diff options
Diffstat (limited to 'ggs')
| -rw-r--r-- | ggs/n1.lyx | 30 | ||||
| -rw-r--r-- | ggs/n2.lyx | 74 | ||||
| -rw-r--r-- | ggs/n3.lyx | 80 | ||||
| -rw-r--r-- | ggs/n4.lyx | 10 | ||||
| -rw-r--r-- | ggs/n5.lyx | 18 | ||||
| -rw-r--r-- | ggs/n6.lyx | 18 | ||||
| -rw-r--r-- | ggs/n7.lyx | 8 | ||||
| -rw-r--r-- | ggs/n8.lyx | 8 | ||||
| -rw-r--r-- | ggs/n9.lyx | 20 |
9 files changed, 133 insertions, 133 deletions
@@ -137,15 +137,15 @@ Entonces, dada una curva \end_inset p.p.a., si -\begin_inset Formula $\mathbf{t}(s):=\alpha'(s)$ +\begin_inset Formula $\mathbf{t}(s)\coloneqq \alpha'(s)$ \end_inset y -\begin_inset Formula $\mathbf{n}(s):=J\mathbf{t}(s)$ +\begin_inset Formula $\mathbf{n}(s)\coloneqq J\mathbf{t}(s)$ \end_inset [...], [...] -\begin_inset Formula $\kappa_{\alpha}(s):=\langle\mathbf{t}'(s),\mathbf{n}(s)\rangle$ +\begin_inset Formula $\kappa_{\alpha}(s)\coloneqq \langle\mathbf{t}'(s),\mathbf{n}(s)\rangle$ \end_inset [...]. @@ -192,12 +192,12 @@ fórmulas de Frenet \end_inset es su vector tangente, [...] -\begin_inset Formula $\kappa(s):=|\mathbf{t}'(s)|$ +\begin_inset Formula $\kappa(s)\coloneqq |\mathbf{t}'(s)|$ \end_inset . [...] -\begin_inset Formula $\mathbf{n}(s):=\frac{\mathbf{t}'(s)}{\kappa(s)}[...],$ +\begin_inset Formula $\mathbf{n}(s)\coloneqq \frac{\mathbf{t}'(s)}{\kappa(s)}[...],$ \end_inset [...] @@ -613,11 +613,11 @@ Para un \end_inset , -\begin_inset Formula $V(t)^{\top}:=\pi_{T_{\alpha(t)}S}V(t)$ +\begin_inset Formula $V(t)^{\top}\coloneqq \pi_{T_{\alpha(t)}S}V(t)$ \end_inset y -\begin_inset Formula $V(t)^{\bot}:=\pi_{(T_{\alpha(t)}S)^{\bot}}V(t)$ +\begin_inset Formula $V(t)^{\bot}\coloneqq \pi_{(T_{\alpha(t)}S)^{\bot}}V(t)$ \end_inset . @@ -772,7 +772,7 @@ Propiedades: Sean \begin_deeper \begin_layout Standard Si -\begin_inset Formula $\pi:=\pi_{T_{\alpha(t)}S}$ +\begin_inset Formula $\pi\coloneqq \pi_{T_{\alpha(t)}S}$ \end_inset , @@ -881,7 +881,7 @@ Sean \end_inset , -\begin_inset Formula $\tilde{\alpha}:=(u,v):=X^{-1}\circ\alpha:I\to U$ +\begin_inset Formula $\tilde{\alpha}\coloneqq (u,v)\coloneqq X^{-1}\circ\alpha:I\to U$ \end_inset y @@ -914,11 +914,11 @@ Demostración: \end_inset , -\begin_inset Formula $p:=\alpha(t)$ +\begin_inset Formula $p\coloneqq \alpha(t)$ \end_inset , -\begin_inset Formula $q:=X^{-1}(p)$ +\begin_inset Formula $q\coloneqq X^{-1}(p)$ \end_inset y @@ -1148,7 +1148,7 @@ E.d.o intrínseca de los campos paralelos: \end_inset , -\begin_inset Formula $(u,v):=X^{-1}\circ\alpha:I\to U$ +\begin_inset Formula $(u,v)\coloneqq X^{-1}\circ\alpha:I\to U$ \end_inset y @@ -1391,11 +1391,11 @@ Sean \end_inset , -\begin_inset Formula $p:=\alpha(a)$ +\begin_inset Formula $p\coloneqq \alpha(a)$ \end_inset , -\begin_inset Formula $q:=\alpha(b)$ +\begin_inset Formula $q\coloneqq \alpha(b)$ \end_inset y @@ -1440,7 +1440,7 @@ La aplicación transporte paralelo \series default es la -\begin_inset Formula $P_{\alpha}:=P_{a}^{b}(\alpha):T_{p}S\to T_{q}S$ +\begin_inset Formula $P_{\alpha}\coloneqq P_{a}^{b}(\alpha):T_{p}S\to T_{q}S$ \end_inset que a cada @@ -195,7 +195,7 @@ Sea \end_inset un cambio de parámetro y -\begin_inset Formula $\alpha:=\gamma\circ h$ +\begin_inset Formula $\alpha\coloneqq \gamma\circ h$ \end_inset , entonces @@ -287,7 +287,7 @@ Si \end_inset es una curva y -\begin_inset Formula $(u,v):=X^{-1}\circ\alpha:I\to U$ +\begin_inset Formula $(u,v)\coloneqq X^{-1}\circ\alpha:I\to U$ \end_inset , @@ -360,7 +360,7 @@ Teorema de Picard en un abierto: \end_inset existe -\begin_inset Formula $K:=[t_{0}-\alpha,t_{0}+\alpha]\times\overline{B}(x_{0},b)\subseteq\Omega$ +\begin_inset Formula $K\coloneqq [t_{0}-\alpha,t_{0}+\alpha]\times\overline{B}(x_{0},b)\subseteq\Omega$ \end_inset tal que @@ -569,7 +569,7 @@ intervalo maximal de existencia Demostración: \series default Sea -\begin_inset Formula ${\cal J}_{p,v}:=\{(I,\alpha)\mid \alpha\mid I\to S\text{ geodésica},0\in I,\alpha(0)=p,\alpha'(0)=v\}$ +\begin_inset Formula ${\cal J}_{p,v}\coloneqq \{(I,\alpha)\mid \alpha\mid I\to S\text{ geodésica},0\in I,\alpha(0)=p,\alpha'(0)=v\}$ \end_inset . @@ -586,7 +586,7 @@ Demostración: \end_inset , -\begin_inset Formula $(u_{0},v_{0}):=X^{-1}(p)$ +\begin_inset Formula $(u_{0},v_{0})\coloneqq X^{-1}(p)$ \end_inset y @@ -619,7 +619,7 @@ Demostración: \end_inset , y entonces -\begin_inset Formula $\alpha(t):=X(u(t),v(t))$ +\begin_inset Formula $\alpha(t)\coloneqq X(u(t),v(t))$ \end_inset es una geodésica con @@ -669,7 +669,7 @@ Sean ahora es abierto y, por el teorema del peine, también conexo, luego es un intervalo. Sea -\begin_inset Formula $A:=\{t\in I_{1}\cap I_{2}\mid \alpha_{1}(t)=\alpha_{2}(t),\alpha'_{1}(t)=\alpha'_{2}(t)\}$ +\begin_inset Formula $A\coloneqq \{t\in I_{1}\cap I_{2}\mid \alpha_{1}(t)=\alpha_{2}(t),\alpha'_{1}(t)=\alpha'_{2}(t)\}$ \end_inset , y queremos ver que @@ -694,7 +694,7 @@ Sean ahora \end_inset , y es cerrado por ser la anti-imagen del 0 por la función continua -\begin_inset Formula $F(t):=\Vert\alpha_{1}(t)-\alpha_{2}(t)\Vert+\Vert\alpha'_{1}(t)+\alpha'_{2}(t)\Vert$ +\begin_inset Formula $F(t)\coloneqq \Vert\alpha_{1}(t)-\alpha_{2}(t)\Vert+\Vert\alpha'_{1}(t)+\alpha'_{2}(t)\Vert$ \end_inset . @@ -742,15 +742,15 @@ Sean ahora \end_inset , y si -\begin_inset Formula $\varepsilon:=\min\{\varepsilon_{1},\varepsilon_{2}\}$ +\begin_inset Formula $\varepsilon\coloneqq \min\{\varepsilon_{1},\varepsilon_{2}\}$ \end_inset , -\begin_inset Formula $(u_{1},v_{1}):=X^{-1}\circ\alpha_{1}$ +\begin_inset Formula $(u_{1},v_{1})\coloneqq X^{-1}\circ\alpha_{1}$ \end_inset y -\begin_inset Formula $(u_{2},v_{2}):=X^{-1}\circ\alpha_{2}$ +\begin_inset Formula $(u_{2},v_{2})\coloneqq X^{-1}\circ\alpha_{2}$ \end_inset , entonces @@ -802,7 +802,7 @@ Así, . Sea entonces -\begin_inset Formula $I_{v}:=\bigcup_{(I,\alpha)\in{\cal J}_{p,v}}I$ +\begin_inset Formula $I_{v}\coloneqq \bigcup_{(I,\alpha)\in{\cal J}_{p,v}}I$ \end_inset , @@ -892,7 +892,7 @@ Demostración: \end_inset con -\begin_inset Formula $\alpha(t):=\gamma_{v}(\lambda t)$ +\begin_inset Formula $\alpha(t)\coloneqq \gamma_{v}(\lambda t)$ \end_inset , claramente @@ -925,7 +925,7 @@ e . Ahora bien, sea -\begin_inset Formula $w:=\lambda v$ +\begin_inset Formula $w\coloneqq \lambda v$ \end_inset y @@ -933,7 +933,7 @@ e \end_inset dada por -\begin_inset Formula $\beta(t):=\gamma_{w}(\frac{1}{\lambda}v)$ +\begin_inset Formula $\beta(t)\coloneqq \gamma_{w}(\frac{1}{\lambda}v)$ \end_inset , por el mismo argumento es @@ -1024,7 +1024,7 @@ Cálculo de \end_inset , [...] -\begin_inset Formula $E(T,\lambda_{k}):=\ker(T-\lambda_{k}I)^{n_{k}}$ +\begin_inset Formula $E(T,\lambda_{k})\coloneqq \ker(T-\lambda_{k}I)^{n_{k}}$ \end_inset , y [...] @@ -1160,15 +1160,15 @@ status open \begin_layout Enumerate Sea -\begin_inset Formula $P:=M_{{\cal CB}}$ +\begin_inset Formula $P\coloneqq M_{{\cal CB}}$ \end_inset , entonces la parte semisimple es -\begin_inset Formula $S:=PS_{0}P^{-1}$ +\begin_inset Formula $S\coloneqq PS_{0}P^{-1}$ \end_inset y la nilpotente es -\begin_inset Formula $N:=A-S$ +\begin_inset Formula $N\coloneqq A-S$ \end_inset . @@ -1421,7 +1421,7 @@ Dado el plano \end_inset dada por -\begin_inset Formula $\gamma(t):=p+tv$ +\begin_inset Formula $\gamma(t)\coloneqq p+tv$ \end_inset . @@ -1430,7 +1430,7 @@ Dado el plano \begin_deeper \begin_layout Standard Tomando la normal -\begin_inset Formula $N(p):=a$ +\begin_inset Formula $N(p)\coloneqq a$ \end_inset , como @@ -1471,7 +1471,7 @@ Dado \end_inset , la geodésica maximal de la esfera -\begin_inset Formula $S:=\mathbb{S}^{2}(r)$ +\begin_inset Formula $S\coloneqq \mathbb{S}^{2}(r)$ \end_inset con condiciones iniciales @@ -1500,11 +1500,11 @@ Dado \begin_deeper \begin_layout Standard Tomando la normal -\begin_inset Formula $N(p):=\frac{p}{r}$ +\begin_inset Formula $N(p)\coloneqq \frac{p}{r}$ \end_inset y llamando -\begin_inset Formula $N(t):=N(\gamma(t))$ +\begin_inset Formula $N(t)\coloneqq N(\gamma(t))$ \end_inset , @@ -1524,7 +1524,7 @@ Tomando la normal \end_inset Si -\begin_inset Formula $c:=\frac{\Vert v\Vert^{2}}{r^{2}}=0$ +\begin_inset Formula $c\coloneqq \frac{\Vert v\Vert^{2}}{r^{2}}=0$ \end_inset , @@ -1579,7 +1579,7 @@ Sean \end_inset , -\begin_inset Formula $S:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=r^{2}\}$ +\begin_inset Formula $S\coloneqq \{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}=r^{2}\}$ \end_inset un cilindro, @@ -1634,7 +1634,7 @@ p_{3}+tv_{3} \end_inset en otro caso, donde -\begin_inset Formula $c:=\frac{\sqrt{\Vert v\Vert^{2}-v_{3}^{2}}}{r}$ +\begin_inset Formula $c\coloneqq \frac{\sqrt{\Vert v\Vert^{2}-v_{3}^{2}}}{r}$ \end_inset , que es una circunferencia horizontal si @@ -1679,7 +1679,7 @@ N(x,y,z)=\frac{\nabla f}{\Vert\nabla f\Vert}=\frac{(2x,2y,0)}{2\sqrt{x^{2}+y^{2} \end_inset Entonces, sean -\begin_inset Formula $N(t):=N(\gamma(t))$ +\begin_inset Formula $N(t)\coloneqq N(\gamma(t))$ \end_inset y @@ -1897,7 +1897,7 @@ Sea triedro de Darboux \series default es la base [...] -\begin_inset Formula $(\alpha'(s),J\alpha'(s):=\alpha'(s)\wedge N(\alpha(s)),N(\alpha(s))\rangle$ +\begin_inset Formula $(\alpha'(s),J\alpha'(s)\coloneqq \alpha'(s)\wedge N(\alpha(s)),N(\alpha(s))\rangle$ \end_inset . @@ -1910,7 +1910,7 @@ triedro de Darboux \end_inset donde -\begin_inset Formula $\kappa_{g}:=\langle\alpha'',J\alpha'\rangle:I\to\mathbb{R}$ +\begin_inset Formula $\kappa_{g}\coloneqq \langle\alpha'',J\alpha'\rangle:I\to\mathbb{R}$ \end_inset , es la @@ -1926,7 +1926,7 @@ curvatura geodésica \end_inset [, y -\begin_inset Formula $\kappa_{n}:=\langle\alpha'',N(\alpha)\rangle$ +\begin_inset Formula $\kappa_{n}\coloneqq \langle\alpha'',N(\alpha)\rangle$ \end_inset es la @@ -1995,7 +1995,7 @@ Si \end_inset es un cambio de parámetro que conserva la orientación con -\begin_inset Formula $\beta:=\alpha\circ h$ +\begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset p.p.a., la curvatura geodésica de @@ -2027,7 +2027,7 @@ Demostración: \end_inset , sea -\begin_inset Formula $s:=h^{-1}(t)$ +\begin_inset Formula $s\coloneqq h^{-1}(t)$ \end_inset , @@ -2080,7 +2080,7 @@ pregeodésica \end_inset tal que -\begin_inset Formula $\beta:=\alpha\circ h$ +\begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset es una geodésica de @@ -2112,7 +2112,7 @@ Sea \end_inset un cambio de parámetro tal que -\begin_inset Formula $\beta:=\alpha\circ h$ +\begin_inset Formula $\beta\coloneqq \alpha\circ h$ \end_inset es una geodésica, entonces @@ -2124,7 +2124,7 @@ Sea \end_inset , luego -\begin_inset Formula $\gamma(s):=\beta(\frac{s}{c})$ +\begin_inset Formula $\gamma(s)\coloneqq \beta(\frac{s}{c})$ \end_inset es una geodésica y