From 9508913fa7536ba1b098870396c764a41cb10ad6 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 2 Jun 2020 23:42:38 +0200 Subject: [PATCH] replace w with omega, replace bold face with underlines, fix some typos in last proof --- ana12.pdf | Bin 221660 -> 163118 bytes ana12.tex | 80 ++++++++++++++++++++++++++++++++++-------------------- 2 files changed, 50 insertions(+), 30 deletions(-) diff --git a/ana12.pdf b/ana12.pdf index 46ae31e589ae7d046fc07ca2d626d5894cfa9c08..6b219250bc6e804610e63c3eca27407f7cb4845f 100644 GIT binary patch delta 90125 zcmcG#Q;??5w(eWDUDajVwr$(&vRz-=RI_$q@zZV&(GNuEJokDuem+w5TkR*oMZMp5G}qD5V94_c|1Hf(=3QoW;;+Fl|& z#dfcPfpik|bh0=-eS_5Jlc=M4^oB#K;&@LUR6myu!O@vHU(RV?nQKwf0i6c5!AX>Y zSPyR=ecgz3^qsL>{kReAcr+Ve8KZ2{_}5rkYt$su9i!JIynw#!>9;m|)kOqWnu1vI zaw<9FV6^IE0z)NP51su;OieF9JieYX(5qicH-FO`sF8k&#!sV05gIeoasL9nEC`+lO@*(MfpT5|mPgqfPFcnHCl%+b?f_zwArF@Hvx%i6vqK~WtHj|x% zXC~PK1XB+Ihq9Coj;a>V%j{(0^E^h*hs@E?PD{0Ke0_H32A{{}lj_6Dfbr@AolPT< 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z1cHD-0cpVh8A2@m(9l~Or}uC>fS_R9iY6zLZhQz5>q60cgz?Ex9>5`zV=8;g6_NGj ztAbDV;Pj|!xEYR*VI5*)_XqB*JMoZEcTIy6T8XBzve8Emt0Y&;)5TOre3t}Zje^ln*8$h4HyXiPbnoozKv z2cqx!D%l{%nbSJOT1VcjALCmx)O>Wz5W+iI|4!w03tG)ok#T0<#w`~15(39^reg!W zQ4WTkBjM9u1J{@;^>ABjrfJY!`WJXh*9y1IA4Ckono2{Pzsd!bJ0h#*t99@Lx$pk> iyONXje{0SSZs7{|bhWW1d_ag_M1YWmMNw0U@P7c)Cu?Z{ diff --git a/ana12.tex b/ana12.tex index 8b5d7d7..07f3ec8 100644 --- a/ana12.tex +++ b/ana12.tex @@ -2,14 +2,22 @@ \begin{document} \newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dv}[2]{\frac{\d #1}{\d #2}} + \section{Totale Differenzierbarkeit} -Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $x\in D$ differenzierbar, falls $f$ in $x$ "gut" linear approximierbar ist, d.h. $\exists a\in \R$ mit $f(x + h) = f(x) + a\cdot h + w(h)$ wobei $\lim\limits_{h\to 0} \frac{w(h)}{|h|} = 0 (f'(x) = a)$. +Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $x\in D$ differenzierbar, falls $f$ in $x$ ,,gut`` linear approximierbar ist, d.h. $\exists a\in \R$ mit $f(x + h) = f(x) + a\cdot h + w(h)$ wobei $\lim\limits_{h\to 0} \frac{w(h)}{|h|} = 0 (f'(x) = a)$. + \begin{definition}[total differenzierbar] - Es sei $D\subset \R^n$ offen und $f\colon D \to \R^m$ eine Abbildung. $f$ heißt im Punkt $x\in D$ \textbf{(total) differenzierbar}, falls es eine \textbf{lineare Abbildung} $A \colon \R^n \to \R^m$ gibt, sodass + Es sei $D\subset \R^n$ offen und $f\colon D \to \R^m$ eine Abbildung. $f$ heißt im Punkt $x\in D$ \underline{(total) differenzierbar}, falls es eine lineare Abbildung $A \colon \R^n \to \R^m$ gibt, sodass \begin{equation}\label{eq:diffbar} \lim\limits_{\stackrel{h\to 0}{h\neq 0}} \frac{f(x + h) - f(x) - A\cdot h}{\norm{h}} = 0 \end{equation} - Oft wird (\ref{eq:diffbar}) durch eine Bedingung an den Rest $w(h),\; w\colon D \to \R^m$ definiert $f(x + h) = f(x) + A\cdot h + w(h)$, wobei $\lim\limits_{h\to 0} \frac{\norm{w(h)}}{\norm{h}} = 0 (\Leftrightarrow (\ref{eq:diffbar}), \Leftrightarrow w(h) = o(\norm{h}))$. Da alle Normen auf $\R^m$ äquivalent sind, ist es gleichgültig, welche Norm man in (\ref{eq:diffbar}) verwendet. $A$ heißt das \textbf{Differential} von $f$ im Punkt $x$. + Oft wird (\ref{eq:diffbar}) durch eine Bedingung an den Rest $\omega(h),\; \omega\colon D \to \R^m$ + definiert: + \[ + f(x + h) = f(x) + A\cdot h + \omega(h) + ,\] wobei $\lim\limits_{h\to 0} \frac{\norm{\omega(h)}}{\norm{h}} = 0 \;(\Leftrightarrow (\ref{eq:diffbar}), \Leftrightarrow \omega(h) = o(\norm{h}))$. Da alle Normen auf $\R^m$ äquivalent sind, ist es gleichgültig, welche Norm man in (\ref{eq:diffbar}) verwendet. + + $A$ heißt das \underline{Differential} von $f$ im Punkt $x$. Schreibweise: \[\d f(x),\; \d f\bigg|_x,\; \d f_x,\; Df(x),\; Df\bigg|_x,\; Df_x,\; \d f(x)\bigg|_{x = x_0},\; Df(x_0).\] \end{definition} \begin{bem} @@ -27,8 +35,8 @@ Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $ \begin{enumerate}[1)] \item Sei $f$ differenzierbar. Dann gilt \begin{align*} \lim\limits_{h_i\to 0} \frac{f(x + h_i \cdot e^{(i)}) - f(x)}{h_i} - &= \lim\limits_{h_i\to 0} \left(Df(x) \cdot e^{(i)} + \frac{w(h_i)}{h_i}\right)\\ - &= Df(x) \cdot e^{(i)} + \underbrace{\lim\limits_{h_i\to 0} \frac{w(h_i)}{h_i}}_{\to 0}\\ + &= \lim\limits_{h_i\to 0} \left(Df(x) \cdot e^{(i)} + \frac{\omega(h_i)}{h_i}\right)\\ + &= Df(x) \cdot e^{(i)} + \underbrace{\lim\limits_{h_i\to 0} \frac{\omega(h_i)}{h_i}}_{\to 0}\\ &= Df(x) \cdot e^{(i)} \end{align*} $ \implies f$ partiell differenzierbar und $Df(x)e^{(i)} = \begin{pmatrix} @@ -43,19 +51,19 @@ Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $ f(x + h) - f(x)&\stackrel{\text{MWS}}{=} h_2 \partial_2 f(x_1 + h_1, x_2 + \theta_2 \cdot h_2) + h_1 \partial_1 f(x_1 + \theta_1 \cdot h_1, x_2)\\ &= h_2(\partial_2f(x_1,x_2) + w_2(h_1,h_2)) + h_1(\partial_1f(x_1, x_2) + w_1(h_1,h_2)), \intertext{wobei} - w_1(h_1, h_2) &= \partial_1f(x_1 + \theta_1 h_1, x_2) - \partial_1 f(x_1, x_2) + \omega_1(h_1, h_2) &= \partial_1f(x_1 + \theta_1 h_1, x_2) - \partial_1 f(x_1, x_2) \intertext{und} - w_2(h_1,h_2) &= \partial_2f(x_1 + h_1, x_2 + \theta_2 h_2) - \partial_2f(x_1,x_2). - \intertext{$\partial_1f(x), \partial_2f(x)$ stetig $\implies \lim\limits_{h\to 0} w_1(h_1,h_2) = 