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\begin_body

\begin_layout Section
Teorema de Liouville
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $f,g:I\to\mathbb{R}$
\end_inset

 diferenciables con 
\begin_inset Formula $f^{2}+g^{2}\equiv1$
\end_inset

, 
\begin_inset Formula $t_{0}\in I$
\end_inset

 y 
\begin_inset Formula $\theta_{0}\in\mathbb{R}$
\end_inset

 con 
\begin_inset Formula $f(t_{0})=\cos\theta_{0}$
\end_inset

 y 
\begin_inset Formula $g(t_{0})=\sin\theta_{0}$
\end_inset

, entonces existe una única función diferenciable 
\begin_inset Formula $\theta:I\to\mathbb{R}$
\end_inset

 con 
\begin_inset Formula $\theta(t_{0})=\theta_{0}$
\end_inset

 y, para todo 
\begin_inset Formula $t\in I$
\end_inset

, 
\begin_inset Formula $f(t)=\cos\theta(t)$
\end_inset

 y 
\begin_inset Formula $g(t)=\sin\theta(t)$
\end_inset

.
 
\series bold
Demostración:
\series default
 Sea
\begin_inset Formula 
\[
\theta(t):=\theta_{0}+\int_{t_{0}}^{t}(f(u)g'(u)-f'(u)g(u))du,
\]

\end_inset


\begin_inset Formula $\theta$
\end_inset

 es derivable una vez por el teorema fundamental del cálculo y su derivada
 es 
\begin_inset Formula ${\cal C}^{\infty}$
\end_inset

, por lo que 
\begin_inset Formula $\theta$
\end_inset

 es diferenciable.
 Sea 
\begin_inset Formula $h:I\to\mathbb{R}$
\end_inset

 dada por 
\begin_inset Formula $h(t)\coloneqq (f(t)-\cos\theta(t))^{2}+(g(t)-\sin\theta(t))^{2}$
\end_inset

, entonces 
\begin_inset Formula $h(t_{0})=0$
\end_inset

 y
\begin_inset Formula 
\begin{align*}
\frac{1}{2}h'= & (f-\cos\theta)(f'+\theta'\sin\theta)+(g-\sin\theta)(g'-\theta'\cos\theta)\\
= & (f-\cos\theta)(f'+(fg'-f'g)\sin\theta)+(g-\sin\theta)(g'-(fg'-f'g)\cos\theta)\\
= & ff'+f(fg'-f'g)\sin\theta-f'\cos\theta-(fg'-f'g)\sin\theta\cos\theta+\\
 & +gg'-g'\sin\theta-g(fg'-f'g)\cos\theta+(fg'-f'g)\sin\theta\cos\theta,
\end{align*}

\end_inset

pero derivando 
\begin_inset Formula $f^{2}+g^{2}=1$
\end_inset

 queda 
\begin_inset Formula $2ff'+2gg'=0$
\end_inset

, 
\begin_inset Formula $ff'+gg'=0$
\end_inset

, luego
\begin_inset Formula 
\begin{align*}
\frac{1}{2}h(t) & =(f(fg'-f'g)-g')\sin\theta+(-f'-g(fg'-f'g))\cos\theta\\
 & =(g'(f^{2}-1)-ff'g)\sin\theta+(f'(-1+g^{2})-fgg')\cos\theta\\
 & =(g^{2}g'-ff'g)\sin\theta+(f^{2}f'-fgg')\cos\theta\\
 & =g(gg'-ff')\sin\theta+f(ff'-gg')\cos\theta=0.
\end{align*}

