Oops

1 files changed, 54 insertions, 54 deletions

diff --git a/gcs/n1.lyx b/gcs/n1.lyx
index 75dd0c2..d5a87e7 100644
--- a/gcs/n1.lyx
+++ b/gcs/n1.lyx
@@ -119,7 +119,7 @@ vector tangente
 \end_inset
 
  a 
-\begin_inset Formula $\alpha':=(\alpha_{1}',\dots,\alpha_{n}'):I\to\mathbb{R}^{n}$
+\begin_inset Formula $\alpha'\coloneqq (\alpha_{1}',\dots,\alpha_{n}'):I\to\mathbb{R}^{n}$
 \end_inset
 
 .
@@ -140,7 +140,7 @@ alabeada
 hélice cilíndrica
 \series default
 , 
-\begin_inset Formula $\alpha(t):=(a\cos t,a\sin t,bt)$
+\begin_inset Formula $\alpha(t)\coloneqq (a\cos t,a\sin t,bt)$
 \end_inset
 
  para ciertos 
@@ -215,7 +215,7 @@ cambio de parámetro
 \end_inset
 
 , y si tenemos una curva 
-\begin_inset Formula $\alpha:=I\to\mathbb{R}^{n}$
+\begin_inset Formula $\alpha\coloneqq I\to\mathbb{R}^{n}$
 \end_inset
 
 , llamamos 
@@ -231,7 +231,7 @@ reparametrización
 \end_inset
 
  a la curva 
-\begin_inset Formula $\beta:=\alpha\circ h:J\to\mathbb{R}^{n}$
+\begin_inset Formula $\beta\coloneqq \alpha\circ h:J\to\mathbb{R}^{n}$
 \end_inset
 
 .
@@ -300,7 +300,7 @@ Sean
 \end_inset
 
 , y una partición 
-\begin_inset Formula $P:=\{a=t_{0}<\dots<t_{m}=b\}$
+\begin_inset Formula $P\coloneqq \{a=t_{0}<\dots<t_{m}=b\}$
 \end_inset
 
 , llamamos 
@@ -316,7 +316,7 @@ longitud
 \end_inset
 
  a 
-\begin_inset Formula $L(\alpha,P):=\sum_{k=1}^{m}|\alpha(t_{k})-\alpha(t_{k-1})|$
+\begin_inset Formula $L(\alpha,P)\coloneqq \sum_{k=1}^{m}|\alpha(t_{k})-\alpha(t_{k-1})|$
 \end_inset
 
 , y longitud de 
@@ -375,7 +375,7 @@ L_{a}^{b}(\alpha)=\int_{a}^{b}|\alpha'(t)|dt.
 Demostración:
 \series default
  Sea 
-\begin_inset Formula $P:=\{a=t_{0}<\dots<t_{m}=b\}\in{\cal P}[a,b]$
+\begin_inset Formula $P\coloneqq \{a=t_{0}<\dots<t_{m}=b\}\in{\cal P}[a,b]$
 \end_inset
 
 , por el teorema de los valores intermedios, en cada 
@@ -391,7 +391,7 @@ Demostración:
 \end_inset
 
 , luego si 
-\begin_inset Formula $f(s_{1},\dots,s_{n}):=|(\alpha'_{1}(s_{1}),\dots,\alpha'_{n}(s_{n}))|$
+\begin_inset Formula $f(s_{1},\dots,s_{n})\coloneqq |(\alpha'_{1}(s_{1}),\dots,\alpha'_{n}(s_{n}))|$
 \end_inset
 
 ,
@@ -490,7 +490,7 @@ Con esto, la longitud de una curva es independiente de su parametrización,
 \end_inset
 
  es un cambio de parámetro que conserva la orientación y 
-\begin_inset Formula $\beta:=\alpha\circ h$
+\begin_inset Formula $\beta\coloneqq \alpha\circ h$
 \end_inset
 
