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z4PK~r{{yzV(Z>p9Ze(+Ga%Ev{3T19&Z(?c+b97;Hba--QW(qMiIWaId3MC~)Peuyh CgJ0MH diff --git a/ana13.tex b/ana13.tex index 87448ef..1049477 100644 --- a/ana13.tex +++ b/ana13.tex @@ -11,7 +11,7 @@ \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{s} \right) \cdot h. \end{salign*} - \item Mittelwertsatz der Differentialrechnung: $\exists \tau \in (0,1)$ sodass: +. \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*} @@ -22,7 +22,7 @@ \end{enumerate} \end{bem} \begin{satz}[Mittelwertsatz] - Seien $D \subset \R^{n}$ 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: + 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(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*} @@ -32,9 +32,9 @@ \end{salign*} \end{satz} \begin{proof} - Sei $f: D \to \R^{m}$. Sei $g_{j}\colon [0,1] \to \R ,\ g_{j}(s) \coloneqq f_{j}(x+sh)$. 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) & \overset{\text{HDI}}{=} \int_{0}^{1} g_{j}'(s) \d{s} \overset{\text{Kettenregel}}{=} \int_{0}^{1} \sum_{i=1}^{n} \pdv{f_j}{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*} @@ -50,7 +50,7 @@ \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 nicht (da $\tau \in [0,1]$ nicht für alle Komponenten gleich gewählt werden kann): + 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*} @@ -75,10 +75,10 @@ \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 $\lambda \cdot x + (1-\lambda)x' \in D$. \\ - Geometrisch: 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{korollar} - Seien $D \subset \R^{n}$ offen, $f: D \to \R^{m}$ stetig differenzierbar. Sei $x \in D$ und $\varepsilon > 0$ sodass $K_{\varepsilon}(x) \subset D$. Dann gilt: + 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*} @@ -97,7 +97,7 @@ Sei $D$ nun konvex. Für $x,y \in D$ gilt dann: $$z = ty + (1-t)x = x + t(y-x) \in D, \ \ \ \ \ t \in [0,1].$$ Sei $g(t) \coloneqq f(x+ t(y-x))$ für $t \in [0,1]$. Dann gilt für $i \in \{1,...,m\}$: \begin{salign*} - f_{i}(y) - f_{i}(x) = g_{i}(1) - g_{i}(0) = \int_{0}^{1} g_{i}'(s) \d{s} = \int_{0}^{1} \sum_{j=1}^{n} \pdv{f_{i}(x+s(y-x))}{x_{j}}(y_{j}-x_{j}) \d{s}. + f_{i}(y) - f_{i}(x) = g_{i}(1) - g_{i}(0) = \int_{0}^{1} g_{i}'(s) \d{s} = \int_{0}^{1} \sum_{j=1}^{n} \pdv{f_{i}(x+s(y-x))}{x_{j}}(y_{i}-x_{i}) \d{s}. \end{salign*} Und damit in Vektorform: \begin{salign*} @@ -110,7 +110,7 @@ \end{salign*} \end{proof} \begin{bem} - Obige Lipschitz-Konstante liefert eine Abschätzung für die Ableitungen / Jacobi-Matrix von $f$. + Obige Lipschitz-Konstante liefert eine Abschätzung für die Ableitungen/Jacobi-Matrix von $f$. \end{bem} \section{Taylor-Entwicklung} @@ -129,8 +129,8 @@ &\Longleftrightarrow \ \ \ f \ \text{ist total differenzierbar in} \ D \ \text{und} \ x \mapsto J_{f}(x) = (\partial_{i}f_{k})_{i,k} \ \text{stetig} \end{salign*} \item Ist $f \in C^{k}(D,\R^{m})$, dann sind die Ableitungen der $k-1$-ten Ordnung $\partial_{i_{k-1}}...