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+\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}} +\newcommand{\dv}[2]{\frac{\d #1}{\d #2}} + +\section{Mittelwertsatz} +\begin{bem} + Reminder: + \begin{enumerate}[(1)] + \item Ist $f: [a,b] \to \R$ differenzierbar, dann gilt (HDI): + \begin{salign*} + f(x+h) - f(x) = \int_{0}^{1} \dv{}{s} f(x+sh) \d{s} &\stackrel{\text{Kettenregel}}{=} \int_{0}^{1} f'(x+sh) \cdot h \d{s} \\ &= \left( \int_{0}^{1} f'(x+sh) \d{x} \right) \cdot h. + \end{salign*} +. \item Mittelwertsatz der Differentialrechnung: $\exists \tau \in (0,1)$ sodass: + \begin{salign*} + f(x+h) - f(x) = f'(x+ \tau h) \cdot h. + \end{salign*} + \item Sei $A = \left(a_{i,j} \right)_{i,j = 1}^{m,n} : [a,b] \to \R^{m \times n}$. Dann sei: + \begin{salign*} + \int_{a}^{b} A(s) \d{s} \coloneqq \left( \int_{a}^{b} a_{i,j}(s) \d{s} \right)_{i,j=1}^{m,n} + \end{salign*} + \end{enumerate} +\end{bem} +\begin{satz}[Mittelwertsatz] + Seien $D \in \R^{n}$ offen, $f: D \to \R$ stetig differenzierbar, sei $x \in D$ und $h \in \R^{n}$ sodass $x + sh \in D$, für $s \in [0,1]$. Dann gilt: + \begin{salign*} + f(x+h) - f(h) = \left( \int_{0}^{1} \nabla f(x+sh) \d{s}, h \right)_{2} = \left( \int_{0}^{1} \nabla f(x+sh) \d{s}\right)^{T} \cdot h. + \end{salign*} + Sei $f: D \to \R^{m}$ stetig differenzierbar, mit Jacobi-Matrix $J_{f}(x)$, dann gilt: + \begin{salign*} + f(x+h) - f(h) = \left( \int_{0}^{1} J_{f}(x + sh) \d{s} \right) h. + \end{salign*} +\end{satz} +\begin{proof} + Sei $f: D \to \R^{m}$. Sei $g_{j}: [0,1] \to \R ,g_{j}(s) \coloneqq f_{j}(x+sh)$. Dann gilt: + \begin{salign*} + f_{j}(x+h) - f_{j}(x) = g_{j}(1) - g_{j}(0) & \ \ \stackrel{\text{MWS}}{=} \ \ \int_{0}^{1} g_{j}'(s) \d{s} \ \ \ \ \stackrel{\text{Kettenregel}}{=} \ \ \ \ \int_{0}^{1} \sum_{i=1}^{n} \pdv{f}{x_{i}}(x+sh) \cdot h_{i} \d{s}. + \end{salign*} + Ist $m = 1$, so gilt: + \begin{salign*} + f(x+h) - f(x) &= \int_{0}^{1} \sum_{i=1}^{n} \pdv{f}{x_{i}}(x+sh) \d{s} \cdot h_{i} = \sum_{i=1}^{n} \left( \int_{0}^{1} \pdv{f_{i}}{x_{i}}(x+sh) \d{s} \right) \cdot h_{i} \\ &= \left( \int_{0}^{1} \nabla f(x+sh) \d{s}, h\right)_{2} + \end{salign*} + Ist $m \geq 2$, so gilt analog zu oben: + \begin{salign*} + f(x+h) - f(x) = \left( \int_{0}^{1} J_{f}(x+sh) \d{s} \right) \cdot h. + \end{salign*} +\end{proof} +\begin{bem} + Für $m = 1$, d.h. $f: \R^{n} \supset D \to \R$ gilt sogar für ein bestimmtest $\tau \in (0,1)$: + \begin{salign*} + f(x+h) - f(x) = \int_{0}^{1} \left( \nabla^{T} f(x+sh) \cdot h \right)\d{s} = \nabla^{T} f(x+\tau