t))$ +\end_inset + +, +\begin_inset Formula +\[ +df_{p}(a)=(f\circ\alpha)'(0)=(\tilde{f}\circ\tilde{\alpha})'(0)=d\tilde{f}_{\tilde{\alpha}(0)}(\tilde{\alpha}'(0))=d\tilde{f}_{q}(u'(0),v'(0)), +\] + +\end_inset + +pero por la regla de la cadena, +\begin_inset Formula $a=\alpha'(0)=u'(0)X_{u}(q)+v'_{0}X_{v}(q)$ +\end_inset + +, luego +\begin_inset Formula $u'(0)$ +\end_inset + + y +\begin_inset Formula $v'(0)$ +\end_inset + + no dependen de la curva elegida y +\begin_inset Formula $f$ +\end_inset + + está bien definida. + Además, por lo anterior, +\begin_inset Formula $df_{p}(v_{1}X_{u}(q)+v_{2}X_{v}(q))=d\tilde{f}_{q}(v_{1},v_{2})$ +\end_inset + +, luego +\begin_inset Formula $df_{p}$ +\end_inset + + es lineal. +\end_layout + +\begin_layout Subsection +Ejemplos +\end_layout + +\begin_layout Standard +Sea +\begin_inset Formula $S$ +\end_inset + + una superficie regular: +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $v\in\mathbb{S}^{2}$ +\end_inset + +, la +\series bold +función altura +\series default + +\begin_inset Formula $h(p):=\langle p,v\rangle$ +\end_inset + + representa la distancia de +\begin_inset Formula $p$ +\end_inset + + al plano ortogonal a +\begin_inset Formula $v$ +\end_inset + + por el origen y es diferenciable en todo +\begin_inset Formula $S$ +\end_inset + + con diferencial +\begin_inset Formula $dh_{p}(u)=\langle u,v\rangle$ +\end_inset + +. + En particular, si +\begin_inset Formula $T_{p}S$ +\end_inset + + es paralelo a +\begin_inset Formula $\text{span}\{v\}^{\bot}$ +\end_inset + + es +\begin_inset Formula $dh_{p}\equiv0$ +\end_inset + +. + +\end_layout + +\begin_deeper +\begin_layout Standard +Es diferenciable por ser polinómica y por tanto diferenciable en todo +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + +. + Su diferencial es +\begin_inset Formula $dh_{p}(w)=:(h\circ\alpha)'(0)=dh_{\alpha(0)}(\alpha'(0))=\langle\alpha'(0),v\rangle=\langle w,v\rangle$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Dado +\begin_inset Formula $p_{0}\in\mathbb{R}^{3}$ +\end_inset + +, la función distancia +\begin_inset Formula $g(p):=|p-p_{0}|$ +\end_inset + + es diferenciable en +\begin_inset Formula $S$ +\end_inset + + si y sólo si +\begin_inset Formula $p_{0}\notin S$ +\end_inset + +, con +\begin_inset Formula $dg_{p}(v)=\frac{\langle v,p-p_{0}\rangle}{|p-p_{0}|}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Como +\begin_inset Formula $g$ +\end_inset + + es diferenciable en todo +\begin_inset Formula $\mathbb{R}^{3}\setminus\{p_{0}\}$ +\end_inset + +, si +\begin_inset Formula $p_{0}\notin S$ +\end_inset + +, +\begin_inset Formula $g$ +\end_inset + + es diferenciable en +\begin_inset Formula $S$ +\end_inset + + con +\end_layout + +\begin_layout Standard +\begin_inset Formula +\begin{align*} +df_{p}(v) & =(f\circ\alpha)'(0)=\frac{d}{dt}|\alpha(t)-p_{0}|(0)=\frac{d}{dt}\sqrt{\langle\alpha(t)-p_{0},\alpha(t)-p_{0}\rangle}(0)\\ + & =\left(t\mapsto\frac{\langle\alpha'(t),\alpha(t)-p_{0}\rangle}{|\alpha(t)-p_{0}|}\right)(0)=\frac{\langle v,p-p_{0}\rangle}{|p-p_{0}|}, +\end{align*} + +\end_inset + +pero si +\begin_inset Formula $p_{0}\in S$ +\end_inset + +, +\begin_inset Formula $g$ +\end_inset + + no es diferenciable en +\begin_inset Formula $S$ +\end_inset + + porque de serlo sería +\begin_inset Formula $df_{p_{0}}(v)=\frac{0}{0}\#$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +La +\series bold +función antípoda +\series default + +\begin_inset Formula $A:\mathbb{S}^{2}\to\mathbb{S}^{2}$ +\end_inset + + dada por +\begin_inset Formula $A(p):=-p$ +\end_inset + + es diferenciable con +\begin_inset Formula $dA_{p}=-1_{T_{p}\mathbb{S}^{2}}$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Es diferenciable porque es polinómica, y dada una curva +\begin_inset Formula $\alpha$ +\end_inset + + apropiada, +\begin_inset Formula $dA_{p}(v)=(A\circ\alpha)'(0)=\frac{d}{dt}(-\alpha(t))(0)=-\alpha'(0)=-v$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Dado +\begin_inset Formula $\theta\in\mathbb{R}$ +\end_inset + +, sea +\begin_inset Formula $\hat{F}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ +\end_inset + + la rotación de ángulo +\begin_inset Formula $\theta$ +\end_inset + + respecto al eje +\begin_inset Formula $z$ +\end_inset + +, dada por +\begin_inset Formula $F(x,y,z)=(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta,z)$ +\end_inset + +, +\begin_inset Formula $F:=\hat{F}|_{\mathbb{S}^{2}}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ +\end_inset + + es diferenciable con +\begin_inset Formula $dF_{p}(v)=\hat{F}(v)$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Es diferenciable por ser la restricción de una función diferenciable en + +\begin_inset Formula $\mathbb{R}^{3}$ +\end_inset + +. + Tomando una curva +\begin_inset Formula $\alpha(t):=(x(t),y(t),z(t))$ +\end_inset + + apropiadamente, +\begin_inset Formula +\begin{align*} +dF_{p}(v) & =(F\circ\alpha)'(0)=\frac{d}{dt}\left(x(t)\cos\theta-y(t)\sin\theta,x(t)\sin\theta+y(t)\cos\theta,z(t)\right)(0)\\ + & =(x'(0)\cos\theta-y'(0)\sin\theta,x'(0)\sin\theta+y'(0)\cos\theta,z'(0))\\ + & =\hat{F}(x'(0),y'(0),z'(0))=\hat{F}(v). +\end{align*} + +\end_inset + + +\end_layout + +\end_deeper +\begin_layout Enumerate +Si +\begin_inset Formula $0\notin S$ +\end_inset + +, +\begin_inset Formula $F:S\to\mathbb{S}^{2}$ +\end_inset + + dada por +\begin_inset Formula $F(p):=p/|p|$ +\end_inset + + es diferenciable con +\begin_inset Formula $dF_{p}(v)=\frac{1}{|p|}v-\frac{\langle p,v\rangle}{|p|^{3}}p$ +\end_inset + +. +\end_layout + +\begin_deeper +\begin_layout Standard +Claramente es diferenciable por ser la restricción de una función diferenciable. + Entonces +\begin_inset Formula $dF_{p}(v)=:(F\circ\alpha)(0)=\frac{d}{dt}\frac{\alpha(t)}{|\alpha(t)|}=\left(t\mapsto\frac{\alpha'(t)|\alpha(t)|-\alpha(t)\langle\alpha(t),\alpha'(t)\rangle/|\alpha(t)|}{|\alpha(t)|^{2}}\right)(0)=\frac{v}{|p|}-\frac{p\langle p,v\rangle}{|p|^{3}}$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Subsection +Diferencial de funciones entre superficies +\end_layout + +\begin_layout Standard +Dadas dos superficies regulares +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + +, +\begin_inset Formula $F:S_{1}\to S_{2}$ +\end_inset + + diferenciable, +\begin_inset Formula $p\in S_{1}$ +\end_inset + +, parametrizaciones +\begin_inset Formula $(U_{1},X_{1})$ +\end_inset + + de +\begin_inset Formula $S_{1}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $(U_{2},X_{2})$ +\end_inset + + de +\begin_inset Formula $S_{2}$ +\end_inset + + en +\begin_inset Formula $F(p)$ +\end_inset + +, +\begin_inset