\end_inset

Para la unicidad, sea 
\begin_inset Formula $\hat{\theta}$
\end_inset

 otra función que cumple las condiciones, 
\begin_inset Formula $\hat{\theta}$
\end_inset

 se diferencia de 
\begin_inset Formula $\theta$
\end_inset

 en cada punto en un múltiplo de 
\begin_inset Formula $2\pi$
\end_inset

, pero como 
\begin_inset Formula $\hat{\theta}-\theta$
\end_inset

 es continua con dominio conexo, su rango debe ser conexo y estar en la
 componente conexa de 
\begin_inset Formula $\{2k\pi\}_{k\in\mathbb{Z}}$
\end_inset

 en la que está 
\begin_inset Formula $(\hat{\theta}-\theta)(t_{0})=0$
\end_inset

, que es 
\begin_inset Formula $\{0\}$
\end_inset

, luego 
\begin_inset Formula $\hat{\theta}=\theta$
\end_inset

.
\end_layout

\begin_layout Standard
Sean 
\begin_inset Formula $S$
\end_inset

 una superficie regular orientada por 
\begin_inset Formula $N$
\end_inset

 y 
\begin_inset Formula $\alpha:I\to S$
\end_inset

 una curva regular, 
\begin_inset Formula $e_{1},e_{2},V\in\mathfrak{X}(\alpha)$
\end_inset

 unitarios con 
\begin_inset Formula $e_{2}(t)=Je_{1}(t)=N(\alpha(t))\wedge e_{1}(t)$
\end_inset

 para todo 
\begin_inset Formula $t\in I$
\end_inset

, entonces 
\begin_inset Formula $(e_{1}(t),e_{2}(t))$
\end_inset

 es una base ortonormal de 
\begin_inset Formula $T_{\alpha(t)}S$
\end_inset

 y existe 
\begin_inset Formula $\theta(t)$
\end_inset

 diferenciable tal que 
\begin_inset Formula $V=\cos\theta e_{1}+\sin\theta e_{2}$
\end_inset

, y decimos que 
\begin_inset Formula $\theta$
\end_inset

 es el 
\series bold
ángulo de rotación
\series default
 de 
\begin_inset Formula $V$
\end_inset

 respecto a 
\begin_inset Formula $e_{1}$
\end_inset

.
\end_layout

\begin_layout Standard
La curvatura geodésica de 
\begin_inset Formula $\alpha$
\end_inset

 (no necesariamente p.p.a.) es
\begin_inset Formula 
\[
\kappa_{g}^{\alpha}(s)=\frac{\langle\alpha''(u),J\alpha'(u)\rangle}{\Vert\alpha'(u)\Vert^{3}}.
\]

\end_inset


\end_layout

\begin_layout Standard

\series bold
Teorema de Liouville:
\series default
 Sean 
\begin_inset Formula $(U,X)$
\end_inset

 una parametrización ortogonal de 
\begin_inset Formula $S$
\end_inset

 con primera forma fundamental 
\begin_inset Formula $E,F,G$
\end_inset

, 
\begin_inset Formula $\alpha:I\to X(U)$
\end_inset

 una curva regular p.p.a.
 con curvatura geodésica 
\begin_inset Formula $\kappa_{g}$
\end_inset

, 
\begin_inset Formula $\tilde{\alpha}\coloneqq (u,v)\coloneqq X^{-1}\circ\alpha:I\to U$
\end_inset

, 
\begin_inset Formula $e_{1}:I\to\mathbb{R}^{3}$
\end_inset

 dado por
\begin_inset Formula 
\[
e_{1}(s):=\frac{1}{\sqrt{E(\tilde{\alpha}(s))}}X_{u}(\tilde{\alpha}(s)),
\]

\end_inset


\begin_inset Formula $\theta:I\to\mathbb{R}$
\end_inset

 el ángulo de rotación de 
\begin_inset Formula $\alpha'$
\end_inset

 respecto a 
\begin_inset Formula $e_{1}$
\end_inset

, 
\begin_inset Formula $\alpha_{v}(u)\coloneqq \beta_{u}(v)\coloneqq X(u,v)$
\end_inset

, 
\begin_inset Formula $(\kappa_{g})_{1}(u,v)$
\end_inset

 la curvatura geodésica de 
\begin_inset Formula $\alpha_{v}$
\end_inset

 en 
\begin_inset Formula $u$
\end_inset

 y 
\begin_inset Formula $(\kappa_{g})_{2}(u,v)$
\end_inset

 la de 
\begin_inset Formula $\beta_{u}$
\end_inset

 en 
\begin_inset Formula $v$
\end_inset

, entonces
\begin_inset Formula 
\[
\kappa_{g}=\theta'+\frac{1}{2\sqrt{EG}}\left(-u'E_{v}(\tilde{\alpha})+v'G_{u}(\tilde{\alpha})\right)=\theta'+(\kappa_{g})_{1}(\tilde{\alpha})\cos\theta+(\kappa_{g})_{2}(\tilde{\alpha})\sin\theta.
\]