es p.p.a. @@ -2134,7 +2134,7 @@ Sea . Sea entonces -\begin_inset Formula $\tilde{h}(s):=h(\frac{s}{c})$ +\begin_inset Formula $\tilde{h}(s)\coloneqq h(\frac{s}{c})$ \end_inset , entonces @@ -110,7 +110,7 @@ aplicación exponencial \end_inset donde -\begin_inset Formula ${\cal D}_{p}:=\{v\in T_{p}S\mid 1\in I_{v}\}$ +\begin_inset Formula ${\cal D}_{p}\coloneqq \{v\in T_{p}S\mid 1\in I_{v}\}$ \end_inset . @@ -278,7 +278,7 @@ Como \end_inset , sea -\begin_inset Formula $\alpha(t):=tw$ +\begin_inset Formula $\alpha(t)\coloneqq tw$ \end_inset , existe @@ -377,11 +377,11 @@ entorno normal \end_inset , sean -\begin_inset Formula $v_{p}:=\exp_{p_{0}}^{-1}(p)\in{\cal U}$ +\begin_inset Formula $v_{p}\coloneqq \exp_{p_{0}}^{-1}(p)\in{\cal U}$ \end_inset y el segmento de geodésica -\begin_inset Formula $\gamma_{p}:=\gamma_{v_{p}}|_{[0,1]}:[0,1]\to V$ +\begin_inset Formula $\gamma_{p}\coloneqq \gamma_{v_{p}}|_{[0,1]}:[0,1]\to V$ \end_inset , entonces @@ -487,7 +487,7 @@ Demostración: \end_inset dada por -\begin_inset Formula $\alpha(t):=v+tw=(1+\lambda t)v$ +\begin_inset Formula $\alpha(t)\coloneqq v+tw=(1+\lambda t)v$ \end_inset , entonces @@ -512,7 +512,7 @@ Para el caso general, sea \end_inset dada por -\begin_inset Formula $\tau(s,t):=s\alpha(t):=s(v+tw)$ +\begin_inset Formula $\tau(s,t)\coloneqq s\alpha(t)\coloneqq s(v+tw)$ \end_inset , para todo @@ -591,7 +591,7 @@ Como . Proyectando el subrecubrimiento -\begin_inset Formula $A:=\bigcup_{i=1}^{k}B_{\infty}((s_{i},0),\varepsilon_{s_{i}})$ +\begin_inset Formula $A\coloneqq \bigcup_{i=1}^{k}B_{\infty}((s_{i},0),\varepsilon_{s_{i}})$ \end_inset en @@ -608,7 +608,7 @@ Como . Sea -\begin_inset Formula $\varepsilon:=\min\{\varepsilon_{s_{1}},\dots,\varepsilon_{s_{k}},\varepsilon'\}$ +\begin_inset Formula $\varepsilon\coloneqq \min\{\varepsilon_{s_{1}},\dots,\varepsilon_{s_{k}},\varepsilon'\}$ \end_inset , para @@ -632,7 +632,7 @@ luego \begin_layout Standard Sea ahora -\begin_inset Formula $\varphi:=\exp_{p}\circ\tau:(-\varepsilon,1+\varepsilon)\times(-\varepsilon,\varepsilon)\to S$ +\begin_inset Formula $\varphi\coloneqq \exp_{p}\circ\tau:(-\varepsilon,1+\varepsilon)\times(-\varepsilon,\varepsilon)\to S$ \end_inset . @@ -733,7 +733,7 @@ pues \end_inset dada por -\begin_inset Formula $\beta_{s}(t):=\exp_{p}(s\alpha(t))$ +\begin_inset Formula $\beta_{s}(t)\coloneqq \exp_{p}(s\alpha(t))$ \end_inset , @@ -909,7 +909,7 @@ Sean \end_inset tal que -\begin_inset Formula ${\cal D}(0,r):=\{v\in T_{p}S\mid \Vert v\Vert<r\}\subseteq{\cal D}_{p}$ +\begin_inset Formula ${\cal D}(0,r)\coloneqq \{v\in T_{p}S\mid \Vert v\Vert<r\}\subseteq{\cal D}_{p}$ \end_inset , llamamos @@ -925,7 +925,7 @@ disco geodésico \end_inset a -\begin_inset Formula $D(p,r):=\exp_{p}({\cal D}(0,r))$ +\begin_inset Formula $D(p,r)\coloneqq \exp_{p}({\cal D}(0,r))$ \end_inset , y si @@ -933,7 +933,7 @@ disco geodésico \end_inset cumple que -\begin_inset Formula ${\cal S}(0,r):=\{v\in T_{p}S\mid \Vert v\Vert=r\}\subseteq{\cal D}_{p}$ +\begin_inset Formula ${\cal S}(0,r)\coloneqq \{v\in T_{p}S\mid \Vert v\Vert=r\}\subseteq{\cal D}_{p}$ \end_inset , llamamos @@ -949,7 +949,7 @@ circunferencia geodésica \end_inset a -\begin_inset Formula $S(p,r):=\exp_{p}({\cal S}(0,r))$ +\begin_inset Formula $S(p,r)\coloneqq \exp_{p}({\cal S}(0,r))$ \end_inset . @@ -1042,7 +1042,7 @@ teorema Demostración: \series default Sea -\begin_inset Formula $v_{p}:=\exp_{p_{0}}^{-1}(p)$ +\begin_inset Formula $v_{p}\coloneqq \exp_{p_{0}}^{-1}(p)$ \end_inset , entonces @@ -1075,11 +1075,11 @@ Sea \begin_layout Standard Sean -\begin_inset Formula $A:=\alpha^{-1}(\{p_{0}\})$ +\begin_inset Formula $A\coloneqq \alpha^{-1}(\{p_{0}\})$ \end_inset y -\begin_inset Formula $t_{0}:=\sup A$ +\begin_inset Formula $t_{0}\coloneqq \sup A$ \end_inset , existe una sucesión @@ -1132,7 +1132,7 @@ Sean \end_inset , basta demostrar la propiedad para -\begin_inset Formula $\alpha:=\alpha'$ +\begin_inset Formula $\alpha\coloneqq \alpha'$ \end_inset . @@ -1148,7 +1148,7 @@ Sean \end_inset y -\begin_inset Formula $\tilde{\alpha}:=\exp_{p_{0}}^{-1}\circ\alpha:[0,1]\to{\cal U}$ +\begin_inset Formula $\tilde{\alpha}\coloneqq \exp_{p_{0}}^{-1}\circ\alpha:[0,1]\to{\cal U}$ \end_inset , que cumple @@ -1165,7 +1165,7 @@ Sean . Sean entonces -\begin_inset Formula $r(t):=\Vert\tilde{\alpha}(t)\Vert$ +\begin_inset Formula $r(t)\coloneqq \Vert\tilde{\alpha}(t)\Vert$ \end_inset y, para @@ -1173,7 +1173,7 @@ Sean \end_inset , -\begin_inset Formula $V(t):=\frac{\tilde{\alpha}(t)}{\Vert\tilde{\alpha}(t)\Vert}$ +\begin_inset Formula $V(t)\coloneqq \frac{\tilde{\alpha}(t)}{\Vert\tilde{\alpha}(t)\Vert}$ \end_inset , de modo que @@ -1392,7 +1392,7 @@ Finalmente, sea . En otro caso, sea -\begin_inset Formula $r^{*}:=\frac{r+\Vert v_{p}\Vert}{2}$ +\begin_inset Formula $r^{*}\coloneqq \frac{r+\Vert v_{p}\Vert}{2}$ \end_inset , de modo que @@ -1400,7 +1400,7 @@ Finalmente, sea \end_inset , y si -\begin_inset Formula $\tilde{\alpha}:=\exp_{p_{0}}^{-1}\circ\alpha$ +\begin_inset Formula $\tilde{\alpha}\coloneqq \exp_{p_{0}}^{-1}\circ\alpha$ \end_inset , como @@ -1422,7 +1422,7 @@ Finalmente, sea es \begin_inset Formula \[ -A:=\{t\in(a,b)\mid \Vert\tilde{\alpha}(t)\Vert=r^{*}\}=\{t\in[a,b]\mid \alpha(t)\in S(p_{0},r^{*})\}\neq\emptyset. +A:=\{t\in(a,b)\mid \Vert\tilde{\alpha}(t)\Vert=r^{*}\}=\{t\in[a,b]\mid\alpha(t)\in S(p_{0},r^{*})\}\neq\emptyset. \] \end_inset @@ -1436,11 +1436,11 @@ Entonces, como \end_inset también lo es y existe -\begin_inset Formula $t^{*}:=\min A$ +\begin_inset Formula $t^{*}\coloneqq \min A$ \end_inset , y llamando -\begin_inset Formula $p^{*}:=\alpha(t^{*})\in S(p_{0},r^{*})$ +\begin_inset Formula $p^{*}\coloneqq \alpha(t^{*})\in S(p_{0},r^{*})$ \end_inset , @@ -1492,7 +1492,7 @@ Sean \end_inset y -\begin_inset Formula $U:=\phi^{-1}({\cal U})$ +\begin_inset Formula $U\coloneqq \phi^{-1}({\cal U})$ \end_inset es abierto en @@ -1504,7 +1504,7 @@ Sean \end_inset dada por -\begin_inset Formula $X(u,v):=\exp_{p_{0}}(\phi(u,v))$ +\begin_inset Formula $X(u,v)\coloneqq \exp_{p_{0}}(\phi(u,v))$ \end_inset es una parametrización llamada @@ -1626,7 +1626,7 @@ Sean \end_inset , -\begin_inset Formula $\ell:=\{\lambda e_{1}\}_{\lambda\geq0}$ +\begin_inset Formula $\ell\coloneqq \{\lambda e_{1}\}_{\lambda\geq0}$ \end_inset , @@ -1642,11 +1642,11 @@ Sean \end_inset -\begin_inset Formula $V_{0}:=\exp_{p_{0}}({\cal U}\setminus\ell)$ +\begin_inset Formula $V_{0}\coloneqq \exp_{p_{0}}({\cal U}\setminus\ell)$ \end_inset y -\begin_inset Formula $U_{0}:=\phi^{-1}({\cal U}\setminus\ell)$ +\begin_inset Formula $U_{0}\coloneqq \phi^{-1}({\cal U}\setminus\ell)$ \end_inset , entonces @@ -1654,7 +1654,7 @@ Sean \end_inset dado por -\begin_inset Formula $X(r,\theta):=\exp_{p_{0}}(\phi(r,\theta))$ +\begin_inset Formula $X(r,\theta)\coloneqq \exp_{p_{0}}(\phi(r,\theta))$ \end_inset es una parametrización llamada @@ -1710,7 +1710,7 @@ teorema \begin_deeper \begin_layout Standard Sea -\begin_inset Formula $v_{\theta}:=(\cos\theta e_{1}+\sin\theta e_{2})$ +\begin_inset Formula $v_{\theta}\coloneqq (\cos\theta