0,\; \lim\limits_{h\to 0} w_2(h_1,h_2) = 0$. Daher gilt} - f(x + h) - f(x) &= h_1\partial_1 f(x) + h_2\partial_2f(x) + h_1w_1(h) + h_2w_2(h)\\ + \omega_2(h_1,h_2) &= \partial_2f(x_1 + h_1, x_2 + \theta_2 h_2) - \partial_2f(x_1,x_2). + \intertext{$\partial_1f(x), \partial_2f(x)$ stetig $\implies \lim\limits_{h\to 0} \omega_1(h_1,h_2) = 0,\; \lim\limits_{h\to 0} \omega_2(h_1,h_2) = 0$. Daher gilt} + f(x + h) - f(x) &= h_1\partial_1 f(x) + h_2\partial_2f(x) + h_1\omega_1(h) + h_2\omega_2(h)\\ &= (\partial f_1(x), \partial f_2(x))\begin{pmatrix} h_1\\h_2 - \end{pmatrix} + (w_1(h), w_2(h))\begin{pmatrix} + \end{pmatrix} + (\omega_1(h), \omega_2(h))\begin{pmatrix} h_1\\h_2 \end{pmatrix}\\ - &= Df(x) \cdot h + \tilde{w}(h) + &= Df(x) \cdot h + \tilde{\omega}(h) \end{salign*} - mit $\lim\limits_{h\to 0} \frac{\norm{\tilde{w}(h)}}{\norm{h}} = 0$. + mit $\lim\limits_{h\to 0} \frac{\norm{\tilde{\omega}(h)}}{\norm{h}} = 0$. Also ist $f$ differenzierbar und $Df(x) = \nabla^Tf(x)$. \end{enumerate} \end{proof} @@ -63,7 +71,7 @@ Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $ stetig partiell differenzierbar $\implies$ (total) differenzierbar $\implies$ partiell differenzierbar. Die umgekehrten Implikationen gelten im Allgemeinen nicht. \end{korrolar} \begin{lemma}[Richtungsableitung]\label{lemma:richtungsableitung} - Sei $D\in \R^n$ offen, $f \colon D \to \R$ im Punkt $x\in D$ differenzierbar. Dann gilt $\forall v \in \R^n$ mit $\norm{v}_2 = 1$ existiert die Ableitung in Richtung $v$ (sog. \textbf{Richtungsableitung}) + Sei $D\in \R^n$ offen, $f \colon D \to \R$ im Punkt $x\in D$ differenzierbar. Dann gilt $\forall v \in \R^n$ mit $\norm{v}_2 = 1$ existiert die Ableitung in Richtung $v$ (sog. \underline{Richtungsableitung}) \[\pdv{f}{v}(x) \coloneqq \lim\limits_{t\searrow 0} \frac{f(x + tv) - f(x)}{t}\] und \[\pdv{f}{v}(x) = (\nabla f(x), v)_2\] \end{lemma} \begin{proof} @@ -96,31 +104,43 @@ Erinnerung (Analysis 1) $f \colon D \to \R,\; D \subset \R$, ist genau dann in $ \begin{proof} Seien $x\in D_g$ und $y = g(x) \in D_f$. Dann gilt nach Voraussetzungen \begin{align*} - g(x + \xi) &= \underbrace{g(x)}_{\eqqcolon y} + \underbrace{D_xg(x) \xi + w_g(\xi)}_{\eqqcolon \eta} &&\text{mit } \lim\limits_{\stackrel{x+\xi \in D_g}{\norm{\xi} \to 0}} \frac{\norm{w_g(\xi)}}{\norm{\xi}} = 0\\ - f(y + \eta) &= f(y) + D_yf(y) \eta + w_f(\eta)&&\text{mit } \lim\limits_{\stackrel{x+\eta \in D_f}{\norm{\eta} \to 0}} \frac{\norm{w_f(\eta)}}{\norm{\eta}} = 0\\ + g(x + \xi) &= \underbrace{g(x)}_{\eqqcolon y} + \underbrace{D_xg(x) \xi + \omega_g(\xi)}_{\eqqcolon \eta} &&\text{mit } \lim\limits_{\stackrel{x+\xi \in D_g}{\norm{\xi} \to 0}} \frac{\norm{\omega_g(\xi)}}{\norm{\xi}} = 0\\ + f(y + \eta) &= f(y) + D_yf(y) \eta + \omega_f(\eta)&&\text{mit } \lim\limits_{\stackrel{x+\eta \in D_f}{\norm{\eta} \to 0}} \frac{\norm{\omega_f(\eta)}}{\norm{\eta}} = 0\\ \end{align*} Dann erhalten wir \begin{salign*} (f\circ g)(x + \xi) &= f(g(x + \xi))\\ &\stackrel{g(x + \xi) = g(x) + \eta = y + \eta}{=} f(y + \eta)\\ - &= f(y) + D_yf(y)\cdot \eta + w_f(\eta)\\ - &= f(y) + D_yf(y) (D_xg(x)\xi + w_g(\xi)) + w_f(\eta)\\ - &= f(y) + D_yf(y)D_x g(x) \cdot \xi + D_yf(y) w_g(\xi) + w_f(\eta)\\ - &= (f\circ g)(x) + \underbrace{D_yf(y)D_xg(x)}_{D_x(f\circ g)} \cdot \xi + w_{f\circ g}(\xi), + &= f(y) + D_yf(y)\cdot \eta + \omega_f(\eta)\\ + &= f(y) + D_yf(y) (D_xg(x)\xi + \omega_g(\xi)) + \omega_f(\eta)\\ + &= f(y) + D_yf(y)D_x g(x) \cdot \xi + D_yf(y) \omega_g(\xi) + \omega_f(\eta)\\ + &= (f\circ g)(x) + \underbrace{D_yf(y)D_xg(x)}_{D_x(f\circ g)} \cdot \xi + \omega_{f\circ g}(\xi), \end{salign*}%eta = D_xg(x)\xi + w_g(\xi) - wobei hier $w_{f\circ g}(\xi) = D_yf(y)w_g(\xi) + w_f(\eta)$. Es genügt also zu zeigen, dass \[\lim\limits_{\norm{\xi}\to 0} \frac{\norm{w_{f\circ g}(\xi)}}{\norm{\xi}} = 0.\] - Aus $\lim\limits_{\norm{\xi}\to 0} \frac{\norm{w_g(\xi)}}{\norm{\xi}} = 0$ folgt sofort $\lim\limits_{\norm{\xi}\to 0} \frac{\norm{D_yf(y)w_g(\xi)}}{\norm{\xi}} = 0$. Wir schließen aber auch, dass es eine Konstante $c >0$ geben muss, sodass $\norm{w_g(\xi)} \leq c \norm{\xi}$. Wegen $\lim\limits_{\norm{\eta}\to 0} \frac{\norm{w_f(\eta)}}{\norm{\eta}} = 0$ muss es ein $\tilde{w_f}(\eta)$ mit $\lim\limits_{\eta\to 0} \tilde{w}_f(\eta) = 0$ geben, sodass $w_f(\eta) = \norm{\eta}\cdot \tilde{w}_f(\eta)$. Mit diesen Aussagen gilt: + wobei hier $\omega_{f\circ g}(\xi) = D_yf(y)\omega_g(\xi) + \omega_f(\eta)$. Es genügt also zu zeigen, dass \[\lim\limits_{\norm{\xi}\to 0} \frac{\norm{\omega_{f\circ g}(\xi)}}{\norm{\xi}} = 0.\] + Aus $\lim\limits_{\norm{\xi}\to 0} \frac{\norm{\omega_g(\xi)}}{\norm{\xi}} = 0$ folgt sofort + $\lim\limits_{\norm{\xi}\to 0} \frac{\norm{D_yf(y)\omega_g(\xi)}}{\norm{\xi}} = 0$. + Wir schließen aber auch, dass es eine Konstante $c >0$ geben muss, sodass $\norm{\omega_g(\xi)} \leq c + \norm{\xi}$. Wegen $\lim\limits_{\norm{\eta}\to 0} \frac{\norm{\omega_f(\eta)}}{\norm{\eta}} = 0$ + muss es ein $\tilde{\omega}_f(\eta)$ mit $\lim\limits_{\eta\to 0} \tilde{\omega}_f(\eta) = 0$ geben, + sodass $\omega_f(\eta) = \norm{\eta}\cdot \tilde{\omega}_f(\eta)$. Mit diesen Aussagen gilt: \begin{salign*} - \norm{w_f(\eta)} %&\stackrel{w_f(\eta) = \norm{\eta} \tilde{w}_f(\eta)}{\le} \norm{D_x