h) \cdot h. + \end{salign*} + Für $m \geq 2$ im Allgemeinen aber (da $\tau \in [0,1]$ nicht für alle Komponenten gleich gewählt werden kann): + \begin{salign*} + f(x+h) - f(x) \neq J_{f}(x + \tau h) \cdot h. + \end{salign*} +\end{bem} +\begin{lemma} + Seien $v: [a,b] \to \R^{n}$ und $A: [a,b] \to \R^{m \times n}$ stetig. Dann gilt: + \begin{salign*} + \norm{\int_{a}^{b} v(s) \d{s} }_{2} \leq \int_{a}^{b} \norm{v(s)}_{2} \d{s}, && \norm{\int_{a}^{b} A(s) \d{s} }_{2} \leq \int_{a}^{b} \norm{A(s)}_{2} \d{s} + \end{salign*} +\end{lemma} +\begin{proof} + Sei $u \in \R^{n}$, $u = \int_{a}^{b} v(s) \d{s} = \begin{pmatrix} + \int_{a}^{b} v_{1}(s) \d{s} \\ \vdots \\ \int_{a}^{b} v_{n}(s) \d{s} + \end{pmatrix}$ und $K = \norm{u}_{2}$. Dann gilt: + \begin{salign*} + K^{2} &\stackrel{\ell_{2} \text{Norm}}{=} \left(u,u\right)_{2} = \left( \int_{a}^{b} v(s) \d{s}, u\right)_{2} = \int_{a}^{b} \left(v(s), u\right)_{2} \d{s} \\ + &\stackrel{\text{CSU}}{\leq} \int_{a}^{b} \norm{v(s)}_{2} \cdot \norm{u}_{2} \d{s} = K \cdot \int_{a}^{b} \norm{v(s)}_{2} \d{s} \\ \hfill \\ + \implies \ & K = \norm{\int_{a}^{b} v(s) \d{s} }_{2} \leq \int_{a}^{b} \norm{v(s)}_{2} \d{s}. + \end{salign*} + Der Beweis für $A(s)$ folgt ganz analog mit $u = \int_{a}^{b} A(s) \d{s} \in \R^{m \times n}$. +\end{proof} +\begin{definition} + $D \subset \mathbb{K}^{n}$ heißt \underline{konvex}, genau dann wenn: für alle $x,x' \in D$ und für alle $\lambda \in [0,1]$ gilt $\lambda \cdot x + (1-\lambda)x' \in D$. \\ + Geometrische: für zwei Punkte in $D$ liegt die Verbindungsstrecke der beiden Punkte stets ganz in $D$. +\end{definition} +\begin{korrolar} + Seien $D \subset \R^{n}$ offen, $f: D \to \R^{n}$ stetig differenzierbar. Sei $x \in D$ und $\varepsilon > 0$ sodass $K_{\varepsilon}(x) \subset D$. Dann gilt: + \begin{salign*} + \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall y \in K_{\varepsilon} + \end{salign*} + mit $M \coloneqq \sup_{z \in K_{\varepsilon}(x)} \norm{J_{f}(z)}_{2}$, das heißt $f$ ist lokal Lipschitz-stetig. \\ + Sei $D$ konvex, dann gilt + \begin{salign*} + \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall x,y \in K_{\varepsilon} + \end{salign*} + mit $M \coloneqq \sup_{z \in D} \norm{J_{f}(z)}_{2}$, das heißt $f$ ist auf $D$ Lipschitz-stetig. +\end{korrolar} +\begin{proof} + Aus obigem Lemma folgt: + \begin{salign*} + \norm{\int_{0}^{2} J_{f}(x+sh) h \d{s} }_{2} &\leq \int_{0}^{1} \norm{J_{f}(x+sh)}_{2} \norm{h}_{2} \d{s} \\ & \leq \int_{0}^{1} \norm{J_{f}(x+sh)}_{2} \norm{h}_{2} \d{s} \leq \sup_{0