diff options
Diffstat (limited to 'fvv1')
| -rw-r--r-- | fvv1/n1.lyx | 32 | ||||
| -rw-r--r-- | fvv1/n2.lyx | 16 | ||||
| -rw-r--r-- | fvv1/n3.lyx | 18 | ||||
| -rw-r--r-- | fvv1/n4.lyx | 4 |
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). |