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zG1(2{WK{!qx$kD7oWcTl`FLx`-M6z~ZTo;$y3B-EIm`sQ10*jUUYT3RB^d`1u#byY zFnggSxyBuIzP|;9;>3`N&xg4`ZPLq3y*Y# 0} \underbrace{\Vert x - x' \Vert}_{\ge 0} \le 0 + \implies \Vert x - x'\Vert = 0 \implies x = x' + .\end{align*} + \item $g(M) = M \implies x^{(k)} = g\left( x^{(k-1)} \right)$, $k \in \N$ ist wohldefiniert, + d.h. $x^{(k)} \in M$, $\forall k \in \N$, falls $x^{(0)} \in M$. + + Z.z.: $x^{(k)}$ konvergiert mit $\displaystyle \lim_{k \to \infty} x^{(k)} \in M$, also + g.z.z.: $\left( x^{(k)} \right)_{k \in \N}$ ist Cauchy-Folge. + \begin{leftright} + \begin{salign*} + \Vert x^{(k+1)} - x^{(k)} \Vert &= \left\Vert g(x^{(k)}) - g(x^{(k-1)}) \right\Vert \\ + &\le L \Vert x^{(k)} - x^{(k-1)} \Vert \\ + &= L \left\Vert g(x^{(k-1)}) - g(x^{(k-2)}) \right\Vert \\ + &\le L \cdot L \cdot \Vert x^{(k-1)} - x^{(k-2)} \Vert\\ + &\le \underbrace{L \cdot \ldots \cdot L}_{k} + \Vert x^{(1)} - x^{(0)} \Vert \\ + &= L^{k} \Vert x^{(1)} - x^{(0)} \Vert + \intertext{Seien $k, m$ beliebig. Dann gilt + $\forall \epsilon > 0$} + \Vert x^{(k+m)} - x^{(k)} \Vert &= \Vert x^{(k+m)} - x^{(k+m-1)} + x^{(k+m-1)} - \ldots x^{(k+1)} - x^{(k)} \Vert \\ + &\le \Vert x^{(k+m)} - x^{(k+m+1)} \Vert + + \ldots + \Vert x^{(k+1)} - x^{(k)} \Vert \\ + &= L^{m-1} \Vert x^{(k+1)} - x^{(k)} \Vert + + L^{m-2} \Vert x^{k+1} - x^{k} \Vert + + \ldots + \Vert x^{(k+1)} - x^{(k)} \Vert \\ + &= (L^{m-1} + L^{m-2} + \ldots + 1) \Vert x^{(k+1)} - x^{(k)} \Vert \\ + &= \frac{1 - L^{m}}{1 - L} \Vert x^{(k+1)} - x^{(k)} \Vert \\ + &\le \frac{1 - L^{m}}{1 - L} L^{k} \Vert x^{(1)} - x^{(0)} \Vert \\ + &\le \frac{L^{k}}{1-L} \Vert x^{(1)} - x^{(0)} \Vert \\ + &\stackrel{L < 1}{<}\epsilon \qquad \text{ für } k \text{ groß genug} + .\end{salign*} + \end{leftright} + Also ist $\left( x^{(k)} \right)_{k \in \N}$ eine Cauchy-Folge in $M$ und es existiert + ein $x^{*} \in M$, s.d. $\left( x^{(k)} \right)_{k \in \N}$ gegen + $x^{*}$ konvergiert. $x^{*}$ ist ein Fixpunkt von $g$, weil + \[ + x^{*} = \lim_{k \to \infty} x^{(k)} = \lim_{k \to \infty} g\left( x^{(k-1)} \right) + \qquad \stackrel{g \text{ stetig}}{=} \qquad + g\left( \lim_{k \to \infty} x^{(k-1)} \right) = g(x^{*}) + .\] + \item Für festes $k \in \N$ gilt + \begin{align*} + \Vert \underbrace{x^{(k+m)}}_{\xrightarrow{m \to \infty} x^{*}} - x^{(k)} \Vert \le \frac{L^{k}}{1-L} \Vert x^{(1)} - x^{(0)} \Vert \implies \Vert x^{*} - x^{(k)} \Vert \le \frac{L^{k}}{1 -L} + \Vert x^{(1)} - x^{(0)} \Vert + .