 , 
@@ -556,7 +556,7 @@ p.p.a.
 \end_inset
 
  es un cambio de parámetro tal que 
-\begin_inset Formula $\beta:=\alpha\circ h$
+\begin_inset Formula $\beta\coloneqq \alpha\circ h$
 \end_inset
 
  es p.p.a, 
@@ -724,7 +724,7 @@ g(t):=\int_{t_{0}}^{t}|\alpha'(u)|du=L_{t_{0}}^{t}(\alpha),
 \end_inset
 
 , luego por el teorema de la función inversa, 
-\begin_inset Formula $J:=g(I)$
+\begin_inset Formula $J\coloneqq g(I)$
 \end_inset
 
  es abierto y 
@@ -733,7 +733,7 @@ g(t):=\int_{t_{0}}^{t}|\alpha'(u)|du=L_{t_{0}}^{t}(\alpha),
 
  es un difeomorfismo.
  Llamando 
-\begin_inset Formula $h:=g^{-1}$
+\begin_inset Formula $h\coloneqq g^{-1}$
 \end_inset
 
 , como 
@@ -769,11 +769,11 @@ catenaria
  distribuida uniformemente, suspendida por sus extremos y sometida a un
  campo gravitatorio uniforme.
  Se expresa como 
-\begin_inset Formula $\alpha(t):=(t,\cosh t)$
+\begin_inset Formula $\alpha(t)\coloneqq (t,\cosh t)$
 \end_inset
 
 , y admite una reparametrización por longitud de arco 
-\begin_inset Formula $\beta(s):=(\arg\sinh s,\sqrt{1+s^{2}})$
+\begin_inset Formula $\beta(s)\coloneqq (\arg\sinh s,\sqrt{1+s^{2}})$
 \end_inset
 
  de igual orientación.
@@ -790,7 +790,7 @@ g(t):=\int_{0}^{t}|\alpha'(u)|du=\int_{0}^{t}|(1,\sinh u)|du=\int_{0}^{t}\cosh u
 \end_inset
 
 entonces 
-\begin_inset Formula $h(s):=g^{-1}(s)=\arg\sinh s$
+\begin_inset Formula $h(s)\coloneqq g^{-1}(s)=\arg\sinh s$
 \end_inset
 
 , luego la reparametrización es 
@@ -803,7 +803,7 @@ entonces
 \end_deeper
 \begin_layout Enumerate
 Dada la circunferencia 
-\begin_inset Formula $\alpha(t):=p+(r\cos t,r\sin t)$
+\begin_inset Formula $\alpha(t)\coloneqq p+(r\cos t,r\sin t)$
 \end_inset
 
  para ciertos 
@@ -815,7 +815,7 @@ Dada la circunferencia
 \end_inset
 
 , la reparametrización por longitud de arco es 
-\begin_inset Formula $\beta(s):=p+(r\cos\frac{s}{r},r\sin\frac{s}{r})$
+\begin_inset Formula $\beta(s)\coloneqq p+(r\cos\frac{s}{r},r\sin\frac{s}{r})$
 \end_inset
 
 .
@@ -831,7 +831,7 @@ g(t):=\int_{0}^{t}|\alpha'(u)|du=\int_{0}^{t}r\,du=rt,
 \end_inset
 
 luego 
-\begin_inset Formula $h(s):=g^{-1}(s)=\frac{s}{r}$
+\begin_inset Formula $h(s)\coloneqq g^{-1}(s)=\frac{s}{r}$
 \end_inset
 
  y la reparametrización es 
@@ -874,11 +874,11 @@ 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
 
  es su 
@@ -1016,7 +1016,7 @@ Si
 radio de curvatura
 \series default
  a 
-\begin_inset Formula $\rho(s):=\frac{1}{|\kappa(s)|}$
+\begin_inset Formula $\rho(s)\coloneqq \frac{1}{|\kappa(s)|}$
 \end_inset
 
