Oops

4 files changed, 35 insertions, 35 deletions

diff --git a/fvv1/n1.lyx b/fvv1/n1.lyx
index e3422b2..8eede13 100644
--- a/fvv1/n1.lyx
+++ b/fvv1/n1.lyx
@@ -137,7 +137,7 @@ espacio normado
 distancia asociada a la norma
 \series default
  a 
-\begin_inset Formula $d(x,y):=\Vert x-y\Vert$
+\begin_inset Formula $d(x,y)\coloneqq \Vert x-y\Vert$
 \end_inset
 
 .
@@ -150,7 +150,7 @@ Ejemplos de normas en
 \end_inset
 
  son las dadas por 
-\begin_inset Formula $\Vert(x_{1},\dots,x_{n})\Vert_{p}:=\sqrt[p]{\sum_{i=1}^{n}|x_{i}|^{p}}$
+\begin_inset Formula $\Vert(x_{1},\dots,x_{n})\Vert_{p}\coloneqq \sqrt[p]{\sum_{i=1}^{n}|x_{i}|^{p}}$
 \end_inset
 
  y
@@ -158,16 +158,16 @@ Ejemplos de normas en
 \end_inset
 
 
-\begin_inset Formula $\Vert(x_{1},\dots,x_{n})\Vert_{\infty}:=\max\{|x_{i}|\}_{i=1}^{n}$
+\begin_inset Formula $\Vert(x_{1},\dots,x_{n})\Vert_{\infty}\coloneqq \max\{|x_{i}|\}_{i=1}^{n}$
 \end_inset
 
 .
  Además, 
-\begin_inset Formula $V:={\cal C}[a,b]:=\{f\mid [a,b]\rightarrow\mathbb{R}\text{ continua}\}$
+\begin_inset Formula $V\coloneqq {\cal C}[a,b]\coloneqq \{f\mid [a,b]\rightarrow\mathbb{R}\text{ continua}\}$
 \end_inset
 
  con 
-\begin_inset Formula $\Vert f\Vert_{\infty}:=\sup\{|f(x)|\}_{x\in[a,b]}$
+\begin_inset Formula $\Vert f\Vert_{\infty}\coloneqq \sup\{|f(x)|\}_{x\in[a,b]}$
 \end_inset
 
  es un espacio normado.
@@ -274,7 +274,7 @@ Definimos la norma de una aplicación
 \end_inset
 
  como 
-\begin_inset Formula $\Vert L\Vert:=\Vert L\Vert_{\Vert\cdot\Vert}^{\Vert\cdot\Vert'}:=\sup\{\Vert L(x)\Vert'\}_{x\in E,\Vert x\Vert\leq1}$
+\begin_inset Formula $\Vert L\Vert\coloneqq \Vert L\Vert_{\Vert\cdot\Vert}^{\Vert\cdot\Vert'}\coloneqq \sup\{\Vert L(x)\Vert'\}_{x\in E,\Vert x\Vert\leq1}$
 \end_inset
 
 , y tenemos como 
@@ -377,7 +377,7 @@ Veamos primero que
 \end_inset
 
 , tomando 
-\begin_inset Formula $\delta:=\frac{\varepsilon}{\Vert L\Vert+1}$
+\begin_inset Formula $\delta\coloneqq \frac{\varepsilon}{\Vert L\Vert+1}$
 \end_inset
 
  entonces 
@@ -423,11 +423,11 @@ Dos normas
 Demostración:
 \series default
  Sean 
-\begin_inset Formula $L:=id_{E}:(E,\Vert\cdot\Vert)\rightarrow(E,\Vert\cdot\Vert')$
+\begin_inset Formula $L\coloneqq id_{E}:(E,\Vert\cdot\Vert)\rightarrow(E,\Vert\cdot\Vert')$
 \end_inset
 
  y 
-\begin_inset Formula $L':=L^{-1}$
+\begin_inset Formula $L'\coloneqq L^{-1}$
 \end_inset
 