\end_inset


\series bold
Demostración:
\series default
 En efecto, 
\begin_inset Formula $e_{1}$
\end_inset

 es tangente y unitario, ya que
\begin_inset Formula 
\begin{align*}
e_{1}(s) & =\frac{X_{u}}{\Vert X_{u}\Vert}(\tilde{\alpha}(s)).
\end{align*}

\end_inset

Entonces 
\begin_inset Formula $e_{2}(s)\coloneqq Je_{1}(s)$
\end_inset

 es también tangente y unitario y ortogonal a 
\begin_inset Formula $\frac{\partial X}{\partial u}$
\end_inset

, luego
\begin_inset Formula 
\[
e_{2}(s)=\frac{X_{v}}{\Vert X_{v}\Vert}(\tilde{\alpha}(s))=\frac{1}{\sqrt{G(\tilde{\alpha}(s))}}X_{v}(\tilde{\alpha}(s)).
\]

\end_inset

Con esto,
\begin_inset Formula 
\begin{align*}
\left\langle \frac{De_{1}}{ds},e_{1}\right\rangle  & =\langle e_{1}',e_{1}\rangle=\frac{1}{2}\frac{d}{ds}\langle e_{1},e_{1}\rangle=0,\\
\left\langle \frac{De_{1}}{ds},e_{2}\right\rangle  & =\langle e_{1}',e_{2}\rangle=\frac{d}{ds}\langle e_{1},e_{2}\rangle-\langle e_{1},e_{2}'\rangle=-\langle e_{1},e_{2}'\rangle=-\left\langle \frac{De_{2}}{ds},e_{1}\right\rangle ,\\
\left\langle \frac{De_{2}}{ds},e_{2}\right\rangle  & =\frac{1}{2}\frac{d}{ds}\langle e_{2},e_{2}\rangle=0,
\end{align*}

\end_inset

luego si 
\begin_inset Formula $\omega\coloneqq \langle e_{1}',e_{2}\rangle=-\langle e_{1},e_{2}'\rangle$
\end_inset


\begin_inset Formula 
\begin{align*}
\frac{De_{1}}{ds}(s) & =\left\langle \frac{De_{1}}{ds},e_{1}\right\rangle e_{1}+\left\langle \frac{De_{2}}{ds},e_{2}\right\rangle e_{2}=\omega(s)e_{2}(s), & \frac{De_{2}}{ds}(s) & =-\omega(s)e_{1}(s).
\end{align*}

\end_inset

Por tanto, como 
\begin_inset Formula $\alpha'=\cos\theta e_{1}+\sin\theta e_{2}$
\end_inset

,
\begin_inset Formula 
\begin{align*}
\frac{D\alpha'}{ds} & =-\theta'\sin\theta e_{1}+\cos\theta\omega e_{2}+\theta'\cos\theta e_{2}-\sin\theta\omega e_{1}=(\theta'+\omega)(\cos\theta e_{2}-\sin\theta e_{1})\\
 & =(\theta'+\omega)J\alpha'(s).
\end{align*}

\end_inset

Por otro lado, 
\begin_inset Formula $\frac{D\alpha'}{ds}(s)=\kappa_{g}(s)J\alpha'(s)$
\end_inset

, luego 
\begin_inset Formula $\kappa_{g}(s)=\theta'(s)+\omega(s)$
\end_inset

.
 Derivando la fórmula de 
\begin_inset Formula $e_{1}$
\end_inset

,
\begin_inset Formula 
\[
e_{1}'=\frac{d}{ds}\left(\frac{1}{\sqrt{E}}\right)X_{u}(\tilde{\alpha})+\frac{1}{\sqrt{E}}\left(u'X_{uu}(\tilde{\alpha})+v'X_{uv}(\tilde{\alpha})\right).
\]