 .
@@ -1042,7 +1042,7 @@ El radio de curvatura de una circunferencia de radio
 \begin_deeper
 \begin_layout Standard
 Sea 
-\begin_inset Formula $\alpha(s):=p+r(\cos\frac{s}{r},\sin\frac{s}{r})$
+\begin_inset Formula $\alpha(s)\coloneqq p+r(\cos\frac{s}{r},\sin\frac{s}{r})$
 \end_inset
 
  con 
@@ -1080,7 +1080,7 @@ La curvatura de una recta es 0.
 \begin_deeper
 \begin_layout Standard
 Sea 
-\begin_inset Formula $\alpha(s):=p+sv$
+\begin_inset Formula $\alpha(s)\coloneqq p+sv$
 \end_inset
 
  para ciertos 
@@ -1109,7 +1109,7 @@ Sea
 \end_deeper
 \begin_layout Enumerate
 La catenaria 
-\begin_inset Formula $\alpha(s):=(\arg\sinh s,\sqrt{1+s^{2}})$
+\begin_inset Formula $\alpha(s)\coloneqq (\arg\sinh s,\sqrt{1+s^{2}})$
 \end_inset
 
  tiene radio de curvatura 
@@ -1277,7 +1277,7 @@ movimiento rígido
 \end_inset
 
  dada por 
-\begin_inset Formula $M(x):=Ax+b$
+\begin_inset Formula $M(x)\coloneqq Ax+b$
 \end_inset
 
  para ciertos 
@@ -1326,7 +1326,7 @@ Sean
 \end_inset
 
  dada por 
-\begin_inset Formula $\varphi(s):=\int_{s_{0}}^{s}\kappa$
+\begin_inset Formula $\varphi(s)\coloneqq \int_{s_{0}}^{s}\kappa$
 \end_inset
 
  y 
@@ -1424,15 +1424,15 @@ Sean
 
 .
  Sean entonces 
-\begin_inset Formula $b:=\beta(s_{0})-A\alpha(s_{0})$
+\begin_inset Formula $b\coloneqq \beta(s_{0})-A\alpha(s_{0})$
 \end_inset
 
 , 
-\begin_inset Formula $Mx:=Ax+b$
+\begin_inset Formula $Mx\coloneqq Ax+b$
 \end_inset
 
  un movimiento rígido y 
-\begin_inset Formula $\gamma:=M\circ\alpha$
+\begin_inset Formula $\gamma\coloneqq M\circ\alpha$
 \end_inset
 
 , y queremos ver que 
@@ -1449,7 +1449,7 @@ Sean
 \end_inset
 
 , luego si 
-\begin_inset Formula $f(s):=\frac{1}{2}|t_{\beta}(s)-t_{\gamma}(s)|^{2}$
+\begin_inset Formula $f(s)\coloneqq \frac{1}{2}|t_{\beta}(s)-t_{\gamma}(s)|^{2}$
 \end_inset
 
 , entonces 
@@ -1516,7 +1516,7 @@ Dados una curva regular
 \end_inset
 
  que preserva la orientación tal que 
-\begin_inset Formula $\beta:=\alpha\circ h$
+\begin_inset Formula $\beta\coloneqq \alpha\circ h$
 \end_inset
 
  es p.p.a., llamamos 
@@ -1565,7 +1565,7 @@ Demostración:
 \end_inset
 
 , luego para 
-\begin_inset Formula $s:=h^{-1}(t)$
+\begin_inset Formula $s\coloneqq h^{-1}(t)$
 \end_inset
 
 ,
@@ -1602,19 +1602,19 @@ Sean
 \end_inset
 
 , 
-\begin_inset Formula $p_{0}:=\alpha(s_{0})$
+\begin_inset Formula $p_{0}\coloneqq \alpha(s_{0})$
 \end_inset
 
 , 
-\begin_inset Formula $\mathbf{t}_{0}:=\mathbf{t}(s_{0})$
+\begin_inset Formula $\mathbf{t}_{0}\coloneqq \mathbf{t}(s_{0})$
 \end_inset
 