 , entonces 
@@ -472,7 +472,7 @@ Si
 \end_inset
 
 , luego 
-\begin_inset Formula $\Vert x\Vert'=\Vert L(x)\Vert'\leq\Vert L\Vert\Vert x\Vert\overset{\beta:=\Vert L\Vert}{=}\beta\Vert x\Vert$
+\begin_inset Formula $\Vert x\Vert'=\Vert L(x)\Vert'\leq\Vert L\Vert\Vert x\Vert\overset{\beta\coloneqq \Vert L\Vert}{=}\beta\Vert x\Vert$
 \end_inset
 
 .
@@ -644,7 +644,7 @@ Demostración:
 \end_inset
 
  y tomando 
-\begin_inset Formula $\delta:=\varepsilon$
+\begin_inset Formula $\delta\coloneqq \varepsilon$
 \end_inset
 
 , si 
@@ -706,7 +706,7 @@ teorema
 , que es continua por ser composición de dos funciones continuas (la identidad
  es continua por la otra cota y la demostración del teorema anterior), entonces
  
-\begin_inset Formula $S:=\{x\in\mathbb{R}^{n}\mid \Vert x\Vert_{1}=1\}$
+\begin_inset Formula $S\coloneqq \{x\in\mathbb{R}^{n}\mid \Vert x\Vert_{1}=1\}$
 \end_inset
 
  es cerrado dentro del compacto 
@@ -719,7 +719,7 @@ teorema
 
  es compacto y alcanza su máximo y su mínimo.
  Sea ahora 
-\begin_inset Formula $\mu:=\min\{\Vert x\Vert\}_{x\in S}>0$
+\begin_inset Formula $\mu\coloneqq \min\{\Vert x\Vert\}_{x\in S}>0$
 \end_inset
 
  (pues 
@@ -1338,7 +1338,7 @@ Dadas
 \end_inset
 
  y existe 
-\begin_inset Formula $l:=\lim_{n}\frac{a_{n}}{b_{n}}\in\mathbb{R}\cup\{+\infty\}$
+\begin_inset Formula $l\coloneqq \lim_{n}\frac{a_{n}}{b_{n}}\in\mathbb{R}\cup\{+\infty\}$
 \end_inset
 
 :
@@ -1403,7 +1403,7 @@ Criterio de la raíz:
 \end_inset
 
  y 
-\begin_inset Formula $a:=\lim_{n}\sqrt[n]{a_{n}}\in\mathbb{R}$
+\begin_inset Formula $a\coloneqq \lim_{n}\sqrt[n]{a_{n}}\in\mathbb{R}$
 \end_inset
 
 :
@@ -1449,7 +1449,7 @@ Criterio del cociente:
 \end_inset
 
  y 
-\begin_inset Formula $a:=\lim_{n}\frac{a_{n+1}}{a_{n}}\in\mathbb{R}$
+\begin_inset Formula $a\coloneqq \lim_{n}\frac{a_{n+1}}{a_{n}}\in\mathbb{R}$
 \end_inset
 
 .
diff --git a/fvv1/n2.lyx b/fvv1/n2.lyx
index 1b761e2..f9a9447 100644
--- a/fvv1/n2.lyx
+++ b/fvv1/n2.lyx
@@ -542,7 +542,7 @@ suma telescópica
 
  se supone lo suficientemente pequeño.
  Ahora llamamos 
-\begin_inset Formula $\varphi_{i}(t):=f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1}+t\vec{e}_{i})$
+\begin_inset Formula $\varphi_{i}(t)\coloneqq f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1}+t\vec{e}_{i})$
 \end_inset
 
 , con lo que 
@@ -550,7 +550,7 @@ suma telescópica
 \end_inset
 
 , y 
-\begin_inset Formula $\Delta_{i}:=\varphi_{i}(h_{i})-\varphi_{i}(0)=f(a+h_{1}\vec{e}_{1}+\dots+h_{i}\vec{e}_{i})-f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1})=\varphi'_{i}(\xi_{i})h_{i}$
+\begin_inset Formula $\Delta_{i}\coloneqq \varphi_{i}(h_{i})-\varphi_{i}(0)=f(a+h_{1}\vec{e}_{1}+\dots+h_{i}\vec{e}_{i})-f(a+h_{1}\vec{e}_{1}+\dots+h_{i-1}\vec{e}_{i-1})=\varphi'_{i}(\xi_{i})h_{i}$
 \end_inset
 
  para algún 
@@ -662,11 +662,11 @@ regla de la cadena
 Demostración:
 \series default
  Sean 
-\begin_inset Formula $L:=df(a):\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
+\begin_inset Formula $L\coloneqq df(a):\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$
 \end_inset
 
  y 
-\begin_inset Formula $S:=dg(f(a)):\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$
+\begin_inset Formula $S\coloneqq dg(f(a)):\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$
 \end_inset
 