\end_inset

Entonces, como 
\begin_inset Formula $X_{uu}(\tilde{\alpha})=\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN$
\end_inset

 y 
\begin_inset Formula $X_{uv}=\Gamma_{12}^{1}X_{u}+\Gamma_{12}^{2}X_{v}+fN$
\end_inset

,
\begin_inset Formula 
\begin{align*}
\omega & =\langle e_{1}',e_{2}\rangle=\frac{1}{\sqrt{EG}}\left\langle u'({\textstyle \Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN)}+v'({\textstyle \Gamma_{12}^{2}X_{u}+\Gamma_{12}^{2}X_{v}+fN}),X_{v}\right\rangle \\
 & =\frac{1}{\sqrt{EG}}(u'\Gamma_{11}^{2}G+v'\Gamma_{12}^{2}G),
\end{align*}

\end_inset

pero como
\begin_inset Formula 
\begin{align*}
\Gamma_{11}^{2} & =\frac{-F\frac{E_{u}}{2}+EF_{u}-E\frac{E_{v}}{2}}{EG-F^{2}}=-\frac{E\frac{E_{v}}{2}}{EG}=-\frac{E_{v}}{2G}, & \Gamma_{12}^{2} & =\frac{-F\frac{E_{v}}{2}+E\frac{G_{u}}{2}}{EG-F^{2}}=\frac{E\frac{G_{u}}{2}}{EG}=\frac{G_{u}}{2G},
\end{align*}

\end_inset

queda 
\begin_inset Formula 
\[
\omega=\frac{1}{2\sqrt{EG}}(-u'E_{v}+v'G_{u}),
\]

\end_inset

 la primera expresión.
 Por otro lado, 
\begin_inset Formula 
\begin{align*}
\alpha'_{v}(u) & =X_{u}, & J\alpha'_{v}(u) & =N\wedge X_{u}=\Vert X_{u}\Vert\frac{X_{v}}{\Vert X_{v}\Vert}=\sqrt{\frac{E}{G}}X_{v}, & \alpha''_{v}(u) & =X_{uu},\\
\beta'_{u}(v) & =X_{v}, & J\beta'_{u}(v) & =N\wedge X_{v}=-\Vert X_{v}\Vert\frac{X_{u}}{\Vert X_{u}\Vert}=-\sqrt{\frac{G}{E}}X_{u}, & \beta''_{u}(v) & =X_{vv},
\end{align*}

\end_inset

y como
\begin_inset Formula 
\[
\Gamma_{22}^{1}=\frac{G(F_{v}-\frac{G_{u}}{2})-F\frac{G_{v}}{2}}{EG-F^{2}}=\frac{-G\frac{G_{u}}{2}}{EG}=-\frac{G_{u}}{2E},
\]

\end_inset

queda
\begin_inset Formula 
\begin{align*}
(\kappa_{g})_{1}(u,v) & =\frac{\langle\alpha''_{v}(u),J\alpha'_{v}(u)\rangle}{\Vert\alpha'_{v}(u)\Vert^{3}}=\frac{\langle\Gamma_{11}^{1}X_{u}+\Gamma_{11}^{2}X_{v}+eN,\sqrt{\frac{E}{G}}X_{v}\rangle}{\Vert X_{u}\Vert^{3}}=\sqrt{\frac{E}{G}}\frac{\Gamma_{11}^{2}G}{E\sqrt{E}}=-\frac{E_{v}}{2E\sqrt{G}},\\
(\kappa_{g})_{2}(u,v) & =\frac{\langle\beta''_{u}(v),J\beta'_{u}(v)\rangle}{\Vert\beta'_{u}(v)\Vert^{3}}=\frac{\langle\Gamma_{22}^{1}X_{u}+\Gamma_{22}^{2}X_{v}+gN,-\sqrt{\frac{G}{E}}X_{u}\rangle}{\Vert X_{v}\Vert^{3}}=-\sqrt{\frac{G}{E}}\frac{\Gamma_{22}^{1}E}{G\sqrt{G}}=\frac{G_{u}}{2G\sqrt{E}}.
\end{align*}