\end{align*} + \end{enumerate} +\end{proof} + +\begin{bem} + Für den Beweis ist wichtig, dass der grundlegende Raum vollständig ist, d.h. dass alle + Cauchy-Folgen in diesem Raum konvergieren. +\end{bem} + +\begin{bem}[Anwendung: Lineare Gleichungssysteme] + $A = \left( a_{ij} \right)_{i,j = 1}^{n} \in \mathbb{K}^{n \times n}$ regulär und $b = (b_i)_{i = 1}^{n} \in \mathbb{K}^{n}$. Da $A$ regulär, hat das LGS $Ax = b$ genau + eine Lösung $x^{*} = A^{-1}b$. Sei $g(x) := x - \sigma (Ax - b)$ mit + $\sigma \in \mathbb{K} \setminus \{0\} $. + + Fixpunktiteration $x^{(k)} = x^{(k-1)} - \sigma (Ax^{(k-1}) - b)$, $k \in \N$ konvergiert, wenn + $g$ kontraktiv ist. Zum Beispiel in $\ell_2$: + \begin{align*} + \Vert g(x) - g(y) \Vert_2 &= \Vert x - \sigma(Ax - b) - y + \sigma(Ay -b)\Vert_2 \\ + &= \Vert x - y - \sigma A(x-y) \Vert_2 \\ + &= \Vert (\mathbb{I} - \sigma A)(x-y) \Vert_2 \\ + &\le \Vert \mathbb{I} - \sigma A \Vert_2 \Vert x - y \Vert_2 + ,\end{align*} d.h. $g$ kontraktiv, falls $\vert \mathbb{I} - \sigma A \Vert_2 < 1$. + + Frage: Wahl von $\sigma$? Wähle $\sigma = \Vert A \Vert_{\infty}^{-1} = \frac{1}{\Vert A \Vert_{\infty}}$, falls $A$ hermitesch und positiv definit (= ,,Richardson Iteration``). Zu überprüfen + $\left\Vert \mathbb{I} - \frac{A}{\Vert A \Vert_{\infty}} \right\Vert_2 < 1$. + Da $A$ positiv definit und hermitesch, sind alle Eigenwerte $\lambda > 0$. Es gilt + $\forall $ EW: $0 < \lambda \le \Vert A \Vert_{\infty}$. Für EW von $\mathbb{I} - \frac{A}{\Vert A\Vert_{\infty}} $ gilt $\mu = 1 - \frac{\lambda}{\Vert A \Vert_{\infty}}$, $\lambda$ Eigenwert von $A$. Also + $0 \le \underbrace{1 - \frac{\lambda}{\Vert A \Vert_{\infty}}}_{= \mu} < 1$, also + $\underbrace{\Big\Vert \underbrace{\mathbb{I} - \frac{A}{\Vert A \Vert_{\infty}}}_{\text{hermitesch}} \Big\Vert_2}_{\text{Spektralnorm}} < 1$. Falls $A$ hermitesch und positiv definit, ist also + die Richardson Iteration konvergent. +\end{bem} + +\begin{definition}[Starke Monotonie] + Eine Funktion $f\colon D \subseteq \R^{n} \to \R^{n}$ heißt stark monoton, wenn + eine Konstante $m > 0$ existiert, s.d. $\forall x, y \in D$ gilt + \[ + (f(x) - f(y), x-y)_2 \ge m \Vert x - y \Vert_2^2 + .\] +\end{definition} + +\begin{bem}[Anwendung: Nichtlineare Gleichungssysteme] + Sei $f\colon D \subseteq \R^{n} \to \R^{n}$ Lipschitz stetig mit $L$ und stark monoton mit $m > 0$. + Betrachte $f(x) = b$, $g(x) := x - \theta (f(x) - b)$. + + Frage: Wahl von $\theta$, s.d. $\forall x ^{(0)} \in D$ die Fixpunktiteration konvergiert? + Es ist + \begin{align*} + \Vert g(x) - g(y) \Vert_2^2 &= \Vert x - \theta (f(x) - b) - y + \theta (f(y) - b) \Vert_2^2 \\ + &= \Vert x - y - \theta (f(x) - f(y)\Vert_2^2 \\ + &= \Vert x - y \Vert_2^2 - 2 \theta (x-y, f(x) - f(y))_2 + + \theta^2 \Vert f(x) - f(y) \Vert_2^2 \\ + &\le \Vert x - y \Vert_2^2 - 2 \theta m \Vert x - y \Vert_2^2 + + \theta^2 L^2 \Vert x- y \Vert_2^2 \\ + &= (1 - 2\theta m + \theta^2 L^2) \Vert x - y \Vert_2^2 + .