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-0,0 +1 @@ +\documentclass{lecture} \begin{document} \newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\dv}[2]{\frac{\d #1}{\d #2}} \def\mathunderline#1#2{\color{#1}\underline{{\color{black}#2}}\color{black}} \section{Extremwertaufgaben} \begin{definition}[lokales 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). \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 $\implies \dv{g_i(t)}{t}\bigg|_{t=0} = 0 \forall i = 1, \dots, n$. $f$ total differenzierbar \[\implies 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \forall i = 1\dots, n \] \end{proof} \begin{bem} Die Umkehrung ist falsch, z.B. $f(x) = x^3,\; f\colon \R \to \R$, hat in $x=0$ $\nabla f(x) = 0$, aber $x = 0$ ist kein Max/Min von $f(x) = x^3$.\\ $f\colon \R^2 \to \R,\; f(x_1, x_2) = x_1x_2$ hat in $x = \begin{pmatrix} 0\\0 \end{pmatrix} \nabla f(x) = 0$, aber $x = 0$ ist kein Max/Min von $f$. \end{bem} \begin{satz}[Hinreichende Bedingung für lokales Extremum] Sei $D\subset \R^n$ offen, $f\in C^2(D,\R)$ und $x\in D$ mit $\nabla f(x) = 0$. Dann gilt: \begin{enumerate} \item $H_f(x)$ positiv definit $\implies x$ striktes lokales Minimum von $f$. \item $H_f(x)$ negativ definit $\implies x$ striktes lokales Maximum von $f$. \item $H_f(x)$ indefinit $\implies x$ kein lokales Extremum. \end{enumerate} \end{satz} \begin{proof} 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 \[(H_f(x)h, h)_2 = \norm{h}^2 \underbrace{(H_f(x)\cdot \frac{h}{\norm h}, \frac{h}{\norm h})_2}_{\geq \alpha}\geq \alpha \norm h^2 > 0\] 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 \[ f(x + th) \oldstackrel{\text{Taylor}}{=} f(x) + \frac{1}{2}t^2\cdot \alpha + \omega_2 (x , th) = f(x) + t^2\left(\frac{\alpha}{2}+ \underbrace{\frac{\omega_2(x,th)}{t^2}}_{\oldstackrel{t\to 0}{\to}0}\right) \oldstackrel{\text{für } 0 < t << 1}{>} f(x) \] 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. \end{enumerate} \end{proof} \begin{bsp} \begin{enumerate} \item $f(x,y) \coloneqq x^2 + 2y^2 \implies \nabla f(x,y) = \begin{pmatrix} 2x\\4y \end{pmatrix} = 0$ für $\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 0\\0 \end{pmatrix}$. $H_f(x,y) = \begin{pmatrix} 2 & 0\\ 0 & 4 \end{pmatrix}$ positiv definit. $\implies \begin{pmatrix} x\\y \end{pmatrix} = 0$ striktes lokales Minimum (sogar global). \begin{tabular}{cll} $f(x)\leq f(y)$ &$\forall y\in D$ & globales Minimum\\ $f(x) < f(y)$ &$\forall y\in D\setminus\{x\}$ &striktes globales Minimum \end{tabular} \item $f_1(x,y) \coloneqq x^2 + y^4,\; f_2(x,y) \coloneqq x^2,\; f_2(x,y)\coloneqq x^2 + y^3$ Es gilt \[ \nabla f_i(0) = 0\in \R^2,\; H_{f_i}(0) = \begin{pmatrix} 2 & 0\\ 0 & 0 \end{pmatrix} \forall i =1,2,3 \] Die Hesse-Matrix ist positiv semidefinit. Es gilt\\ \begin{tabular}{lcl} für $f_1$:& Punkt 0 ist ein& \underline{striktes} lokales Maximum,\\ 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} \begin{figure} \begin{tikzpicture}[scale=0.8] \begin{axis} [ domain=-1:1, samples=20, grid = major ] \addplot3 [surf] {x^2 + 2*y^2}; \end{axis} \end{tikzpicture} \begin{tikzpicture}[scale=0.8] \begin{axis} [ domain=-1:1, samples=20, grid = major ] \addplot3 [surf] {x^2 - y^2}; \end{axis} \end{tikzpicture} \caption{Links: $f(x,y) = x^2 +2y^2$ ist ein Paraboloid, Rechts: $f(x,y) = x^2-y^2$ ist eine Sattelfläche} \end{figure} \end{bsp} \section{Implizite Funktionen und Umkehrabbildung.