\end_inset

Con esto, 
\begin_inset Formula $E_{v}=-2E\sqrt{G}(\kappa_{g})_{1}$
\end_inset

 y 
\begin_inset Formula $G_{u}=2G\sqrt{E}(\kappa_{g})_{2}$
\end_inset

, luego
\begin_inset Formula 
\[
\kappa_{g}=\theta'+\frac{1}{2\sqrt{EG}}\left(u'2E\sqrt{G}(\kappa_{g})_{1}+v'2G\sqrt{E}(\kappa_{g})_{2}\right)=\theta'+u'\sqrt{E}(\kappa_{g})_{1}+\sqrt{G}v'(\kappa_{g})_{2},
\]

\end_inset

y queda ver que 
\begin_inset Formula $u'\sqrt{E}=\cos\theta$
\end_inset

 y 
\begin_inset Formula $v'\sqrt{G}=\sin\theta$
\end_inset

, pero
\begin_inset Formula 
\begin{align*}
\alpha' & =(X\circ\tilde{\alpha})'=u'X_{u}+v'X_{v}\\
 & =\cos\theta e_{1}+\sin\theta e_{2}=\cos\theta\frac{1}{\sqrt{E}}X_{u}+\sin\theta\frac{1}{\sqrt{G}}X_{v},
\end{align*}

\end_inset

y usando que 
\begin_inset Formula $(X_{u},X_{v})$
\end_inset

 es base despejamos y se obtiene el resultado.
\end_layout

\begin_layout Section
Teorema de rotación de las tangentes
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $S$
\end_inset

 una superficie regular, un 
\series bold
polígono curvado
\series default
 es la imagen 
\begin_inset Formula $\Gamma$
\end_inset

 de un segmento de curva 
\begin_inset Formula $\alpha:[0,\ell]\to S$
\end_inset

 regular a trozos p.p.a.
 (en cada trozo) 
\series bold
cerrado
\series default
 (
\begin_inset Formula $\alpha(0)=\alpha(\ell)$
\end_inset

) y 
\series bold
simple
\series default
 
\begin_inset Formula $(\forall s,s'\in[0,\ell],(\alpha(s)=\alpha(s')\implies s=s'\lor\{s,s'\}=\{0,\ell\})$
\end_inset

).
 Si 
\begin_inset Formula $\Gamma$
\end_inset

 es la frontera de una región 
\begin_inset Formula $R$
\end_inset

 de 
\begin_inset Formula $S$
\end_inset

 simplemente conexa, 
\begin_inset Formula $\alpha$
\end_inset

 está 
\series bold
positivamente orientada
\series default
 si, para 
\begin_inset Formula $s\in[0,\ell]$
\end_inset

 que no sea un vértice, 
\begin_inset Formula $J\alpha'(s)$
\end_inset

 apunta al interior de 
\begin_inset Formula $R$
\end_inset

 (
\begin_inset Formula $\exists\delta>0:\forall t\in(0,\delta),\alpha(s)+tJ\alpha'(s)\in R$
\end_inset

).
 
\end_layout

\begin_layout Standard
Sea 
\begin_inset Formula $0=s_{0}<\dots<s_{k}=\ell$
\end_inset

 una partición en la que los 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 con 
\begin_inset Formula $i\in\{1,\dots,k-1\}$
\end_inset

 son los vértices de 
\begin_inset Formula $\alpha$
\end_inset

, la 
\series bold
velocidad que llega
\series default
 a un vértice 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 es 
\begin_inset Formula $\alpha'_{-}(s_{i})$
\end_inset

, que en 
\begin_inset Formula $\alpha(\ell)$
\end_inset

 es 
\begin_inset Formula $\alpha'_{-}(\ell)\coloneqq \lim_{s\to\ell^{-}}\alpha'(s)$
\end_inset

, y la 
\series bold
velocidad que sale
\series default
 es 
\begin_inset Formula $\alpha'_{+}(s_{i})$
\end_inset

, que en 
\begin_inset Formula $\alpha(0)$
\end_inset

 es 
\begin_inset Formula $\alpha'_{+}(0)=\lim_{s\to0^{+}}\alpha'(s)$
\end_inset

.
 El 
\series bold
ángulo exterior
\series default
 en un 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 es el único 
\begin_inset Formula $\theta\in(-\pi,\pi]$
\end_inset

 tal que 
\begin_inset Formula 
\[
\alpha'_{+}(s_{i})=\cos\theta\alpha'_{-}(s_{i})=\sin\theta J\alpha'_{-}(s_{i}),
\]