e_{1}+\sin\theta e_{2})$ \end_inset , de modo que @@ -1830,7 +1830,7 @@ Para un \begin_deeper \begin_layout Standard Sean -\begin_inset Formula $\overline{X}(u,v):=\exp_{p_{0}}(ue_{1}+ve_{2})$ +\begin_inset Formula $\overline{X}(u,v)\coloneqq \exp_{p_{0}}(ue_{1}+ve_{2})$ \end_inset la parametrización normal centrada en @@ -1846,7 +1846,7 @@ Sean \end_inset los parámetros de su primera forma fundamental, como -\begin_inset Formula $X(r,\theta)=\overline{X}(r_{\theta}):=\overline{X}(r\cos\theta,r\sin\theta)$ +\begin_inset Formula $X(r,\theta)=\overline{X}(r_{\theta})\coloneqq \overline{X}(r\cos\theta,r\sin\theta)$ \end_inset , se tiene @@ -1989,7 +1989,7 @@ Fijado \end_inset , sea -\begin_inset Formula $u(r):=\sqrt{G(r,\theta)}$ +\begin_inset Formula $u(r)\coloneqq \sqrt{G(r,\theta)}$ \end_inset , de modo que @@ -2212,11 +2212,11 @@ Demostración: \end_inset , -\begin_inset Formula $V_{1}:=D(p_{1},\varepsilon)$ +\begin_inset Formula $V_{1}\coloneqq D(p_{1},\varepsilon)$ \end_inset y -\begin_inset Formula $V_{2}:=D(p_{2},\varepsilon)$ +\begin_inset Formula $V_{2}\coloneqq D(p_{2},\varepsilon)$ \end_inset , entonces @@ -2252,11 +2252,11 @@ Sean ahora \end_inset una isometría lineal dada por -\begin_inset Formula $\tilde{\varphi}(e_{1}):=f_{1}$ +\begin_inset Formula $\tilde{\varphi}(e_{1})\coloneqq f_{1}$ \end_inset y -\begin_inset Formula $\tilde{\varphi}(e_{2}):=f_{2}$ +\begin_inset Formula $\tilde{\varphi}(e_{2})\coloneqq f_{2}$ \end_inset , entonces @@ -134,7 +134,7 @@ segmento de curva diferenciable a trozos \end_inset , -\begin_inset Formula $\alpha_{i}:=\alpha|_{[t_{i-1},t_{i}]}$ +\begin_inset Formula $\alpha_{i}\coloneqq \alpha|_{[t_{i-1},t_{i}]}$ \end_inset es un segmento de curva diferenciable. @@ -258,7 +258,7 @@ Demostración: \begin_layout Standard Primero vemos que -\begin_inset Formula $A:=\{q\in S\mid \Omega(p,q)\neq\emptyset\}=S$ +\begin_inset Formula $A\coloneqq \{q\in S\mid \Omega(p,q)\neq\emptyset\}=S$ \end_inset viendo que es abierto, cerrado y no vacío. @@ -468,7 +468,7 @@ Por la desigualdad de Cauchy-Schwarz, sean \end_inset y -\begin_inset Formula $v:=\frac{\overrightarrow{q-p}}{\Vert\overrightarrow{q-p}\Vert}$ +\begin_inset Formula $v\coloneqq \frac{\overrightarrow{q-p}}{\Vert\overrightarrow{q-p}\Vert}$ \end_inset , entonces @@ -682,7 +682,7 @@ Existe un entorno \end_inset estrellado respecto al 0 con -\begin_inset Formula $\exp_{p}:{\cal U}\to(V:=\exp_{p}({\cal U}))$ +\begin_inset Formula $\exp_{p}:{\cal U}\to(V\coloneqq \exp_{p}({\cal U}))$ \end_inset difeomorfismo, luego existe @@ -750,7 +750,7 @@ Queremos ver que \end_inset , existe -\begin_inset Formula $t^{*}:=\inf\{t\in[a,b]\mid \alpha(t)\notin D(p,r^{*})\}$ +\begin_inset Formula $t^{*}\coloneqq \inf\{t\in[a,b]\mid \alpha(t)\notin D(p,r^{*})\}$ \end_inset , pero @@ -157,7 +157,7 @@ Demostración: . Sea -\begin_inset Formula $w:=\alpha'(0)$ +\begin_inset Formula $w\coloneqq \alpha'(0)$ \end_inset , la geodésica maximal @@ -225,11 +225,11 @@ Demostración: . Sean -\begin_inset Formula $\tilde{\alpha}(t):=(\exp_{p_{0}}|_{{\cal U}})^{-1}(\alpha(t))\in{\cal U}$ +\begin_inset Formula $\tilde{\alpha}(t)\coloneqq (\exp_{p_{0}}|_{{\cal U}})^{-1}(\alpha(t))\in{\cal U}$ \end_inset y -\begin_inset Formula $A:=\{t\in[0,1]\mid \tilde{\alpha}(t)=tw\}$ +\begin_inset Formula $A\coloneqq \{t\in[0,1]\mid \tilde{\alpha}(t)=tw\}$ \end_inset , queremos ver que @@ -488,7 +488,7 @@ realiza la distancia \end_inset , -\begin_inset Formula $\alpha_{i}:=\alpha|_{[t_{i-1},t_{i}]}$ +\begin_inset Formula $\alpha_{i}\coloneqq \alpha|_{[t_{i-1},t_{i}]}$ \end_inset es diferenciable e @@ -525,7 +525,7 @@ Demostración: \end_inset por entornos convexos con cada -\begin_inset Formula $\alpha_{i}:=\alpha|_{[t_{i-1},t_{i}]}$ +\begin_inset Formula $\alpha_{i}\coloneqq \alpha|_{[t_{i-1},t_{i}]}$ \end_inset diferenciable con imagen en @@ -797,7 +797,7 @@ Demostración: \end_inset una sucesión de Cauchy y -\begin_inset Formula $A:=\{p_{n}\}_{n}$ +\begin_inset Formula $A\coloneqq \{p_{n}\}_{n}$ \end_inset su conjunto de puntos, para @@ -839,7 +839,7 @@ Demostración: . Tomando -\begin_inset Formula $r:=\max\{r_{0},d(p,p_{1}),\dots,d(p,p_{N-1})\}$ +\begin_inset Formula $r\coloneqq \max\{r_{0},d(p,p_{1}),\dots,d(p,p_{N-1})\}$ \end_inset , es @@ -931,7 +931,7 @@ completa \end_inset tales que -\begin_inset Formula $\gamma:=\gamma_{v}$ +\begin_inset Formula $\gamma\coloneqq \gamma_{v}$ \end_inset no está definida en todo @@ -1218,7 +1218,7 @@ Como . Sea -\begin_inset Formula $v:=\gamma'(0)$ +\begin_inset Formula $v\coloneqq \gamma'(0)$ \end_inset , entonces @@ -102,7 +102,7 @@ variación \end_inset con -\begin_inset Formula $\phi_{0}(u):=\phi(u,0)=\alpha(u)$ +\begin_inset Formula $\phi_{0}(u)\coloneqq \phi(u,0)=\alpha(u)$ \end_inset para todo @@ -118,7 +118,7 @@ Para \end_inset , llamamos -\begin_inset Formula $\alpha_{t}:=(u\mapsto\phi(u,t)):[a,b]\to S$ +\begin_inset Formula $\alpha_{t}\coloneqq (u\mapsto\phi(u,t)):[a,b]\to S$ \end_inset y @@ -139,7 +139,7 @@ curvas de la variación \end_inset , llamamos -\begin_inset Formula $\beta_{u}:=(t\mapsto\phi(u,t)):(-\varepsilon,\varepsilon)\to S$ +\begin_inset Formula $\beta_{u}\coloneqq (t\mapsto\phi(u,t)):(-\varepsilon,\varepsilon)\to S$ \end_inset y @@ -269,7 +269,7 @@ funcional longitud de arco \end_inset dada por -\begin_inset Formula $L(t):=L(\alpha_{t})$ +\begin_inset Formula $L(t)\coloneqq L(\alpha_{t})$ \end_inset . @@ -422,7 +422,7 @@ de modo que sea lo mayor posible. Si -\begin_inset Formula $\varepsilon_{0}:=\inf_{u[a,b]}\delta_{u}=0$ +\begin_inset Formula $\varepsilon_{0}\coloneqq \inf_{u[a,b]}\delta_{u}=0$ \end_inset , entonces existe una sucesión @@ -732,7 +732,7 @@ Caracterización variaciones de las geodésicas: \end_inset el campo tangente dado por -\begin_inset Formula $Z(s):=-(s^{2}-s(a+b)+ab)\frac{D\alpha'}{ds}(s)$ +\begin_inset Formula $Z(s)\coloneqq -(s^{2}-s(a+b)+ab)\frac{D\alpha'}{ds}(s)$ \end_inset , si existe una variación @@ -761,7 +761,7 @@ Caracterización variaciones de las geodésicas: . Además, -\begin_inset Formula $f(s):=s^{2}-s(a+b)+ab$ +\begin_inset Formula $f(s)\coloneqq s^{2}-s(a+b)+ab$ \end_inset es una parábola que vale 0 en @@ -866,7 +866,7 @@ No recuerdo haber visto este teorema. \end_inset existe -\begin_inset Formula $\varepsilon:=\min_{s\in[a,b]}\varepsilon_{s}>0$ +\begin_inset Formula $\varepsilon\coloneqq \min_{s\in[a,b]}\varepsilon_{s}>0$ \end_inset , @@ -882,7 +882,7 @@ No recuerdo haber visto este teorema. \end_inset como -\begin_inset Formula $\phi(s,t):=\gamma_{Z(s)}(t)$ +\begin_inset Formula $\phi(s,t)\coloneqq \gamma_{Z(s)}(t)$ \end_inset . @@ -156,7 +156,7 @@ Demostración: \end_inset , -\begin_inset Formula $h:=\overline{X}^{-1}\circ X$ +\begin_inset Formula $h\coloneqq \overline{X}^{-1}\circ X$ \end_inset la reparametrización y @@ -260,7 +260,7 @@ Dada una función \end_inset como -\begin_inset Formula $\det(d\phi)(p):=\det(J\phi_{p})$ +\begin_inset Formula $\det(d\phi)(p)\coloneqq \det(J\phi_{p})$ \end_inset . @@ -273,7 +273,7 @@ soporte \end_inset es -\begin_inset Formula $\text{sop}f:=\overline{\{x\in