 , tenemos que 
@@ -686,7 +686,7 @@ y queremos ver que
 \end_inset
 
 Si llamamos 
-\begin_inset Formula $\eta:=f(a+h)-f(a)$
+\begin_inset Formula $\eta\coloneqq f(a+h)-f(a)$
 \end_inset
 
 , que tiende a 0 por la continuidad de 
@@ -897,17 +897,17 @@ to por abiertos de
 \end_inset
 
  y 
-\begin_inset Formula $\{B_{i}\}_{i=1}^{k}\mid =\{B(x_{i},\frac{\delta_{x_{i}}}{2})\}_{i=1}^{k}$
+\begin_inset Formula $\{B_{i}\}_{i=1}^{k}\coloneqq \{B(x_{i},\frac{\delta_{x_{i}}}{2})\}_{i=1}^{k}$
 \end_inset
 
  un subrecubrimiento finito del que suponemos que no podemos quitar ninguna
  bola.
  Ahora llamamos 
-\begin_inset Formula $x_{0}:=a$
+\begin_inset Formula $x_{0}\coloneqq a$
 \end_inset
 
  y 
-\begin_inset Formula $x_{k+1}:=b$
+\begin_inset Formula $x_{k+1}\coloneqq b$
 \end_inset
 
  y suponemos 
diff --git a/fvv1/n3.lyx b/fvv1/n3.lyx
index 91f5019..b6138cf 100644
--- a/fvv1/n3.lyx
+++ b/fvv1/n3.lyx
@@ -343,12 +343,12 @@ Demostración:
 \end_inset
 
 , y consideramos 
-\begin_inset Formula $\Delta_{t,s}:=f(t,s)-f(t,y_{0})-f(x_{0},s)+f(x_{0},y_{0})$
+\begin_inset Formula $\Delta_{t,s}\coloneqq f(t,s)-f(t,y_{0})-f(x_{0},s)+f(x_{0},y_{0})$
 \end_inset
 
 .
  Si ahora llamamos 
-\begin_inset Formula $F_{\overline{s}}(\overline{t}):=f(\overline{t},\overline{s})-f(\overline{t},y_{0})$
+\begin_inset Formula $F_{\overline{s}}(\overline{t})\coloneqq f(\overline{t},\overline{s})-f(\overline{t},y_{0})$
 \end_inset
 
 , vemos que 
@@ -360,12 +360,12 @@ Demostración:
 \end_inset
 
  y que entonces 
-\begin_inset Formula $\Delta_{t,s}=F_{s}(t)-F_{s}(x_{0})=F'_{\overline{s}}(\xi_{t,s})(t-x_{0})=\left(\frac{\partial f}{\partial x}(\xi_{t,s},s)-\frac{\partial f}{\partial x}(\xi_{t,s},y_{0})\right)(t-x_{0})\overset{\Phi(\overline{s}):=\frac{\partial f}{\partial x}(\xi_{t,s},\overline{s})}{=}(\Phi(s)-\Phi(y_{0}))(t-x_{0})\overset{\Phi\text{ derivable por hipótesis}}{=}\Phi'(\eta_{t,s})(s-y_{0})(t-x_{0})=\frac{\partial^{2}f}{\partial x\partial y}(\xi_{t,s},\eta_{t,s})(s-y_{0})(t-x_{0})$
+\begin_inset Formula $\Delta_{t,s}=F_{s}(t)-F_{s}(x_{0})=F'_{\overline{s}}(\xi_{t,s})(t-x_{0})=\left(\frac{\partial f}{\partial x}(\xi_{t,s},s)-\frac{\partial f}{\partial x}(\xi_{t,s},y_{0})\right)(t-x_{0})\overset{\Phi(\overline{s})\coloneqq \frac{\partial f}{\partial x}(\xi_{t,s},\overline{s})}{=}(\Phi(s)-\Phi(y_{0}))(t-x_{0})\overset{\Phi\text{ derivable por hipótesis}}{=}\Phi'(\eta_{t,s})(s-y_{0})(t-x_{0})=\frac{\partial^{2}f}{\partial x\partial y}(\xi_{t,s},\eta_{t,s})(s-y_{0})(t-x_{0})$
 \end_inset
 