\end_inset

 que en 
\begin_inset Formula $\alpha(0)=\alpha(\ell)$
\end_inset

 es el que cumple 
\begin_inset Formula $\alpha'_{+}(0)=\cos\theta\alpha'_{-}(\ell)+\sin\theta J\alpha'_{-}(\ell)$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Teorema de rotación de las tangentes:
\series default
 Sean 
\begin_inset Formula $(U,X)$
\end_inset

 una parametrización ortogonal de una superficie 
\begin_inset Formula $S$
\end_inset

, 
\begin_inset Formula $\alpha:[0,\ell]\to X(U)$
\end_inset

 una parametrización positivamente orientada de la frontera 
\begin_inset Formula $\Gamma$
\end_inset

 de una región 
\begin_inset Formula $R$
\end_inset

 de 
\begin_inset Formula $S$
\end_inset

, 
\begin_inset Formula $0=s_{0}<\dots<s_{k}=\ell$
\end_inset

 una partición en la que los 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 son los vértices de 
\begin_inset Formula $\alpha$
\end_inset

, 
\begin_inset Formula $\varepsilon_{i}$
\end_inset

 el ángulo exterior de 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 y 
\begin_inset Formula $\theta_{i}$
\end_inset

 el ángulo de rotación de la velocidad de 
\begin_inset Formula $\alpha_{i}\coloneqq \alpha|_{[s_{i-1},s_{i}]}$
\end_inset

 respecto a 
\begin_inset Formula $e_{1}(s)\coloneqq X_{u}(X^{-1}(\alpha(s)))/\sqrt{E(s)}$
\end_inset

, entonces
\begin_inset Formula 
\[
\sum_{i=1}^{k}(\theta_{i}(s_{i})-\theta_{i}(s_{i-1}))+\sum_{i=1}^{k}\varepsilon_{i}=2\pi.
\]

\end_inset


\end_layout

\begin_layout Section
Teorema de Gauss-Bonnet
\end_layout

\begin_layout Standard

\series bold
Teorema de Green:
\series default
 Sea 
\begin_inset Formula $\tilde{\alpha}\coloneqq (u,v):[0,\ell]\to\mathbb{R}^{2}$
\end_inset

 una parametrización positivamente orientada de la frontera de un 
\begin_inset Formula $\Omega\subseteq\mathbb{R}^{2}$
\end_inset

 acotado y 
\begin_inset Formula $P,Q:\overline{\Omega}\to\mathbb{R}$
\end_inset

 diferenciables,
\begin_inset Formula 
\[
\iint_{\Omega}\left(\frac{\partial Q}{\partial u}-\frac{\partial P}{\partial v}\right)du\,dv=\int_{\partial\Omega}(P(\tilde{\alpha})u'+Q(\tilde{\alpha})v')ds:=\sum_{i=1}^{k}\int_{s_{i-1}}^{s_{i}}(P(\tilde{\alpha})u'+Q(\tilde{\alpha})v')ds,
\]

\end_inset

donde 
\begin_inset Formula $0=s_{0}<\dots<s_{k}=\ell$
\end_inset

 es una partición de 
\begin_inset Formula $[0,\ell]$
\end_inset

 tal que los 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 son los vértices de 
\begin_inset Formula $\alpha$
\end_inset

