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-rw-r--r--ggs/n2.lyx74
1 files changed, 37 insertions, 37 deletions
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
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-26 23:32:39 +0000