 .
  Permutando los papeles de las dos coordenadas (definiendo 
-\begin_inset Formula $\sigma_{\overline{t}}(\overline{s}):=f(\overline{t},\overline{s})-f(x,\overline{s})$
+\begin_inset Formula $\sigma_{\overline{t}}(\overline{s})\coloneqq f(\overline{t},\overline{s})-f(x,\overline{s})$
 \end_inset
 
 ) obtenemos que 
@@ -448,7 +448,7 @@ teorema
 Demostración:
 \series default
  Sea 
-\begin_inset Formula $R(h):=f(a+h)-f(a)-df(a)(h)-\frac{1}{2}d^{2}f(a)(h,h)$
+\begin_inset Formula $R(h)\coloneqq f(a+h)-f(a)-df(a)(h)-\frac{1}{2}d^{2}f(a)(h,h)$
 \end_inset
 
 , y hemos de ver que 
@@ -631,7 +631,7 @@ Si
 \end_inset
 
 -espacio vectorial con 
-\begin_inset Formula $k:=\dim_{K}(V)<+\infty$
+\begin_inset Formula $k\coloneqq \dim_{K}(V)<+\infty$
 \end_inset
 
  y 
@@ -751,7 +751,7 @@ Podemos suponer que alcanza un máximo.
 \end_inset
 
  definimos 
-\begin_inset Formula $\varphi_{i}(t):=f(a_{1},\dots,a_{i-1},t,a_{i+1},\dots,a_{m})$
+\begin_inset Formula $\varphi_{i}(t)\coloneqq f(a_{1},\dots,a_{i-1},t,a_{i+1},\dots,a_{m})$
 \end_inset
 
 , fijado 
@@ -848,7 +848,7 @@ suponiendo
 \end_inset
 
  dada por 
-\begin_inset Formula $\Phi(u):=d^{2}f(a)(u,u)=\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)u_{i}u_{j}$
+\begin_inset Formula $\Phi(u)\coloneqq d^{2}f(a)(u,u)=\sum_{i,j}\frac{\partial f}{\partial x_{i}\partial x_{j}}(a)u_{i}u_{j}$
 \end_inset
 
  es continua, luego 
@@ -914,7 +914,7 @@ Como
 \end_inset
 
  y definimos 
-\begin_inset Formula $\varphi(t):=a+tu$
+\begin_inset Formula $\varphi(t)\coloneqq a+tu$
 \end_inset
 
  como la función 
diff --git a/fvv1/n4.lyx b/fvv1/n4.lyx
index f95baae..2edff4c 100644
--- a/fvv1/n4.lyx
+++ b/fvv1/n4.lyx
@@ -341,7 +341,7 @@ gradiente
 \end_inset
 
  al vector 
-\begin_inset Formula $\nabla f(a):=\left(\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{\partial x_{n}}\right)\in\mathbb{R}^{n}$
+\begin_inset Formula $\nabla f(a)\coloneqq \left(\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{\partial x_{n}}\right)\in\mathbb{R}^{n}$
 \end_inset
 
 , la matriz de la diferencial de 
@@ -495,7 +495,7 @@ teorema de los multiplicadores de Lagrange
 \end_inset
 
 , entonces 
-\begin_inset Formula $\nabla f(a)\in\text{span}(\nabla g_{1}(a),\dots,\nabla g_{k}(a)):=<\nabla g_{1}(a),\dots,\nabla g_{k}(a)>$
+\begin_inset Formula $\nabla f(a)\in\text{span}(\nabla g_{1}(a),\dots,\nabla g_{k}(a))\coloneqq <\nabla g_{1}(a),\dots,\nabla g_{k}(a)>$
 \end_inset
 
  (el espacio generado por los vectores).