.
\end_layout

\begin_layout Standard

\series bold
Versión local del teorema de Gauss-Bonnet:
\series default
 Sean 
\begin_inset Formula $(U,X)$
\end_inset

 una parametrización ortogonal positiva de 
\begin_inset Formula $S$
\end_inset

, 
\begin_inset Formula $\alpha:[0,\ell]\to X(U)$
\end_inset

 una parametrización positivamente orientada de la frontera de una región
 
\begin_inset Formula $R$
\end_inset

 de 
\begin_inset Formula $S$
\end_inset

, 
\begin_inset Formula $0=s_{0}<\dots<s_{k}=\ell$
\end_inset

 una partición en la que los 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

 son los vértices de 
\begin_inset Formula $\alpha$
\end_inset

 y 
\begin_inset Formula $\varepsilon_{i}$
\end_inset

 el ángulo exterior de 
\begin_inset Formula $\alpha(s_{i})$
\end_inset

, entonces 
\begin_inset Formula 
\[
\int_{R}K\,dS+\int_{\partial R}\kappa_{g}ds+\sum_{i=1}^{k}\varepsilon_{i}=2\pi.
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
sremember{TS}
\end_layout

\end_inset


\end_layout

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

 es un complejo simplicial 
\begin_inset Formula $n$
\end_inset

-dimensional con 
\begin_inset Formula $i_{k}$
\end_inset

 
\begin_inset Formula $k$
\end_inset

-símplices para cada 
\begin_inset Formula $k\in\{0,\dots,n\}$
\end_inset

, el 
\series bold
número
\series default
 o 
\series bold
característica de Euler
\series default
 de 
\begin_inset Formula $T$
\end_inset

 es 
\begin_inset Formula $\chi(T)\coloneqq i_{0}-i_{1}+\dots+(-1)^{n}i_{n}$
\end_inset

.
 [...] El 
\series bold
número de Euler
\series default
 [o 
\series bold
característica de Euler-Poincaré
\series default
] de un espacio triangulable 
\begin_inset Formula $X$
\end_inset

, 
\begin_inset Formula $\chi(X)$
\end_inset

, es el de cualquier complejo simplicial cuyo poliedro es homeomorfo a 
\begin_inset Formula $X$
\end_inset

, y es un invariante topológico.
 [...]
\end_layout

\begin_layout Standard
El 
\series bold
género
\series default
 de una superficie compacta 
\begin_inset Formula $M$
\end_inset

, o el número de 
\series bold
agujeros
\series default
, es
\begin_inset Formula 
\[
g(M):=\begin{cases}
\frac{1}{2}(2-\chi(M)), & M\text{ orientable};\\
2-\chi(M), & M\text{ no orientable}.
\end{cases}
\]

\end_inset


\end_layout

\begin_layout Standard
Tenemos 
\begin_inset Formula $g(\mathbb{S}^{2})=0$
\end_inset

, [...] si 
\begin_inset Formula $T_{1},\dots,T_{n}$
\end_inset

 son toros, 
\begin_inset Formula $g(T_{1}\sharp\dots\sharp T_{n})=n$
\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
Versión global del teorema de Gauss-Bonnet:
\series default
 Sean 
\begin_inset Formula $(U,X)$
\end_inset

 una parametrización ortogonal positiva de una superficie orientada 
\begin_inset Formula $S$
\end_inset

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

 una región de 
\begin_inset Formula $S$
\end_inset

 cuya frontera es la unión disjunta de los 
\series bold
polígonos curvados
\series default
 
\begin_inset Formula $\Gamma_{1},\dots,\Gamma_{n}$
\end_inset

, 
\begin_inset Formula $\alpha_{i}:[0,\ell_{i}]\to S$
\end_inset

 una parametrización positivamente orientada de 
\begin_inset Formula $\alpha_{i}$
\end_inset

 y 
\begin_inset Formula $\varepsilon_{i1},\dots,\varepsilon_{ik_{i}}$
\end_inset

 los ángulos exteriores de los vértices de 
\begin_inset Formula $\alpha_{i}$
\end_inset

 (incluyendo 
\begin_inset Formula $\alpha_{i}(0)$
\end_inset

), entonces
\begin_inset Formula 
\[
\int_{R}K\,dS+\sum_{i=1}^{n}\int_{\Gamma_{i}}\kappa_{g}^{\alpha_{i}}ds+\sum_{i=1}^{n}\sum_{j=1}^{k_{i}}\varepsilon_{i}=2\pi{\cal X}(R),
\]

\end_inset

siendo 
\begin_inset Formula ${\cal X}(R)$
\end_inset

 el número de Euler de 
\begin_inset Formula $R$
\end_inset

.
\end_layout

\end_body
\end_document