\end{align*} + Die Fixpunktiteration konvergiert, falls $1 - 2 \theta m + \theta^2L^2 < 1$, d.h. + für $\theta \in \left( 0, \frac{2m}{L^2} \right) $. Dann existiert + ein $\displaystyle x^{*} = \lim_{k \to \infty} x^{(k)}$ mit $g(x^{*}) = x^{*}$. Ist + $x^{*}$ eindeutig? Seien $x, x'$ zwei Lösungen. Dann ist + \begin{salign*} + 0 &= (\underbrace{f(x) - b + b -f(x')}_{= 0}, x - x')_2 \\ + &= (f(x) - f(x'), x-x')_2 \\ + &\stackrel{f\text{ stark monoton}}{\ge} m \Vert x - x' \Vert_2^2 \\ + &> 0 + .\end{salign*} + Also $x = x'$, damit ist $x^{*}$ eindeutig. +\end{bem} + +\end{document} diff --git a/ana11.pdf b/ana11.pdf new file mode 100644 index 0000000000000000000000000000000000000000..87eaf13995f85ffe703cbfdc30f8023cd64f6af2 GIT binary patch literal 168120 zcmb5VW02+Dvn5=%ZM)0tvVF?7*=5_dZQHhO+jbYaT=hOP@4XWf@$ZQH%=z^Dn7K~w zTx;#zNiHuUM#n_Y217o#G`t4$lZcVX*3c4$hX;lMU}9tDWKP8Vlbwy|fB#?@#4N0w zOdN<9#HiD|`1oKPog7RItYO?XwzVc5a5*rxFMoKrC=ZL=`H&@7NJ`n$ zjE~fJM>gaYrfe}S-^V?!B@cVO;{kmFb@Z1UjU^hh8zzGJN521j-hw@QNS)2CzcBR` ztIUj0-JRC$b$9>ss#HTG8@}S^?XH6@Ruq?<5@KFhnX0xtTS>@ankbQ|VUM+J(*M$M(f}J6omeb*8Z6>WcM8 zmBnFDLMbXtahM7oYAAH>C{sXj0kbgPaWeT#`Om7%i1G>KQeqOZ>R{v{Dvj!9(twGd z_nB{Xn;#xC*B8y-j&~rMEghB}p|yJ5mMv z185m`oN#St5Mxg8vSs*U+KMxfbj1Mz$5sWTVoaSZ17d<{i^)V8GOS58;NY6zywQbU 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+++ b/ana11.tex @@ -0,0 +1,217 @@ +\documentclass{lecture} + +\begin{document} + +\chapter{Differenzierbare Funktionen in \texorpdfstring{$\R^{n}$}{R\unichar{"207F}}} + +Betrachte die Abbildung $f \colon D \subseteq \R^{n} \to \R^{n}$. Die Stetigkeitsdefinition von $f$ entspricht +der Stetigkeit von $f$ in $\R$. Aber: Differenzierbarkeit einer Funktion $g\colon I \subseteq \R \to \R$, +\[ + g'(x_0) := \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} +.\] Für $h \in \R^{n}$ \underline{nicht sinnvoll}. + +\section{Partielle Differenzierbarkeit} + +\begin{definition}[Partielle Ableitung] + Sei $D \subseteq \R^{n}$ offen und $f\colon D \to \R$. + \begin{itemize} + \item $f$ heißt + im Punkt $x \in D$ partiell differenzierbar nach $i$-ter Koordinatenrichtung, falls der + Grenwert + \[ + \partial_i f(x) := \lim_{h \to 0} \frac{f\left( x + h \cdot e^{(i)} \right) - f(x)}{h} + .