From 67ef5ab22631ab6ab40de5ba0cb6cec19b5ae220 Mon Sep 17 00:00:00 2001 From: salagne Date: Wed, 17 Jun 2020 12:18:41 +0200 Subject: [PATCH] minor corrections --- ana10.pdf | Bin 145988 -> 145988 bytes ana12.pdf | Bin 162826 -> 162652 bytes ana12.tex | 2 +- ana14.pdf | Bin 254831 -> 197296 bytes ana14.tex | 36 +++++++++++++++++++++++------------- 5 files changed, 24 insertions(+), 14 deletions(-) diff --git a/ana10.pdf b/ana10.pdf index ea312a8ced4be0e465c4c8d58692c4a00a5e6643..98005e121f9ca0dfcef52ceeda7ec3991bbfa036 100644 GIT binary patch delta 114 zcmX@|h2zKe|zTEFq-=Ce` z+S#7z?yI}1t7l^#P=4K^VAHBe$*^*=@uJesEDWxoasr%kU^s4m&h(Hv5M#0mI;2O_ z19~6p;ixHjS8xBGxBM8MC@`*x7IMkWM-`Rw6k@_-oNh+h z^1EnQ@DJBNQTzkc3DNcXd*fFF0?M_t2)o`MK|Sj7tn?X&l~AKTGoz^mxV*1MEIKFG zJ6uRXY-$Omu5N}KIm(2PSchMs`I|^|snxbGQflNG^IhbW6-7G}k!(fa%8=P+^WAD@ zwZ)HYr8j~zoz!W_IfC@UeMu}J?j9lxE=(nKN+LI!AJ)Y~4^wHHeu5)uj+lxH0(#lT z8@1b6{onUiJM{MEY1L{2`2Jp+`*Np!eAY5bH|%@HR6Woav>4iea4W#pnCG1Mm58Z* 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Maximum/Minimum] - Sei $D \subset \R^n$ offen, $f\colon D\to \R$. Ein Punkt $x\in D$ heißt lokales \underline{Minimum(Maximum)} von $f$, falls eine Umgebung $K_\delta(x)\subset \R^n$ existiert mit \[f(x)\leq f(y),\; \forall y\in K_\delta(x)\cap D\] ($f(x)\geq f(y),\; \forall y\in K_\delta(x)\cap D$). Falls \[f(x) < f(y), \forall y \in K_\delta(x)\cap D\setminus\{x\}\] ($f(x) > f(y)$), dann heißt $x$ \underline{striktes} lokales Minimum (Maximum). + Sei $D \subset \R^n$ offen, $f\colon D\to \R$. Ein Punkt $x\in D$ heißt lokales \underline{Minimum (Maximum)} von $f$, falls eine Umgebung $K_\delta(x)\subset \R^n$ existiert mit \[f(x)\leq f(y),\; \forall y\in K_\delta(x)\cap D\] ($f(x)\geq f(y),\; \forall y\in K_\delta(x)\cap D$). Falls \[f(x) < f(y), \forall y \in K_\delta(x)\cap D\setminus\{x\}\] ($f(x) > f(y)$), dann heißt $x$ \underline{striktes} lokales Minimum (Maximum). \end{definition} \begin{satz}[Notwendige Bedingung für lokales Extremum (Min oder Max)] Sei $D\subset \R^n$ offen, $f\colon D\to \R$ stetig differenzierbar und $x\in D$ ein lokales Extremum von $f$. Dann gilt : $\nabla f(x) = 0$. \end{satz} \begin{proof} - Für $i = 1,\dots,n$, betrachte $g_i(t)\coloneqq f(x + te_i)$. Da $D$ offen sind alle $g_i$ auf einem Intervall $(-\delta, \delta)$ definiert für ein $\delta > 0$, und differenzierbar. $g_i(t)$ hat in $t = 0$ ein lokales Minimum/Maximum $\implies \dv{g_i(t)}{t}\bigg|_{t=0} = 0 \forall i = 1, \dots, n$. + Für $i = 1,\dots,n$, betrachte $g_i(t)\coloneqq f(x + te_i)$. Da $D$ offen sind alle $g_i$ auf einem Intervall $(-\delta, \delta)$ definiert für ein $\delta > 0$, und differenzierbar. $g_i(t)$ hat in $t = 0$ ein lokales Minimum/Maximum $\implies \dv{g_i(t)}{t}\bigg|_{t=0} = 0 \quad \forall i = 1, \dots, n$. $f$ total differenzierbar - \[\implies 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \forall i = 1\dots, n + \[\implies 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \quad \forall i = 1,\dots, n \] \end{proof} \begin{bem} @@ -35,17 +35,17 @@ Sei $D\subset \R^n$ offen, $f\colon D\to \R$ stetig differenzierbar und $x\in D$ Nach Taylor gilt lokal um $x$: \[f(x+h) = f(x) + (\nabla f(x), h)_2 + \frac{1}{2}(H_f(x)h, h)_2 + \omega_2(x,h)\] mit $\frac{\omega_2(x,h)}{\norm{h}^2} \oldstackrel{h \to 0}{\to} 0$ \begin{enumerate} - \item Sei $H_f(x)$ positiv definit. Betrachte $\min_{\norm{h}=1} (H_f(x)h, h)_2$. Die Menge $\{h\in \R^n\mid\norm{h} = 1\}$ ist kompakt $\implies (H_f(x)h, h)_2$. nimmt ihr Minimum auf $\{h\in \R^n\mid \norm{h} =1\}$ als stetige Funktion an. $\implies \alpha \coloneqq \min_{\norm{h} = 1} (H_f(x)h, h)_2 > 0$, da $H_f(x)$ positiv definit ist. Sei $h\in \R^n\setminus\{0\}$ beliebig. Dann gilt - \[(H_f(x)h, h)_2 = \norm{h}^2 \underbrace{(H_f(x)\cdot \frac{h}{\norm h}, \frac{h}{\norm h})_2}_{\geq \alpha}\geq \alpha \norm h^2 > 0\] + \item Sei $H_f(x)$ positiv definit. Betrachte $\min_{\norm{h}=1} (H_f(x)h, h)_2$. Die Menge $\{h\in \R^n\mid\norm{h} = 1\}$ ist kompakt $\implies (H_f(x)h, h)_2$ nimmt ihr Minimum auf $\{h\in \R^n\mid \norm{h} =1\}$ als stetige Funktion an. $\implies \alpha \coloneqq \min_{\norm{h} = 1} (H_f(x)h, h)_2 > 0$, da $H_f(x)$ positiv definit ist. Sei $h\in \R^n\setminus\{0\}$ beliebig. Dann gilt + \[(H_f(x)h, h)_2 = \norm{h}^2 \underbrace{\left(H_f(x)\cdot \frac{h}{\norm h}, \frac{h}{\norm h}\right)_2}_{\geq \alpha}\geq \alpha \norm h^2 > 0\] Wähle $\delta > 0$ klein, sodass $\forall \norm h < \delta$ gilt $|\omega_2(x,h)| \leq \frac{\alpha}{4}\norm h^2$ (weil $\omega_2(x,h) = o(\norm h^2)$). Damit gilt $\forall h, \norm h < \delta$ \[f(x+h) = f(x) + (\underbrace{\nabla f(x)}_{=0}, h)_2 + \frac{1}{2}(H_f(x)h, h)_2 + \omega_2(x,h) \geq f(x) + \frac{\alpha}{2}\norm h^2 - \frac{\alpha}{4}\norm h^2 > f(x)\] $\implies x$ striktes lokales Minimum von $f$. \item Ersetze $f$ durch $-f$, dann 1) \item $\exists h \in \R^n$ mit $(H_f(x)h, h)_2 = \alpha > 0$, sodass \[ - f(x + th) \oldstackrel{\text{Taylor}}{=} f(x) + \frac{1}{2}t^2\cdot \alpha + \omega_2 (x , th) = f(x) + t^2\left(\frac{\alpha}{2}+ \underbrace{\frac{\omega_2(x,th)}{t^2}}_{\oldstackrel{t\to 0}{\to}0}\right) \oldstackrel{\text{für } 0 < t << 1}{>} f(x) + f(x + th) \oldstackrel{\text{Taylor}}{=} f(x) + \frac{1}{2}t^2\cdot \alpha + \omega_2 (x , th) = f(x) + t^2\left(\frac{\alpha}{2}+ \underbrace{\frac{\omega_2(x,th)}{t^2}}_{\oldstackrel{t\to 0}{\longrightarrow}0}\right) \oldstackrel{\text{für } 0 < t \ll 1}{>} f(x) \] - Außerdem $\exists \eta \in \R^n$ mit $(H_f(x)\eta, \eta)_2 = \beta < 0$. Analog $\implies f(x + t\eta) \leq f(x) + \beta\frac{t^2}{4}< f(x)$ für $0 < t << 1$. $\implies f(x)$ kein Maximum/Minimum. + Außerdem $\exists \eta \in \R^n$ mit $(H_f(x)\eta, \eta)_2 = \beta < 0$. Analog $\implies f(x + t\eta) \leq f(x) + \beta\frac{t^2}{4}< f(x)$ für $0 < t \ll 1$. $\implies f(x)$ kein Maximum/Minimum. \end{enumerate} \end{proof} \begin{bsp} @@ -66,7 +66,17 @@ Sei $D\subset \R^n$ offen, $f\colon D\to \R$ stetig differenzierbar und $x\in D$ $f(x)\leq f(y)$ &$\forall y\in D$ & globales Minimum\\ $f(x) < f(y)$ &$\forall y\in D\setminus\{x\}$ &striktes globales Minimum \end{tabular} - \item $f_1(x,y) \coloneqq x^2 + y^4,\; f_2(x,y) \coloneqq x^2,\; f_2(x,y)\coloneqq x^2 + y^3$ + \item $f(x,y) \coloneqq x^2 - y^2 \implies \nabla f(x,y) = \begin{pmatrix} + 2x\\-2y + \end{pmatrix} = 0$ für $\begin{pmatrix} + x\\y + \end{pmatrix} = \begin{pmatrix} + 0\\0 + \end{pmatrix}$. + $H_f(x,y) = \begin{pmatrix} + 2 & 0\\ 0 & -2 + \end{pmatrix}$ indefinit. $\implies 0 \in \R^2$ kein lokales Extremum. + \item $f_1(x,y) \coloneqq x^2 + y^4,\; f_2(x,y) \coloneqq x^2,\; f_3(x,y)\coloneqq x^2 + y^3$ Es gilt \[ \nabla f_i(0) = 0\in \R^2,\; H_{f_i}(0) = \begin{pmatrix} @@ -79,7 +89,7 @@ Sei $D\subset \R^n$ offen, $f\colon D\to \R$ stetig differenzierbar und $x\in D$ \begin{tabular}{lcl} für $f_1$:& Punkt 0 ist ein& \underline{striktes} lokales Maximum,\\ - ür $f_2$:& Punkt 0 ist ein& \underline{lokales Minimum}, aber \underline{nicht strikt},\\ + für $f_2$:& Punkt 0 ist ein& \underline{lokales Minimum}, aber \underline{nicht strikt},\\ für $f_3$:& Punkt 0 ist ein& Sattelpunkt. \end{tabular} \end{enumerate} @@ -122,7 +132,7 @@ Frage: \underline{Umkehrabbildung:} Auflösen von $x = g(y)$, d.h. \end{array}\right\}\tag{*}\label{star} \end{equation} $n$ Gleichungen, $n$ Unbekannte $y_1,\dots, y_n$. -Gesucht: Abb $f$ mit $y = f(x)$ ($f=g^{-1})$, s.d. $(x,f(x))$ Gleichung \eqref{star} löst (lokal um $(x_0,y_0=f(x_0))$). +Gesucht: Abbildung $f$ mit $y = f(x)$ ($f=g^{-1})$, s.d. $(x,f(x))$ Gleichung \eqref{star} löst (lokal um $(x_0,y_0=f(x_0))$). \paragraph{Implizite Funktion} $m$ Gleichungen, $m$ Unbekannte $y_1,\dots, y_m$. \begin{equation} @@ -164,7 +174,7 @@ Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung v \[ D_yF(x,y) =\begin{pmatrix} \pdv{F_1}{y_1} &\dots &\pdv{F_1}{y_m}\\ - \vdots & & \vdots\\ + \vdots & \ddots & \vdots\\ \pdv{F_m}{y_1}&\dots&\pdv{F_m}{y_m} \end{pmatrix} \] @@ -198,7 +208,7 @@ Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung v Für $k\in \N$ gilt \[y^{(k)}(x) = G(x,y^{(k-1)}(x)) = y^{(k-1)}(x)-J_y^{-1}\underbrace{F(x,y^{(k-1)}(x))}_{\text{stetig}}\] Induktiv folgt $y^{(k)}(x)$ stetig in $x\in K_2^x(0)$. Definiere $f(x)=y(x),\; f\colon K_s^x(0) \to K_r^y(0)$ und nach Konstruktion gilt \[G(x,f(x)) = f(x),\quad x\in K_s^x(0).\] - Aus der Abschätzung $\norm{y(x)-y^{(k)}(x)}\leq 2^{-k-1}\cdot r,\; x\in K_s^x(0)$ folgt, dass $y^{(k)}(x)\oldstackrel{k\to \infty}{\to y(x)}$ gleichmäßig auf $K_s^x(0)$ konvergiert $\implies y(x)$ stetig. + Aus der Abschätzung $\norm{y(x)-y^{(k)}(x)}\leq 2^{-k-1}\cdot r,\; x\in K_s^x(0)$ folgt, dass $y^{(k)}(x)\oldstackrel{k\to \infty}{\longrightarrow} y(x)$ gleichmäßig auf $K_s^x(0)$ konvergiert $\implies y(x)$ stetig. $\implies$ 1. Behauptung für $(\hat x, \hat y) = (0,0)$ mit $U(\hat x)\coloneqq K_s^x(0)$ und $U(\hat y)\coloneqq K_r^y(0)$. \item Die Eindeutigkeit von $y = f(x)$ folgt nun aus dem Banachschen Fixpunktsatz. Für $x\in K_s(\hat x)$ ist der Fixpunkt der Gleichung $G(x,y) = y$ eindeutig bestimmt. \end{enumerate}