g(x)\xi + w_g(\xi)} \tilde{w}_f (\eta)\\ - &\le \norm{D_x g(x)\xi + w_g(\xi)} \tilde{w}_f (\eta)\\ - &\le \left(\norm{D_xg(x)} \norm{\xi} + \norm{w_g(\xi)}\right) \cdot \norm{\tilde{w}_f(\eta)}\\ - &\le \left(\norm{D_xg(x)} + c\right)\norm{\xi} \cdot \norm{\tilde{w}_f(\eta)}\\ - \implies \frac{\norm{w_f(\eta)}}{\norm{\xi}} &= \left(\norm{D_xg(x)} + c\right) \cdot \norm{\tilde{w}_f(\eta)} - \intertext{Wegen $\lim\limits_{\xi\to 0} \eta = \lim\limits_{\xi\to 0} D_xg(x)\xi + w_g(\xi) = 0$ und der Stetigkeit von $\tilde{w}_f$ gilt} - \lim\limits_{\xi\to 0} \frac{\norm{w_f(\eta)}}{\norm{\xi}} &= \lim\limits_{\xi\to 0} \left(\norm{D_xg(x)} + c\right) \cdot \norm{\tilde{w}_f(\eta)} = \left(\norm{D_xg(x)} + c\right) \cdot \tilde{w}_f\left(\lim\limits_{\xi\to 0} \eta\right) = 0 + \norm{\omega_f(\eta)} %&\stackrel{w_f(\eta) = \norm{\eta} \tilde{w}_f(\eta)}{\le} \norm{D_x g(x)\xi + w_g(\xi)} \tilde{w}_f (\eta)\\ + &= \norm{D_x g(x)\xi + \omega_g(\xi)} \norm{\tilde{\omega}_f (\eta)}\\ + &\le \left(\norm{D_xg(x)} \norm{\xi} + \norm{\omega_g(\xi)}\right) \cdot \norm{\tilde{\omega}_f(\eta)}\\ + &\le \left(\norm{D_xg(x)} + c\right)\norm{\xi} \cdot \norm{\tilde{\omega}_f(\eta)}\\ + \implies \frac{\norm{\omega_f(\eta)}}{\norm{\xi}} &\le \left(\norm{D_xg(x)} + c\right) + \cdot \norm{\tilde{\omega}_f(\eta)} + \intertext{Wegen $\lim\limits_{\xi\to 0} \eta = \lim\limits_{\xi\to 0} D_xg(x)\xi + \omega_g(\xi) = 0$ + folgt $\displaystyle \lim_{\xi \to 0} \tilde{\omega}_f(\eta) = 0$ und damit} + 0 \le \lim\limits_{\xi\to 0} \frac{\norm{\omega_f(\eta)}}{\norm{\xi}} + &\le \lim\limits_{\xi\to 0} \left(\norm{D_xg(x)} + c\right) \cdot \norm{\tilde{\omega}_f(\eta)} + = \left(\norm{D_xg(x)} + c\right) \cdot \norm{ \lim_{\xi \to 0} \tilde{\omega}_f(\eta)} + = 0 \end{salign*} - Insgesamt erhalten wir - \[\lim\limits_{\xi\to 0} \frac{\norm{w_{f\circ g}(\xi)}}{\norm{\xi}} = \lim\limits_{\norm{\xi}\to 0} \frac{\norm{D_yf(y)w_g(\xi)}}{\norm{\xi}} + \lim\limits_{\xi\to 0} \frac{\norm{w_f(\eta)}}{\norm{\xi}} = 0\] + Insgesamt erhalten wir + \[\lim\limits_{\norm{\xi}\to 0} \frac{\norm{\omega_{f\circ g}(\xi)}}{\norm{\xi}} + \le \lim\limits_{\norm{\xi}\to 0} \frac{\norm{D_yf(y)\omega_g(\xi)}}{\norm{\xi}} + + \lim\limits_{\xi\to 0} \frac{\norm{\omega_f(\eta)}}{\norm{\xi}} = 0\] \end{proof} \begin{bem}[Komponentenweise für $i = 1,\dots,m$ und $j = 1, \dots, r$] \[D_xh(x) = D_yf(y) D_xg(x) \Leftrightarrow \underbrace{\pdv{h_j}{x_i}(x)}_{\partial_i(f\circ g)_j} = \sum_{i = 1}^{n} \pdv{f_j}{y_k}(g_1(x)\dots, g_n(x))\cdot \pdv{g_k}{x_i}(x_1, \dots, x_m)\]