\] existiert mit $e^{(i)} := $ $i$-te Spalte der $n \times n$ Einheitsmatrix. + Schreibweise auch $\frac{\partial f}{\partial x_i}(x)$ oder $\frac{\partial f(x)}{\partial x_i}$. + \item $f$ heißt partiell differenzierbar in $x \in D$, falls $\partial_i f(x)$ für alle $1 \le i \le n$ + existieren. + \item Sind $\partial_i f(x)$ $\forall i$ stetig, dann heißt $f$ stetig partiell differenzierbar. + \item Falls $f \colon D \subseteq \R^{n} \to \R^{m}$, dann heißt $f$ (stetig) partiell differenzierbar, + falls alle Komponentenfunktionen $f_j$ $(1 \le j \le m)$ (stetig) partiell differenzierbar + in $x \in D$ sind, d.h. wenn $\partial_i f_j(x)$ $\forall i=1,\ldots,n$, $j = 1,\ldots,m$ existieren. + \end{itemize} +\end{definition} + +\begin{bem}[Interpretation als gewöhnliche Ableitung] + Sei $f(x) = f(x_1, \ldots, x_n)$. Definiere $\tilde{f}(\xi) = f(x_1, \ldots, x_{i-1}, \xi, x_{i+1}, \ldots, x_n)$, d.h. $x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n$ fest. Dann ist + \[ + \partial_i f(x) = \frac{\d{\tilde{f}(\xi)}}{\d\xi} + .\] + D.h. für partielle Ableitungen + gelten analoge Regeln, wie für die gewöhnliche Ableitung, insbesondere Produktregel, Quotientenregel + und auch Kettenregel. +\end{bem} + +\begin{bsp} + Die Funktion + \[ + r(x) := \Vert x \Vert_2 = \sqrt{\sum_{i=1}^{n} x_i^2} + .\] ist in $D = \R^{n} \setminus \{0\} $ stetig partiell differenzierbar mit den partiellen + Ableitungen + \begin{align*} + \partial_i r (x_1, \ldots, x_i, \ldots, x_n) + \quad \qquad \stackrel{\text{gew. Kettenregel}}{=} \quad \qquad + \frac{1}{2} \frac{1}{\sqrt{\sum_{i=1}^{n} x_i^2} } 2x_i + = \frac{x_i}{\Vert x \Vert_2} + .\end{align*} + Sei $F\colon \R_+ \to \R$ beliebige differenzierbare Funktion. Dann ist $f(x) = F(r(x))$ auf + $\R^{n} \setminus \{0\} $ definiert und partiell differenzierbar. + \begin{align*} + \partial_i f(x) = \frac{\d F(r(x))}{\d y} \partial_i r(x) = F'(r(x)) \frac{x_i}{\Vert x\Vert_{2}} + = F'(r(x)) \frac{x_i}{r(x)} + .\end{align*} +\end{bsp} + +\begin{bem} + Für $n =1$ gilt: $f$ differenzierbar $\implies$ $f$ stetig. Für $n > 1$ und $f$ partiell + differenzierbar, folgt i.A. nicht, dass $f$ stetig ist. +\end{bem} + +\begin{satz} + Sei $D \subseteq \R^{n}$ offen und $f\colon D \to \R$. Für $x \in D$ gelte: $\exists K_r(x) \subseteq D$, + s.d. die partiellen Ableitungen $\partial_i f(y)$, $i = 1, \ldots, n$ beschränkt sind $\forall y \in K_r(x)$, + d.h. + \[ + \sup_{x \in K_r(x)} | \partial_i f(y) | \le M, \quad i = 1, \ldots, n + .