} Frage: \underline{Umkehrabbildung:} Auflösen von $x = g(y)$, d.h. \begin{equation} \left\{\begin{array}{rl} x_1&=g_1(y_1,\dots,y_n)\\ \vdots&\\ x_n&=g_n(y_1,\dots,y_n)\\ \end{array}\right\} \Leftrightarrow \left\{\begin{array}{rl} 0&=x_1-g_1(y_1,\dots,y_n)\\ \vdots&\\ 0&=x_n-g_n(y_1,\dots,y_n)\\ \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 $(x_0,y_0=f(x_0))$). \paragraph{Implizite Funktion} $m$ Gleichungen, $m$ Unbekannte $y_1,\dots, y_m$. \begin{equation} \left.\begin{array}{rl} 0&=F_1(x_1,\dots,x_n,y_1,\dots,y_m)\\ \vdots&\\ 0&=F_m(x_1,\dots, x_n,y_1,\dots,y_m) \end{array}\right\}\tag{**}\label{doublestar} \end{equation} $0 = F(x,y)$ Auflösen nach $y$, d.h. Gesucht: Abbildung $f$, s.d. $y = f(x)$ mit $(x,f(x))$ löst \eqref{doublestar} (lokal um eine Lösung $(x_0,y_0):F(x_0,y_0) = 0$) \begin{bsp} $m=1,\;n=1,\; F(x,y) = x^2 + y^2-1$\\\begin{minipage}[c]{0.35\textwidth} \begin{tikzpicture}[scale=0.8] \draw[->] (0,-1.5) -- node[right,pos=.9] {$(x_0,y_0)$} (0,1.5); \draw[->] (-1.5,0) -- node[below,pos=1.3] {$F(x,y) = 0$} (1.5,0); \draw (0,0) circle (1cm); \draw[fill] (0,1) circle (2pt); \draw[fill=red,draw=red] (0,0) circle (2pt); \node[color=red] at (.3,0) {)}; \node[color=red] at (-.3,0) {(}; \draw[fill=blue,draw=blue] (1,0) circle (2pt); \node[color=blue] at (1.3,0) {)}; \node[color=blue] at (.7,0) {(}; \end{tikzpicture} \end{minipage}\begin{minipage}[c]{0.65\textwidth} An der Stelle $\mathunderline{red}{x_0 = 0,\; y_0 = 1}$ gilt $F(0,1)=0$,\\ also $F(x,y) = 0 \Leftrightarrow y^2 = 1-x^2$. \[\implies \mathunderline{red}{f(x)\coloneqq \sqrt{1-x^2}}\text{ für } |x| < 1\] $\mathunderline{red}{\text{ erfüllt }F(x,f(x)) = 0},\; |x| < 1$. \end{minipage}\\Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung von $x_0 = 1$, sodass \[\mathunderline{blue}{\exists f\colon U(x_0) \to \R\text{ mit }F(x,f(x)) = 0}.\] \end{bsp} \begin{satz}[Satz über implizite Funktionen] Sei $D^x \subset \R^n$ offen, $D^y \subset \R^m$ offen, $F^1 \in C^1 (D^x\times D^y,\R^m)$ (stetig differenzierbar) und $(\hat{x}, \hat{y})\in D^x\times D^y$ mit $F(\hat x, \hat y) = 0$. Die $m\times m$ Matrix \[ D_yF(x,y) =\begin{pmatrix} \pdv{F_1}{y_1} &\dots &\pdv{F_1}{y_m}\\ \vdots & & \vdots\\ \pdv{F_m}{y_1}&\dots&\pdv{F_m}{y_m} \end{pmatrix} \] sei im Punkt $(\hat x, \hat y)$ invertierbar. Dann gilt: \begin{enumerate} \item $\exists$ offene Umgebungen $U(\hat x) \subset D^x,\; U(\hat y)\subset D^y$ um $\hat x$ und $\hat y$ und $\exists$ eine stetige Funktion $f\colon U(\hat x) \to U(\hat y)$, s.d. \[F(x,f(x)) = 0\quad\forall x\in U(\hat x)\] \item $f$ ist eindeutig bestimmt, d.h. $F(x,y) = 0$ für $(x,y) \in U(\hat x)\times U(\hat y)\Leftrightarrow y = f(x)$ \item $f$ ist in $\hat x$ stetig differenzierbar und $J_f(\hat x) = D_xf(\hat x)\in \R^{m \times n}$ ist \[D_xf(\hat x) = -(D_yF(\hat x, \hat y))^{-1}D_xF(\hat x, \hat y).