\partial_{i_{1}}f: D \to \R^{m}$ total differenzierbar, weil stetig partiell differenzierbar. Also ist $\partial_{i_{k-1}}...\partial_{i_{1}}f$ stetig und damit sind alle Ableitungen $k-1$-ter Ordnung stetig. Ananolg folgt induktiv, dass alle Ableitungen $j$-ter Ordnung mit $j\leq k$ stetig auf $D$ sind. - \item Seien $D \subset \R^{n}$ offen, $f: D \to \R^{m}$. Existieren $\partial_{i}f, \partial_{j}f$ und $\partial_{j}\partial_{i}f$ auf $D$ \ ($i,j \in \{1,...,n\}$) \ und $\partial_{j}\partial_{i}f$ stetig in $a \in D$. Dann existiert $\partial_{i}\partial_{j}f$ und es gilt $$ \partial_{i}\partial_{j}f(a) = \partial_{j}\partial_{i}f(a).$$ - \item Seien $D \subset \R^{n}$ und $f \in C^{k}(D, \R^{m})$. Sei $\pi \in \mathcal{S}_{k}$ eine Permutation, dann gilt: $$ \partial_{i_{k}}...\partial_{i_{1}}f = \partial_{i_{\pi(k)}}...\partial_{i_{\pi(1)}}f, \ \ \ \ \ \ \ \forall i_{1},...,i_{k} \in \{1,...,n\}.$$ + \item Seien $D \subset \R^{m}$ offen, $f: D \to \R^{m}$. Existieren $\partial_{i}f, \partial_{j}f$ und $\partial_{j}\partial_{i}f$ auf $D$ \ ($i,j \in \{1,...,n\}$) \ und $\partial_{j}\partial_{i}f$ stetig in $a \in D$. Dann existiert $\partial_{i}\partial_{j}f$ und es gilt $$ \partial_{i}\partial_{j}f(a) = \partial_{j}\partial_{i}f(a).$$ + \item Seien $D \subset \R^{m}$ und $f \in C^{k}(D, \R^{m})$. Sei $\pi \in \mathcal{S}_{k}$ eine Permutation, dann gilt: $$ \partial_{i_{k}}...\partial_{i_{1}}f = \partial_{i_{\pi(k)}}...\partial_{i_{\pi(1)}}f, \ \ \ \ \ \ \ \forall i_{1},...,i_{k} \in \{1,...,n\}.$$ \end{enumerate} Reminder - Taylor-Entwicklung in $\R$: \begin{enumerate}[(1)] @@ -148,4 +148,149 @@ \end{salign*} \end{enumerate} \end{bem} +\begin{satz}[Taylor-Formel] + Seien $D \subset \R^{n}$ offen, $x \in D$, $h \in \R^{n}$ mit $\{x+th \ | \ t \in [0,1] \} \subset D$ und $f \in C^{r+1}(D,\R)$. Dann existiert ein $\theta \in [0,1]$ sodass gilt: + \begin{salign*} + f(x+h) &= \underbrace{f(x) + \sum_{k=1}^{r} \frac{1}{k!} \sum_{i_{1},...,i_{k} = 1}^{n} \partial_{i_{k}}...\partial_{i_{1}}f(x) \cdot h_{i_{1}} \cdot...\cdot h_{i_{k}}}_{\text{Taylor-Polynom}} \\ &+ \underbrace{\frac{1}{(r+1)!} \sum_{i_{1},...,i_{r+1} = 1}^{n} \partial_{i_{r+}}...\partial_{i_{1}}f(x+\theta h) \cdot h_{i_{1}} \cdot...\cdot h_{i_{r+1}}}_{\text{Restglied}} + \end{salign*} +\end{satz} +\begin{proof} + Sei $g \colon [0,1] \to \R, g(t) \coloneqq f(x+th)$. Es gilt $g \in C^{r+1}([0,1], \R)$ und + \begin{equation} \label{eq::taylor-hilfsfkt} + \frac{\text{d}^{k} g}{\d t^{k}} =\sum_{i_{1},...,i_{k}=1}^{n} \frac{\partial^{k}f}{\partial x_{i_{k}}...