\] Dann gilt $f$ stetig in Punkt $x$. +\end{satz} + +\begin{proof} + Sei $n = 2$ und $y = (y_1, y_2) \in K_r(x)$. Es ist + \begin{salign*} + f(y_1, y_2) - f(x_1, x_2) &= f(y_1, y_2) - f(x_1, y_2) + f(x_1, y_2) - f(x_1, x_2) + \intertext{Mit $y_2$ fest, folgt mit dem Mittelwertsatz der Differentialrechnung: + $\exists \xi = \xi(y_2)$ zwischen $y_1$ und $x_1$, also} + f(y_1, y_2) - f(x_1, y_2) &\stackrel{\text{MWS}}{=} \partial_1 f(\xi, y_2) (y_1 - x_1) + \intertext{Analog für $x_1$ fest und $\zeta(x_1)$ zwischen $y_2$ und $x_2$} + f(x_1, y_2) - f(x_1, x_2) &\stackrel{\text{MWS}}{=} \partial_2 f(x_1, \zeta) (y_2 - x_2) + \intertext{Dann folgt} + |f(y) - f(y)| &\le \underbrace{|\partial_1 f(\xi, y_2)|}_{\le M} |y_1 - x_1| + + \underbrace{|\partial_2 f(x_2, \zeta)|}_{\le M} |y_2 - x_2| \\ + &\le M (|y_1 - x_1| + |y_2 - x_2|) \\ + &= M \Vert y - x \Vert_1 + .\end{salign*} + Sei $\epsilon \in (0, r]$ beliebig, dann gilt für $\delta := \frac{\epsilon}{M}$ + \[ + \Vert y -x \Vert_1 \le \delta \implies | f(y) - f(y) | \le \epsilon +.\] Also $f$ stetig in $x$ und wegen $|f(y) - f(x)| \le M \Vert y - x \Vert_1$ sogar Lipschitz-stetig. +Für $n > 2$ analog. +\end{proof} + +\begin{definition} + Sei $f\colon D \subseteq \R^{n} \to \R$ partiell differenzierbar mit $\partial_i f \colon D \to \R$. + \begin{itemize} + \item Falls $\partial_i f$ partiell differenzierbar sind, dann heißt $f$ zweimal + partiell differenzierbar mit den partiellen Ableitungen zweiter Ordnung + \[ + \partial_i \partial_j f(x) := \frac{\partial^2 f(x)}{\partial x_i \partial x_j} + := \frac{\partial}{\partial x_i} \left( \frac{\partial f(x)}{\partial x_j} \right) + .\] + \item $f$ heißt $k$-mal stetig partiell differenzierbar, falls alle partiellen Ableitungen + $k$-ter Ordnung von $f$ existieren und stetig sind. + \end{itemize} +\end{definition} + +\begin{bem} + Im Allgemeinen ist $\partial_i \partial_j f(x) \neq \partial_j \partial_i f(x)$! +\end{bem} + +\begin{satz}[Vertauschbarkeit der Differentiationsreihenfolge] + Sei $D \subseteq \R^{n}$ offen und $f\colon D \to \R$ zweimal stetig differenzierbar in + einer Umgebung $K_r(x) \subseteq D$ eines Punktes $x \in D$. Dann gilt + \[ + \partial_i \partial_j f(x) = \partial_j \partial_i f(x), \qquad \forall i,j=1,\ldots,n + .\] Allgemein: Für eine $k$-mal stetig partiell differenzierbare Funktion ist die Reihenfolge + der partiellen Ableitungen vertauschbar. +\end{satz} + +\begin{proof} + \begin{enumerate}[1)] + \item Sei $n = 2$ und + \[ + A := \underbrace{f(x_1 + h_1, x_2 + h_2) - f(x_1 + h_1, x_2)}_{= \varphi(x_1 + h_1)} + - \underbrace{f(x_1, x_2 + h_2) + f(x_1, x_2)}_{= \varphi(x_1)} + .