From c34b47089a133e58032fe4ea52f61efacaf5f548 Mon Sep 17 00:00:00 2001
From: Juan Marin Noguera <juan@mnpi.eu>
Date: Sun, 4 Dec 2022 22:49:17 +0100
Subject: Oops

---
 ggs/n3.lyx | 80 +++++++++++++++++++++++++++++++-------------------------------
 1 file changed, 40 insertions(+), 40 deletions(-)

(limited to 'ggs/n3.lyx')

diff --git a/ggs/n3.lyx b/ggs/n3.lyx
index f553749..28601db 100644
--- a/ggs/n3.lyx
+++ b/ggs/n3.lyx
@@ -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 
-- 
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