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/n1.lyx | 30 +++++++++++------------
 ggs/n2.lyx | 74 ++++++++++++++++++++++++++++-----------------------------
 ggs/n3.lyx | 80 +++++++++++++++++++++++++++++++-------------------------------
 ggs/n4.lyx | 10 ++++----
 ggs/n5.lyx | 18 +++++++-------
 ggs/n6.lyx | 18 +++++++-------
 ggs/n7.lyx |  8 +++----
 ggs/n8.lyx |  8 +++----
 ggs/n9.lyx | 20 ++++++++--------
 9 files changed, 133 insertions(+), 133 deletions(-)

(limited to 'ggs')

diff --git a/ggs/n1.lyx b/ggs/n1.lyx
index b2f3a82..3cd6e5a 100644
--- a/ggs/n1.lyx
+++ b/ggs/n1.lyx
@@ -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 
diff --git a/ggs/n2.lyx b/ggs/n2.lyx
index 09e8555..6c3a57b 100644
--- a/ggs/n2.lyx
+++ b/ggs/n2.lyx
@@ -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 
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 
diff --git a/ggs/n4.lyx b/ggs/n4.lyx
index f6e22f9..dab3566 100644
--- a/ggs/n4.lyx
+++ b/ggs/n4.lyx
@@ -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 
diff --git a/ggs/n5.lyx b/ggs/n5.lyx
index 00374b2..30b502f 100644
--- a/ggs/n5.lyx
+++ b/ggs/n5.lyx
@@ -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
diff --git a/ggs/n6.lyx b/ggs/n6.lyx
index d6b961e..7993f48 100644
--- a/ggs/n6.lyx
+++ b/ggs/n6.lyx
@@ -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
 
 .
diff --git a/ggs/n7.lyx b/ggs/n7.lyx
index 142e739..309418d 100644
--- a/ggs/n7.lyx
+++ b/ggs/n7.lyx
@@ -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 
diff --git a/ggs/n8.lyx b/ggs/n8.lyx
index 4bda440..141b0b3 100644
--- a/ggs/n8.lyx
+++ b/ggs/n8.lyx
@@ -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 
diff --git a/ggs/n9.lyx b/ggs/n9.lyx
index 5d0cf43..3f506e8 100644
--- a/ggs/n9.lyx
+++ b/ggs/n9.lyx
@@ -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
 
 .
-- 
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