\] Definiere $\varphi(x) := f(x, x_2 + h_2) - f(x, x_2)$. Dann ist + $A = \varphi(x_1 + h_1) - \varphi(x_1)$. Mit dem MWS bezügl. $x_1$ folgt + \begin{salign*} + \varphi(x_1 + h_1) - \varphi(x_1) &= \varphi'(x_1 + \theta_1) \cdot h_1, \quad \theta_1 \in (0, h_1) + \intertext{Für $\varphi'$ gilt} + \varphi'(x_1) &= \partial_1 f(x_1, x_2 + h_2) - \partial_1 f(x_1, x_2) \\ + &\stackrel{\text{MWS bezügl. } x_2}{=} + \partial_2 (\partial_1 f(x_1, x_2 + \theta_1')) \cdot h_2, \quad \theta_1' \in (0, h_2) + \intertext{Dann folgt} + \varphi'(x_1 + \theta_1) &= \partial_2 (\partial_1 f(x_1 + \theta_1, x_2 + \theta_1')) h_2 + .\end{salign*} + Und damit ist + \begin{salign*} + A &= \varphi'(x_1 + \theta_1) h_1 = \partial_2 (\partial_1 f(x_1 + \theta_1, x_2 + \theta_1')) + \cdot h_1 \cdot h_2 + \intertext{Analog definiere $\psi(x) := f(x_1 + h_1, x) - f(x_1, x)$, dann} + A &= \psi(x_2 + h_2) - \psi(x_2) \\ + &\stackrel{\text{wie oben}}{=} h_2 \psi'(x_2 + \theta_2) \\ + &= h_1 \cdot h_2 \partial_ + (\partial_2 f(x_1 + \partial_2, x_2 + \partial_2')), \quad \theta_2 \in (0, h_1), + \theta_2' \in (0, h_2) + .\end{salign*} + Also folgt + \[ + \partial_2 \partial_1 f(x_1 + \theta_1, x_2 + \theta_1') = \frac{A}{h_1 \cdot h_2} + = \partial_1 \partial_2 f(x_1 + \theta_2, x_2 + \theta_2') + .\] + Die partiellen Ableitungen $\partial_2 \partial_1 f$ und + $\partial_1 \partial_2 f$ sind stetig in $K_r(x)$, also gilt + für $h_1, h_2 \to 0$, d.h. $x_1 + \theta_1 \to x_1, x_2 + \theta_1' \to x_2, \ldots$ + \begin{salign*} + \partial_2 \partial_1 f(x_1 + \theta_1, x_2 + \theta_1') &\xrightarrow{h_1, h_2 \to 0} + \partial_2 \partial_1 f(x) \\ + \partial_1 \partial_2 f(x_1 + \theta_2, x_2 + \theta_2') + &\xrightarrow{h_1, h_2 \to 0} \partial_1 \partial_2 f(x) + .\end{salign*} + Also $\partial_1 \partial_2 f(x) = \partial_2 \partial_1 f(x)$. Für $n > 2$ analog. + \item Sei $f$ k-mal stetig differenzierbar. Dann folgt durch Induktion nach $k$ + \[ + \partial_1 \ldots \partial_k f(x) = \partial_{i_1} \ldots \partial_{i_k} f(x) + .\] für jede Permutation ($i_1, \ldots, i_k)$ von $(1 \ldots k)$. + \end{enumerate} +\end{proof} + +\begin{definition}[Begriffe der Vektoranalysis] + \begin{itemize} + \item \underline{Gradient}: Sei $D \subseteq \R^{n}$ offen und $f\colon D \to \R$ eine partiell + differenzierbare Funktion. Der Vektor + \[ + \text{grad} f(x) := \nabla f(x) := \begin{pmatrix} \partial_1 f(x) \\ \vdots \\ \partial_n f(x) \end{pmatrix} \in \R^{n} + \] heißt der Gradient von $f$ in $x \in D$. + \item \underline{Hesse-Matrix}: Sei $D \subseteq \R^{n}$ offen, $f\colon D \to \R$ + eine zweimal partiell differenzierbare Funktion. Die Matrix + \[ + H_f(x) := \nabla^2 f(x) := (\partial_i \partial_j f(x))_{i,j=1}^{n} \in \R^{n \times n} + .