\] \end{enumerate} \end{satz} \begin{proof} \begin{enumerate} \item O.B.d.A. sei $(\hat x, \hat y) = (0,0)$ (sonst betrachte $F(x,y) - F(\hat x, \hat y)$). Die Matrix $J_y \coloneqq D_yF(0,0)$ ist regulär. Definiere $G\colon D^x\times D^y \to \R^m,\; G(x,y) \coloneqq y - J^{-1}_yF(x,y)$. $G$ ist stetig differenzierbar und erfüllt: $G(0,0) = 0$ und $F(x,y) = 0 \Leftrightarrow G(x,y) = y$. Jacobi-Matrix von $G(x,y)$ bzgl. $y$: \[D_yG(x,y) = \mathbb{I} - J_y^{-1} D_yF(x,y) \text{ und insb. } D_yG(0,0) = \mathbb{I} - J_y^{-1}J_y = 0.\] $F$ stetig differenzierbar $\implies D_yG(x,y)$ stetig $\implies \exists K_r^x(0) \times K_r^y(0) \subset D^x \times D^y$ mit Radius $r$, sodass $\norm{D_yG(x,y)}_2\leq \frac{1}{2},\; (x,y)\in K_r^x(0)\times K_r^y(0)$. $G(0,0) = 0\implies \exists K_s^x(0)\subset K_r^x(0)$ mit Radius $0< s\le r$ sodass \[\norm{G(x,0)}_2 \le \frac{1}{2} r,\; x\in K_s^x(0)\] Ziel: Konstruiere $f\colon K_s^x(0)\to K_r^y(0)$ stetig mit $G(x,f(x)) = f(x)\; (\Leftrightarrow F(x,f(x)) = 0)$. Betrachte Fixpunktgleichung \[G(x,y) = y,\; x\in K_s^x(0).\] Für $(x,y_1),\; (x,y_2)\in K_s^x(0)\times K_r^y(0)$ gilt \[\norm{G(x,y_1) - G(x,y_2)}_2 \oldstackrel{\text{MWS}}{\le} \sup_{(x,y)\in K_s^x(0)\times K_r^y(0)}\norm{D_yG(x,y)}_2 \cdot \norm{y_1-y_2}_2 \le \frac{1}{2}\norm{y_1-y_2}_2.\] Sei $y\in \overline{K_r^y(0)}$ \[\norm{G(x,y)}_2 \le \norm{G(x,y)-G(x,0)}_2 + \norm{G(x,0)}_2 \le \frac{1}{2}\norm{y}_2 + \frac{1}{2}r\le r\] d.h. $G(x,\cdot)$ ist eine Selbstabbildung der abgeschlossenen Kugler $\overline{K_r^y(0)}$.\\ Außerdem, $\norm{G(x,y_1) -G(x,y_2)}_2 \le \frac{1}{2}\norm{y_1-y_2}\implies G(x,\cdot)$ ist eine Kontraktion mit Lipschitz-Konstante $L=\frac{1}{2}$. Aus dem Banachschen Fixpunktsatz folgt $\forall x \in K_s^x(0) \exists!$ ein Fixpunkt $y(x)\in K_r^y(0)$ von $G(x,\cdot)$: \[y(x) = \lim\limits_{k\to \infty} y^{(k)}(x),\; y^{(k)}(x)=G(x,y^{(k-1)}(x)),\; k\in \N\] mit Startpunkt $y^{(0)}(x)\coloneqq 0$. Es gilt die Fehlerabschätzung $\forall x \in K_s^x(0)$: \[\norm{y(x)-y^{(k)}(x)}_2 \le 2^{-k}\norm{y^{1}(x)-y^{(0)}(x)}_2 = 2^{-k}\norm{G(x,0)-0}_2 \le 2^{-k} \frac{1}{2}r\] 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. $\implies$ 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} \end{proof} \end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index 3406a51ccc781f1e707e081741539e86b61beb9c..124bf23b10e73f6751723eb188cbfe9754b4751f 100644 GIT binary patch delta 608898 zcmV)QK(xPw{U)FKFOVbwGM9mr1S)@xQ%!H3Fc7@IU*V%Uc*EKln_Js_wTV(i374wv zfxc&WPF_Lcqf-8TcX84vZKdP_v&-zvGKQo2<{CveZF13xT%5mp|4?s^&U7izfks=1 zx)s`1fJG&&HBezC>Yey1-}lYoVTx0x3>x_VHp_6e1PM+Rp*2-oMW9*yMy3Dkf!!&b)q=hNKmqsY<0W~J6t2j^FjL7>j zwsX?_#r4pyGYWHh6LfyyppXWnX4w3gtXxD0y3y9U-PZ z;dj%#Fy6t^N3{=e7Y;ut3MYTV*ugH47OqQZ2i8@z0wq}Zw8H1O9g_XWrfHwed+NiU zTCD!6vO)`v<6Wg)lD-(GrAgYCOLqw4v-%xBg8Q}mHUo`doPD`_iZ3>OGPw%l9QwcW z<76Ib@N;Fn$*{|J!fFJ@=YUJC3{j--;ro85Q>OGE_mB2H1Hxi?#Q zSN^Mf8E4hzHlW%!<(*0Yu~*AHf87M{XTQS~#M4MeG_o^bFO(Qj>0Tv4sHyN@20HSK zEHQZyGR6FDzHk?pCSAIl+`kPN^^0ai{l+$5RcVpi0hD5eedaz$MDK%yBUoriG2%iu zUq$d;f+>Y*>aR7^+_Xzz?GoT3$?6SBf4RP9oL@63)5N%EuLG*=lHl*He;r~gX4 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