Formula $q_{1}:=X_{1}^{-1}(p)$ +\end_inset + + y +\begin_inset Formula $q_{2}:=X_{2}^{-1}(F(p))$ +\end_inset + +, la matriz asociada a +\begin_inset Formula $dF_{p}$ +\end_inset + + respecto de las bases +\begin_inset Formula ${\cal B}_{1}:=((X_{1})_{u}(q_{1}),(X_{1})_{v}(q_{1}))$ +\end_inset + + y +\begin_inset Formula ${\cal B}_{2}:=((X_{2})_{u}(q_{2}),(X_{2})_{v}(q_{2}))$ +\end_inset + + es el jacobiano de la expresión en coordenadas de +\begin_inset Formula $F$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + +. + +\series bold +Demostración: +\series default + Sea +\begin_inset Formula $v:=v_{1}(X_{1})_{u}+v_{2}(X_{1})_{v}$ +\end_inset + +, de modo que +\begin_inset Formula $[v]_{{\cal B}_{1}}=(v_{1},v_{2})$ +\end_inset + +, entonces +\begin_inset Formula $dF_{p}(v)=d(F\circ X_{1})_{q_{1}}(v_{1},v_{2})$ +\end_inset + +, pero la expresión en coordenadas +\begin_inset Formula $\tilde{F}:=X_{2}^{-1}\circ F\circ X_{1}:U_{1}\to U_{2}$ +\end_inset + + cumple +\begin_inset Formula $d\tilde{F}_{q_{1}}=d(X_{2}^{-1})_{F(p)}\circ d(F\circ X_{1})_{q_{1}}=(d(X_{2})_{F(p)})^{-1}\circ d(F\circ X_{1})_{q_{1}}$ +\end_inset + +, de modo que +\begin_inset Formula $[dF_{p}(v)]_{{\cal B}_{2}}=M_{{\cal B}_{2}{\cal C}}dF_{p}(v)=(d(X_{2})_{F(p)})^{-1}(d(F\circ X_{1})_{q_{1}}(v))=d\tilde{F}_{q_{1}}(v)$ +\end_inset + +. +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +La demostración original era más larga porque la primera la hacía solo para + funciones reales ( +\begin_inset Formula $\mathbb{R}$ +\end_inset + + en vez de +\begin_inset Formula $\mathbb{R}^{n}$ +\end_inset + +). + Esta puede que no sea válida +\emph on +\lang english +though +\emph default +\lang spanish +. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Regla de la cadena: +\series default + Sean +\begin_inset Formula $S_{1}$ +\end_inset + +, +\begin_inset Formula $S_{2}$ +\end_inset + + y +\begin_inset Formula $S_{3}$ +\end_inset + + superficies regulares y +\begin_inset Formula $F:S_{1}\to S_{2}$ +\end_inset + + y +\begin_inset Formula $G:S_{2}\to S_{3}$ +\end_inset + + diferenciables, para +\begin_inset Formula $p\in S_{1}$ +\end_inset + +, +\begin_inset Formula $d(G\circ F)_{p}=dG_{F(p)}\circ dF_{p}$ +\end_inset + +. + +\series bold +Demostración: +\series default + Ya hemos visto que +\begin_inset Formula $G\circ F$ +\end_inset + + es diferenciable. + Sean +\begin_inset Formula $\alpha:I\to S_{1}$ +\end_inset + + una curva regular con +\begin_inset Formula $\alpha(0)=p$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v$ +\end_inset + + y +\begin_inset Formula $\beta:=F\circ\alpha$ +\end_inset + +, +\begin_inset Formula $\beta$ +\end_inset + + es una curva regular en +\begin_inset Formula $S_{2}$ +\end_inset + + con +\begin_inset Formula $\beta(0)=F(p)$ +\end_inset + + y +\begin_inset Formula $\beta'(0)=dF_{\alpha(0)}(\alpha'(0))=dF_{p}(v)$ +\end_inset + +, luego +\begin_inset Formula +\[ +d(G\circ F)_{p}(v)=(G\circ F\circ\alpha)'(0)=(G\circ\beta)'(0)=dG_{\beta(0)}\beta'(0)=dG_{F(p)}(dF_{p}(v)). +\] + +\end_inset + + +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +sremember{FVV3} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de la función inversa: +\series default + Sean +\begin_inset Formula $f:D\subseteq\mathbb{R}^{d}\to\mathbb{R}^{d}$ +\end_inset + + una aplicación de clase +\begin_inset Formula ${\cal C}^{k}$ +\end_inset + + con +\begin_inset Formula $k\geq1$ +\end_inset + + y +\begin_inset Formula $x_{0}\in D$ +\end_inset + + tal que +\begin_inset Formula $df(x_{0})$ +\end_inset + + es no singular, existen [...] +\begin_inset Formula ${\cal U}\in{\cal E}(x_{0})$ +\end_inset + + [y] +\begin_inset Formula ${\cal V}\in{\cal E}(y_{0}:=f(x_{0}))$ +\end_inset + + tales que +\begin_inset Formula $f:{\cal U}\to{\cal V}$ +\end_inset + + es biyectiva y inversa es +\begin_inset Formula ${\cal C}^{k}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +eremember +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard + +\series bold +Teorema de la función inversa: +\series default + Sean +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + superficies regulares, +\begin_inset Formula $F:S_{1}\to S_{2}$ +\end_inset + + +\begin_inset Formula ${\cal C}^{1}$ +\end_inset + + y +\begin_inset Formula $p\in S_{1}$ +\end_inset + + con +\begin_inset Formula $dF_{p}$ +\end_inset + + biyectiva, si existen parametrizaciones de +\begin_inset Formula $S_{1}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y de +\begin_inset Formula $S_{2}$ +\end_inset + + en +\begin_inset Formula $F(p)$ +\end_inset + + +\begin_inset Formula ${\cal C}^{1}$ +\end_inset + +, +\begin_inset Formula $F$ +\end_inset + + es un difeomorfismo local en +\begin_inset Formula $p$ +\end_inset + +. + +\series bold +Demostración: +\series default + Tomamos parametrizaciones +\begin_inset Formula ${\cal C}^{1}$ +\end_inset + + +\begin_inset Formula $(U_{1},X_{1})$ +\end_inset + + de +\begin_inset Formula $S_{1}$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $(U_{2},X_{2})$ +\end_inset + + de +\begin_inset Formula $S_{2}$ +\end_inset + + en +\begin_inset Formula $F(p)$ +\end_inset + +, +\begin_inset Formula $q_{1}:=X_{1}^{-1}(p)$ +\end_inset + +, +\begin_inset Formula $q_{2}:=X_{2}^{-1}(F(p))$ +\end_inset + +, +\begin_inset Formula $V_{1}:=X_{1}(U_{1})$ +\end_inset + + y +\begin_inset Formula $V_{2}:=X_{2}(U_{2})$ +\end_inset + +. + Sean +\begin_inset Formula $\tilde{F}$ +\end_inset + + la expresión en coordenadas de +\begin_inset Formula $F$ +\end_inset + + respecto a +\begin_inset Formula $(U_{1},X_{1})$ +\end_inset + + y +\begin_inset Formula $(U_{2},X_{2})$ +\end_inset + + y +\begin_inset Formula $U:=X_{1}^{-1}(F^{-1}(V_{2}))$ +\end_inset + + el dominio de +\begin_inset Formula $\tilde{F}$ +\end_inset + +, +\begin_inset Formula $d\tilde{F}_{q_{1}}$ +\end_inset + + es un isomorfismo lineal, pues su matriz jacobiana es la de +\begin_inset Formula $dF_{p}$ +\end_inset + + en una cierta base, y aplicando el teorema de la función inversa a +\begin_inset Formula $\tilde{F}:U\to\tilde{F}(U)$ +\end_inset + +, existen abiertos +\begin_inset Formula $\tilde{U}_{1}\subseteq U$ +\end_inset + + de +\begin_inset Formula $q_{1}$ +\end_inset + + y +\begin_inset Formula $\tilde{U}_{2}\subseteq U_{2}$ +\end_inset + + de +\begin_inset Formula $q_{2}$ +\end_inset + + tales que +\begin_inset Formula $\tilde{F}:\tilde{U}_{1}\to\tilde{U}_{2}$ +\end_inset + + es un difeomorfismo. + Sea +\begin_inset Formula $V:=X_{1}(\tilde{U}_{1})$ +\end_inset + +, +\begin_inset Formula $F|_{V}:=(X_{2}\circ\tilde{F}\circ X_{1}^{-1})|_{V}:V\to F(V)$ +\end_inset + + es un difeomorfismo por ser composición de difeomorfismos. +\end_layout + +\begin_layout Standard +Como +\series bold +teorema +\series default +, sean +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + + superficies regulares, si +\begin_inset Formula $F$ +\end_inset + + es un difeomorfismo local en +\begin_inset Formula $p\in S_{1}$ +\end_inset + + entre +\begin_inset Formula $S_{1}$ +\end_inset + + y +\begin_inset Formula $S_{2}$ +\end_inset + +, +\begin_inset Formula $dF_{p}:T_{p}S_{1}\to T_{F(p)}S_{2}$ +\end_inset + + es un isomorfismo lineal. + En efecto, tomando entornos +\begin_inset Formula $V_{1}\subseteq S_{1}$ +\end_inset + + de +\begin_inset Formula $p$ +\end_inset + + y +\begin_inset Formula $V_{2}\subseteq S_{2}$ +\end_inset + + de +\begin_inset Formula $F(p)$ +\end_inset + + tales que +\begin_inset Formula $F|_{V_{1}}:V_{1}\to V_{2}$ +\end_inset + + es un difeomorfismo, y por la regla de la cadena +\begin_inset Formula $1_{T_{p}S_{1}}=1_{T_{p}V_{1}}=d(F^{-1}\circ F)_{p}=d(F^{-1})_{F_{p}}\circ dF_{p}$ +\end_inset + + y análogamente +\begin_inset Formula $dF_{p}\circ d(F^{-1})_{F(p)}=1_{T_{F(p)}S_{2}}$ +\end_inset + +, luego +\begin_inset Formula $dF_{p}$ +\end_inset + + es invertible y por tanto un isomorfismo lineal. +\end_layout + +\begin_layout Standard +Dadas una superficie regular +\begin_inset Formula $S$ +\end_inset + + y una función diferenciable +\begin_inset Formula $f:S\to\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $p\in S$ +\end_inset + + es un +\series bold +punto crítico +\series default + de +\begin_inset Formula $f$ +\end_inset + + si +\begin_inset Formula $df_{p}\equiv0$ +\end_inset + +. + Entonces: +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $f$ +\end_inset + + es constante, todos los puntos son críticos. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $f(p)\equiv k$ +\end_inset + +, +\begin_inset Formula $df_{p}(v)=(f\circ\alpha)'(0)=\frac{dk}{dt}(0)=0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Enumerate +Si todos los puntos son críticos y +\begin_inset Formula $S$ +\end_inset + + es conexa, +\begin_inset Formula $f$ +\end_inset + + es constante. +\end_layout + +\begin_deeper +\begin_layout Standard +Sean +\begin_inset Formula $a\in f(S)$ +\end_inset + + y +\begin_inset Formula $A:=\{p\in S:f(p)=a\}\neq\emptyset$ +\end_inset + +, pues +\begin_inset Formula $a\in A$ +\end_inset + +. + Como +\begin_inset Formula $A=f^{-1}(\{a\})$ +\end_inset + +, +\begin_inset Formula $A$ +\end_inset + + es cerrado, y solo queda ver que +\begin_inset Formula $A$ +\end_inset + + es abierto para ver que +\begin_inset Formula $A=S$ +\end_inset + +. + Sean +\begin_inset Formula $p\in A$ +\end_inset + +, +\begin_inset Formula $(U,X)$ +\end_inset + + una parametrización de +\begin_inset Formula $S$ +\end_inset + + en +\begin_inset Formula $p$ +\end_inset + + en la que podemos suponer que +\begin_inset Formula $U$ +\end_inset + + es conexo, +\begin_inset Formula $V:=X(U)$ +\end_inset + + y +\begin_inset Formula $q\in U$ +\end_inset + +, +\begin_inset Formula $d(f\circ X)_{q}=df_{X(q)}\circ dX_{q}\equiv0$ +\end_inset + +, pues +\begin_inset Formula $df_{p}\equiv0$ +\end_inset + + para todo +\begin_inset Formula $p\in S$ +\end_inset + +, luego +\begin_inset Formula $f\circ X:U\to\mathbb{R}$ +\end_inset + + es constante en +\begin_inset Formula $a$ +\end_inset + + y +\begin_inset Formula $f|_{V}$ +\end_inset + + es constante en +\begin_inset Formula $a$ +\end_inset + +, de modo que +\begin_inset Formula $V\subseteq A$ +\end_inset + + y +\begin_inset Formula $A$ +\end_inset + + es abierto. +\end_layout + +\end_deeper +\begin_layout Enumerate +Los extremos relativos son puntos críticos. +\end_layout + +\begin_deeper +\begin_layout Standard +Sea +\begin_inset Formula $p_{0}\in S$ +\end_inset + + un máximo relativo de +\begin_inset Formula $f$ +\end_inset + + (para un mínimo se hace por simetría), de modo que un entorno +\begin_inset Formula $V\subseteq S$ +\end_inset + + de +\begin_inset Formula $p_{0}$ +\end_inset + + con +\begin_inset Formula $f(p_{0})\geq f(p)$ +\end_inset + + para todo +\begin_inset Formula $p\in V$ +\end_inset + +. + Para calcular +\begin_inset Formula $f_{p_{0}}(v)$ +\end_inset + +, sean +\begin_inset Formula $\alpha:I\to V$ +\end_inset + + una curva regular con +\begin_inset Formula $\alpha(0)=p_{0}$ +\end_inset + + y +\begin_inset Formula $\alpha'(0)=v\in T_{p_{0}}S$ +\end_inset + + y +\begin_inset Formula $h:=f\circ\alpha:I\to\mathbb{R}$ +\end_inset + +, +\begin_inset Formula $h(t)\leq h(0)$ +\end_inset + + para todo +\begin_inset Formula $t\in I$ +\end_inset + +, luego +\begin_inset Formula $h'(0)=0$ +\end_inset + + y +\begin_inset Formula $df_{p_{0}}(v)=(f\circ\alpha)'(0)=h'(0)=0$ +\end_inset + +. +\end_layout + +\end_deeper +\begin_layout Section Primera forma fundamental \end_layout @@ -3880,16 +5171,68 @@ Por tanto \end_layout \begin_layout Standard -\begin_inset Note Note -status open +El área del toro +\begin_inset Formula $X(u,v):=((r\cos u+a)\cos v,(r\cos u+a)\sin v,r\sin u)$ +\end_inset -\begin_layout Plain Layout -TODO Toro de revolución -\end_layout + es +\begin_inset Formula $4\pi^{2}ar$ +\end_inset + +. + +\series bold +Demostración: +\series default + Se tiene +\begin_inset Formula +\begin{align*} +X_{u}(u,v) & =(-r\sin u\cos v,-r\sin u\sin v,r\cos u),\\ +X_{v}(u,v) & =(-(r\cos u+a)\sin v,(r\cos u+a)\cos v,0), +\end{align*} \end_inset +luego los coeficientes de la primera forma fundamental son +\begin_inset Formula $E=r^{2}$ +\end_inset + +, +\begin_inset Formula $F=0$ +\end_inset + y +\begin_inset Formula $G=(r\cos u+a)^{2}$ +\end_inset + +, y +\begin_inset Formula $\sqrt{EG-F^{2}}=r(r\cos u+a)$ +\end_inset + +. + La parametrización dada con el abierto +\begin_inset Formula $U:=(0,2\pi)\times(0,2\pi)$ +\end_inset + + no cubre todo el toro, pero si definimos la región +\begin_inset Formula $R_{\varepsilon}:=X((\varepsilon,2\pi-\varepsilon)\times(\varepsilon,2\pi-\varepsilon))\subseteq X(U)$ +\end_inset + +, +\begin_inset Formula +\begin{align*} +A(R_{\varepsilon}) & =\iint_{X^{-1}(R_{\varepsilon})}r(a+r\cos u)du\,dv=\int_{\varepsilon}^{2\pi-\varepsilon}\int_{\varepsilon}^{2\pi-\varepsilon}r(a+r\cos u)dv\,du\\ + & =2r(\pi-\varepsilon)\int_{\varepsilon}^{2\pi-\varepsilon}(a+r\cos u)du=2r(\pi-\varepsilon)\left(a(2\pi-2\varepsilon)+r(\sin(2\pi-\varepsilon)-\sin\varepsilon)\right)\\ + & =4r(\pi-\varepsilon)(a(\pi-\varepsilon)+r\sin\varepsilon), +\end{align*} + +\end_inset + +y tomando límites, +\begin_inset Formula $A(\mathbb{T}^{2})=A(R_{0})=4\pi^{2}ar$ +\end_inset + +. \end_layout \end_body |