\] heißt Hesse-Matrix von $f$ im Punkt $x \in D$. + \item \underline{Jacobi-Matrix}: Sei $D \subseteq \R^{n}$ offen, $f\colon D \to \R^{m}$ + eine partiell differenzierbare Vektorfunktion. Die Matrix + \[ + J_f(x) := \begin{pmatrix} \partial_1 f_1 & \ldots &\partial_n f_1 \\ + \vdots & & \vdots \\ + \partial_1 f_m & \ldots & \partial_n f_m + \end{pmatrix} \in \R^{m \times n} + .\] heißt Funktionalmatrix oder auch Jacobimatrix von $f$ in $x \in D$. + + Schreibweise: $J_f(x) = f'(x) = \left( \nabla f(x) \right)^{T}$. + \end{itemize} +\end{definition} + +\begin{bsp} + $r(x) = \Vert x \Vert_2$. + \[ + \nabla r(x) = \begin{pmatrix} \vdots \\ \partial_i r(x) \\ \vdots \end{pmatrix} + = \begin{pmatrix} \vdots \\ \frac{x_i}{r(x)} \\ \vdots \end{pmatrix} \in \R^{n} + .\] Für die Hesse-Matrix $\nabla ^2 r(x) = \left( \partial_j \frac{x_i}{r(x)} \right)_{i,j=1}^{n}$ folgt + \[ + \partial_j \left( \frac{x_i}{r(x)}\right) + = \begin{cases} + \frac{r(x) - x_i \frac{x_i}{r(x)}}{r(x)^2} = \frac{1}{r(x)} - \frac{x_i^2}{r(x)^2} & i = j \\ + - \frac{x_i \frac{x_j}{r(x)}}{r(x)^2} = - \frac{x_i x_j}{r(x)^{3}} & i \neq j + \end{cases} + .\] Diese Hesse-Matrix ist die Jacobi-Matrix der Vektorfunktion $v(x) = \frac{x}{r(x)} = \begin{pmatrix} \vdots \\ \frac{x_i}{r(x)} \\ \vdots \end{pmatrix} $. +\end{bsp} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index 0972e3dbd5ffa17754908dd1b8d1adb6a37f6897..7c474b2e7f1cc20683451387ce43d1681a8ff577 100644 GIT binary patch delta 350319 zcmV)oK%Bpc#3s&_BM&7|L`E$!E;R}z(A34)3X=ih6PJpG1uTDskGl@SFbqZaeuc+m zh>4w;k`cs1z*0dr#88QcDo`u6UHE%a%ErKAUs*o)+LL%F2<}~G;SLryfBPb4lQnyw z9GS#Jya7u<@&SY`iercuIH`GEK3`qcY2(PLtu{PO^&EGJ79heP!UJs_<1|SljLx}{ zF52dX2~E(l{fIYQtsYuLmHoA?>T(?IHP2Mj^_4cHYE7yf62J?33CTG}7ryr#o8xsf zWXpC~`+02@vo{0=Jy4e`vjrcQ%?|_z12Hf-mvQO?DSzcyOLN;c5Wf3Y@JJnq;Qg4M z+B`IA+L?^2$)R=*R0UHQQB;Sbk~Y6S3xcFcijq?&brRAuFN=#OFSw2EQ&ulSEm}c>zScS%=b?dhC z-Hg$ASO4MRi$k%g+^+BgD8(AbEG$SwZ$ZKlY&4-5abcQO5v)rvr7%ssS~JZ}y9L&6 z0UB9eA4tdR?KR{4nknTn8{WP0{jepfV0X~8C4YSXVU`FbVgwJ2ALhZ0IqC#UbVx>5 z2{D8P``q^8u~}$imS?sM2OQd`Ob^?}!lZ9uq7cIdrmLrU8iwfJ2!%9b?fa{VDJW7> zw*%6jU>j1!%j1a%gJ!qV+X=WeVyc_zOWE?Fv73BM0V;;Q2@w!mLXB}{4X!T>@1>Q^ 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RFu7HF>b>v1_o4Hr{vRc&1Oos7 diff --git a/analysisII.tex b/analysisII.tex index e348298..c474825 100644 --- a/analysisII.tex +++ b/analysisII.tex @@ -25,5 +25,7 @@ \input{ana7.tex} \input{ana8.tex} \input{ana9.tex} +\input{ana10.tex} +\input{ana11.tex} \end{document}