 , 
-\begin_inset Formula $\mathbf{n}_{0}:=\mathbf{n}(s_{0})$
+\begin_inset Formula $\mathbf{n}_{0}\coloneqq \mathbf{n}(s_{0})$
 \end_inset
 
 , 
-\begin_inset Formula $\ell:=p_{0}+\langle\mathbf{t}_{0}\rangle$
+\begin_inset Formula $\ell\coloneqq p_{0}+\langle\mathbf{t}_{0}\rangle$
 \end_inset
 
  y 
@@ -1634,7 +1634,7 @@ distancia orientada
 \end_inset
 
  a 
-\begin_inset Formula $\text{dist}(p,\ell):=\langle p-p_{0},\mathbf{n}_{0}\rangle$
+\begin_inset Formula $\text{dist}(p,\ell)\coloneqq \langle p-p_{0},\mathbf{n}_{0}\rangle$
 \end_inset
 
 .
@@ -1647,11 +1647,11 @@ distancia orientada
 \end_inset
 
  en dos semiplanos 
-\begin_inset Formula $H^{+}:=\{p\mid \text{dist}(p,\ell)\geq0\}$
+\begin_inset Formula $H^{+}\coloneqq \{p\mid \text{dist}(p,\ell)\geq0\}$
 \end_inset
 
  y 
-\begin_inset Formula $H^{-}:=\{p\mid \text{dist}(p,\ell)\leq0\}$
+\begin_inset Formula $H^{-}\coloneqq \{p\mid \text{dist}(p,\ell)\leq0\}$
 \end_inset
 
 , de modo que 
@@ -1685,7 +1685,7 @@ Si
 \begin_deeper
 \begin_layout Standard
 Sea 
-\begin_inset Formula $f(s):=\text{dist}(\alpha(s),\ell)=\langle\alpha(s)-p_{0},\mathbf{n}_{0}\rangle$
+\begin_inset Formula $f(s)\coloneqq \text{dist}(\alpha(s),\ell)=\langle\alpha(s)-p_{0},\mathbf{n}_{0}\rangle$
 \end_inset
 
 , entonces 
@@ -1929,7 +1929,7 @@ Sean
 \end_inset
 
  y 
-\begin_inset Formula $f(s):=\text{dist}(\alpha(s),\ell)-\text{dist}(\beta(s),\ell)=\langle\alpha(s)-p_{0},\mathbf{n}_{0}\rangle-\langle\beta(s)-p_{0},\mathbf{n}_{0}\rangle=\langle\alpha(s)-\beta(s),\mathbf{n}_{0}\rangle$
+\begin_inset Formula $f(s)\coloneqq \text{dist}(\alpha(s),\ell)-\text{dist}(\beta(s),\ell)=\langle\alpha(s)-p_{0},\mathbf{n}_{0}\rangle-\langle\beta(s)-p_{0},\mathbf{n}_{0}\rangle=\langle\alpha(s)-\beta(s),\mathbf{n}_{0}\rangle$
 \end_inset
 
 , entonces 
@@ -2065,7 +2065,7 @@ curvatura
 \end_inset
 
  a 
-\begin_inset Formula $\kappa(s):=|\mathbf{t}'(s)|$
+\begin_inset Formula $\kappa(s)\coloneqq |\mathbf{t}'(s)|$
 \end_inset
 
 .
@@ -2147,7 +2147,7 @@ torsión
 \end_inset
 
  dada por 
-\begin_inset Formula $\tau(s):=\langle\mathbf{t}(s)\land\mathbf{n}'(s),\mathbf{n}(s)\rangle=\langle\mathbf{b}'(s),\mathbf{n}(s)\rangle$
+\begin_inset Formula $\tau(s)\coloneqq \langle\mathbf{t}(s)\land\mathbf{n}'(s),\mathbf{n}(s)\rangle=\langle\mathbf{b}'(s),\mathbf{n}(s)\rangle$
 \end_inset
 