D\mid f(x)\neq0\}}$ +\begin_inset Formula $\text{sop}f\coloneqq \overline{\{x\in D\mid f(x)\neq0\}}$ \end_inset . @@ -318,7 +318,7 @@ Demostración \end_inset y -\begin_inset Formula $(U,\overline{X}:=\phi\circ X)$ +\begin_inset Formula $(U,\overline{X}\coloneqq \phi\circ X)$ \end_inset una parametrización de @@ -210,7 +210,7 @@ variación \end_inset tal que, llamando -\begin_inset Formula $\Phi_{t}(q):=\Phi(q,t)$ +\begin_inset Formula $\Phi_{t}(q)\coloneqq \Phi(q,t)$ \end_inset , @@ -383,7 +383,7 @@ y por continuidad existe \end_inset y entonces tomaríamos -\begin_inset Formula $\varepsilon:=\min_{(u,v)\in\text{sop}\varphi}\varepsilon_{u,v}$ +\begin_inset Formula $\varepsilon\coloneqq \min_{(u,v)\in\text{sop}\varphi}\varepsilon_{u,v}$ \end_inset . @@ -424,7 +424,7 @@ Sean \end_inset y -\begin_inset Formula $A(t):=A(R_{t}):=A(\Phi_{t}(X^{-1}(R)))$ +\begin_inset Formula $A(t)\coloneqq A(R_{t})\coloneqq A(\Phi_{t}(X^{-1}(R)))$ \end_inset , entonces @@ -589,7 +589,7 @@ Demostramos el contrarrecíproco. \end_inset es una región, de modo que llamando -\begin_inset Formula $\varphi:=H\circ X:R\to\mathbb{R}$ +\begin_inset Formula $\varphi\coloneqq H\circ X:R\to\mathbb{R}$ \end_inset , como @@ -161,7 +161,7 @@ Demostración: \end_inset dada por -\begin_inset Formula $h(t):=(f(t)-\cos\theta(t))^{2}+(g(t)-\sin\theta(t))^{2}$ +\begin_inset Formula $h(t)\coloneqq (f(t)-\cos\theta(t))^{2}+(g(t)-\sin\theta(t))^{2}$ \end_inset , entonces @@ -345,7 +345,7 @@ Teorema de Liouville: \end_inset , -\begin_inset Formula $\tilde{\alpha}:=(u,v):=X^{-1}\circ\alpha:I\to U$ +\begin_inset Formula $\tilde{\alpha}\coloneqq (u,v)\coloneqq X^{-1}\circ\alpha:I\to U$ \end_inset , @@ -373,7 +373,7 @@ e_{1}(s):=\frac{1}{\sqrt{E(\tilde{\alpha}(s))}}X_{u}(\tilde{\alpha}(s)), \end_inset , -\begin_inset Formula $\alpha_{v}(u):=\beta_{u}(v):=X(u,v)$ +\begin_inset Formula $\alpha_{v}(u)\coloneqq \beta_{u}(v)\coloneqq X(u,v)$ \end_inset , @@ -425,7 +425,7 @@ e_{1}(s) & =\frac{X_{u}}{\Vert X_{u}\Vert}(\tilde{\alpha}(s)). \end_inset Entonces -\begin_inset Formula $e_{2}(s):=Je_{1}(s)$ +\begin_inset Formula $e_{2}(s)\coloneqq Je_{1}(s)$ \end_inset es también tangente y unitario y ortogonal a @@ -451,7 +451,7 @@ Con esto, \end_inset luego si -\begin_inset Formula $\omega:=\langle e_{1}',e_{2}\rangle=-\langle e_{1},e_{2}'\rangle$ +\begin_inset Formula $\omega\coloneqq \langle e_{1}',e_{2}\rangle=-\langle e_{1},e_{2}'\rangle$ \end_inset @@ -709,7 +709,7 @@ velocidad que llega \end_inset es -\begin_inset Formula $\alpha'_{-}(\ell):=\lim_{s\to\ell^{-}}\alpha'(s)$ +\begin_inset Formula $\alpha'_{-}(\ell)\coloneqq \lim_{s\to\ell^{-}}\alpha'(s)$ \end_inset , y la @@ -814,11 +814,11 @@ Teorema de rotación de las tangentes: \end_inset el ángulo de rotación de la velocidad de -\begin_inset Formula $\alpha_{i}:=\alpha|_{[s_{i-1},s_{i}]}$ +\begin_inset Formula $\alpha_{i}\coloneqq \alpha|_{[s_{i-1},s_{i}]}$ \end_inset respecto a -\begin_inset Formula $e_{1}(s):=X_{u}(X^{-1}(\alpha(s)))/\sqrt{E(s)}$ +\begin_inset Formula $e_{1}(s)\coloneqq X_{u}(X^{-1}(\alpha(s)))/\sqrt{E(s)}$ \end_inset , entonces @@ -842,7 +842,7 @@ Teorema de Gauss-Bonnet Teorema de Green: \series default Sea -\begin_inset Formula $\tilde{\alpha}:=(u,v):[0,\ell]\to\mathbb{R}^{2}$ +\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 @@ -987,7 +987,7 @@ característica de Euler \end_inset es -\begin_inset Formula $\chi(T):=i_{0}-i_{1}+\dots+(-1)^{n}i_{n}$ +\begin_inset Formula $\chi(T)\coloneqq i_{0}-i_{1}+\dots+(-1)^{n}i_{n}$ \end_inset . |