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a/ana13.tex b/ana13.tex index c91ca5d..95d6259 100644 --- a/ana13.tex +++ b/ana13.tex @@ -9,7 +9,7 @@ \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. + 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{s} \right) \cdot h. \end{salign*} . \item Mittelwertsatz der Differentialrechnung: $\exists \tau \in (0,1)$ sodass: \begin{salign*} @@ -24,17 +24,17 @@ \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. + f(x+h) - f(x) = \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. + f(x+h) - f(x) = \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: + Sei $f: D \to \R^{m}$. Sei $g_{j}\colon [0,1] \to \R ,g_{j}(s) \coloneqq f_{j}(x+sh)$. 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}. + 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_j}{x_{i}}(x+sh) \cdot h_{i} \d{s}. \end{salign*} Ist $m = 1$, so gilt: \begin{salign*} @@ -46,7 +46,7 @@ \end{salign*} \end{proof} \begin{bem} - Für $m = 1$, d.h. $f: \R^{n} \supset D \to \R$ gilt sogar für ein bestimmtest $\tau \in (0,1)$: + Für $m = 1$, d.h. $f: \R^{n} \supset D \to \R$ gilt sogar für ein bestimmtes $\tau \in (0,1)$: \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*} @@ -56,6 +56,7 @@ \end{salign*} \end{bem} \begin{lemma} + \label{lemma:dreieck-integrale} 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} @@ -74,7 +75,7 @@ \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 $D$. + Geometrisch: 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: @@ -84,27 +85,28 @@ 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} + \norm{f(y) - f(x)}_{2} \leq M \cdot \norm{y-x}_{2} \ \ \ \ \ \forall x,y \in D \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: + Aus Lemma \ref{lemma:dreieck-integrale} 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_{0fqnDM+VucY~yKNJ=9hAoZT-^L>AR z%yq4M-D~fC@3q%nbIr_|hE@8GP5PQjWKvt~lMe{3c9Kf3tm6kyOjNy&$Cd_&RukRA zLx+$fhmhf+7XtJ`Y#KraSW)1GK+SUmR@jOYnJ{j^jowwe;-Xyy_-jHcXo7mgsgBG{ 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