 .
@@ -2321,7 +2321,7 @@ status open
 \end_inset
 
 Si 
-\begin_inset Formula $\alpha(s):=p+sv$
+\begin_inset Formula $\alpha(s)\coloneqq p+sv$
 \end_inset
 
 , 
@@ -2472,7 +2472,7 @@ Si
 \end_inset
 
  y, si 
-\begin_inset Formula $f(s):=\langle\alpha(s),\mathbf{b}(s)\rangle$
+\begin_inset Formula $f(s)\coloneqq \langle\alpha(s),\mathbf{b}(s)\rangle$
 \end_inset
 
 , 
@@ -2513,7 +2513,7 @@ Sean
 \end_inset
 
  un cambio de parámetro que conserva la orientación y tal que 
-\begin_inset Formula $\beta:=\alpha\circ h$
+\begin_inset Formula $\beta\coloneqq \alpha\circ h$
 \end_inset
 
  es p.p.a., definimos la curvatura de 
@@ -2521,11 +2521,11 @@ Sean
 \end_inset
 
  como 
-\begin_inset Formula $\kappa_{\alpha}(t):=\kappa_{\beta}(h^{-1}(t))$
+\begin_inset Formula $\kappa_{\alpha}(t)\coloneqq \kappa_{\beta}(h^{-1}(t))$
 \end_inset
 
  y, si esta no se anula, la torsión como 
-\begin_inset Formula $\tau_{\alpha}(t):=\tau_{\beta}(h^{-1}(t))$
+\begin_inset Formula $\tau_{\alpha}(t)\coloneqq \tau_{\beta}(h^{-1}(t))$
 \end_inset
 
 .
@@ -2542,7 +2542,7 @@ Sean
 Demostración:
 \series default
  Sea 
-\begin_inset Formula $s:=h^{-1}(t)$
+\begin_inset Formula $s\coloneqq h^{-1}(t)$
 \end_inset
 
 ,
@@ -2792,7 +2792,7 @@ Sea entonces
 \end_inset
 
  la curva dada por 
-\begin_inset Formula $\alpha(s):=\int_{s_{0}}^{s}\mathbf{t}(u)du$
+\begin_inset Formula $\alpha(s)\coloneqq \int_{s_{0}}^{s}\mathbf{t}(u)du$
 \end_inset
 
 , para todo 
@@ -2883,15 +2883,15 @@ Sean
 
 .
  Sean entonces 
-\begin_inset Formula $b:=\beta(s_{0})-A\alpha(s_{0})$
+\begin_inset Formula $b\coloneqq \beta(s_{0})-A\alpha(s_{0})$
 \end_inset
 
 , 
-\begin_inset Formula $M(x):=Ax+b$
+\begin_inset Formula $M(x)\coloneqq Ax+b$
 \end_inset
 
  un movimiento rígido y 
-\begin_inset Formula $\gamma:=M\circ\alpha$
+\begin_inset Formula $\gamma\coloneqq M\circ\alpha$
 \end_inset
 
 , y queremos ver que 
@@ -2918,7 +2918,7 @@ Se tiene
 \end_inset
 
 Sea ahora 
-\begin_inset Formula $f(s):=\frac{1}{2}(|\mathbf{t}_{\beta}(s)-\mathbf{t}_{\gamma}(s)|^{2}+|\mathbf{n}_{\beta}(s)-\mathbf{n}_{\gamma}(s)|^{2}+|\mathbf{b}_{\beta}(s)-\mathbf{b}_{\gamma}(s)|^{2})$
+\begin_inset Formula $f(s)\coloneqq \frac{1}{2}(|\mathbf{t}_{\beta}(s)-\mathbf{t}_{\gamma}(s)|^{2}+|\mathbf{n}_{\beta}(s)-\mathbf{n}_{\gamma}(s)|^{2}+|\mathbf{b}_{\beta}(s)-\mathbf{b}_{\gamma}(s)|^{2})$
 \end_inset
 
 , entonces