\partial x_{i_{1}}}(x+th) \cdot h_{i_{1}} \cdot ... \cdot h_{i_{k}}. + \end{equation} + Wir zeigen (\ref{eq::taylor-hilfsfkt}) durch Induktion nach $k$. \\ + (IA) \ $k=1$ \ Es ist $g \in C^{1}$ und nach Kettenregel gilt: $$ \frac{\d{g}}{\d{t}}(t) = \frac{\d}{\d{t}}f(x+th) = \sum_{i=1}^{n} \pdv{f(x+th)}{x_{i}} \cdot h_{i}. $$ + (IV) Für ein $k \in \N$ gilt \ref{eq::taylor-hilfsfkt}. \\ + (IS) \ $k \mapsto k+1$ \ Nach Induktionsvoraussetzung gilt: $$\frac{\text{d}^{k} g(t)}{\d t^{k}} =\sum_{i_{1},...,i_{k}=1}^{n} \frac{\partial^{k}f(x+th)}{\partial x_{i_{k}}...\partial x_{i_{1}}} \cdot h_{i_{1}} \cdot ... \cdot h_{i_{k}}.$$ + Aus $f \in C^{k+1}(D, \R)$ folgt, dass $\frac{\text{d}^{k} g}{\d{t^{k}}} \in C^{1}([0,1], \R)$, womit nach Kettenregel folgt: + \begin{salign*} + \frac{\text{d}^{k+1} g(t)}{\d{t^{k+1}}} &= \frac{\d}{\d{t}} \left( \sum_{i_{1},...,i_{k}=1}^{n} \frac{\partial^{k}f(x+th)}{\partial x_{i_{k}}...\partial x_{i_{1}}} \cdot h_{i_{1}} \cdot ... \cdot h_{i_{k}} \right) \\ + &= \sum_{j=1}^{n} \pdv{}{x_{j}} \left( \sum_{i_{1},...,i_{k}=1}^{n} \frac{\partial^{k}f(x+th)}{\partial x_{i_{k}}...\partial x_{i_{1}}} \cdot h_{i_{1}} \cdot ... \cdot h_{i_{k}} \right) \cdot h_{j} \\ + &= \sum_{i_{1},...,i_{k+1}=1}^{n} \frac{\partial^{k+1}f(x+th)}{\partial x_{i_{k+1}}...\partial x_{i_{1}}} \cdot h_{i_{1}} \cdot ... \cdot h_{i_{k+1}}. + \end{salign*} + Ferner liefert die Taylor-Formel in $\R$ für $g$ die Existenz eines $\theta \in [0,1]$ sodass gilt: + \begin{salign*} + g(1) - g(0) = f(x+h) - f(x) = \left(\sum_{k=1}^{r} \frac{g^{(k)}(0)}{k!} \right) + \frac{g^{(r+1)}(\theta)}{(r+1)!}. + \end{salign*} + Einsetzen von (\ref{eq::taylor-hilfsfkt}) liefert die Behauptung. +\end{proof} +\begin{bem} + Um die Notation vieler, verschieden indizierter partieller Ableitungen zu erleichtern, führen wir die \underline{Multiindex-Notation} ein: \\ + Für $\alpha = (\alpha_{1},...,\alpha_{n}) \in \N_{0}^{n}$ seien: + \begin{salign*} + \vert \alpha \vert \ &\coloneqq \ \alpha_{1} + ... +\alpha_{n} \in \N_{0} \\ + \partial^{\alpha}f \ &\coloneqq \ \partial_{1}^{\alpha_{1}}...\partial_{n}^{\alpha_{n}}f = \frac{\partial^{\vert \alpha \vert}f}{\partial x_{1}^{\alpha_{1}} ... \partial x_{n}^{\alpha_{n}}}, & \ \text{für} \ f \in C^{\vert \alpha \vert}(D, \R), \\ + \alpha ! \ &\coloneqq \ \alpha_{1} ! \cdot ... \cdot \alpha_{n} ! \in \N_{0}, \\ + h^{\alpha} \ &\coloneqq \ h_{1}^{\alpha_{1}} \cdot ... \cdot h_{n}^{\alpha_{n}} \in \R, & \ \text{für} \ h = (h_{1},...,h_{n})^{T} \in \R^{n}. + \end{salign*} +\end{bem} +\begin{bsp} + Seien $n = 3$, $\alpha= (\alpha_{1},\alpha_{2},\alpha_{3})$. Multiindizes der Ordnung $\vert \alpha \vert = 2$ sind: \\ + \begin{table}[h] + \centering + \setlength{\tabcolsep}{8pt} + \renewcommand{\arraystretch}{1.6} + \begin{tabular}{c|c|c|c|c|c|c} + $\alpha$ & $(2,0,0)$ & $(0,2,0)$ & $(0,0,2)$ & $(1,1,0)$ & $(1,0,1)$ & $(0,1,1)$ \\ + \hline + $\alpha !$ & $2! \cdot 0! \cdot 0! = 2$ & $2$ & $2$ & $1$, & $1$ & $1$ \\ + \hline + $\partial^{\alpha}f$ & $\frac{\partial^{2}f}{\partial x_{1}^{2}}$ & $\frac{\partial^{2}f}{\partial x_{2}^{2}}$ & $\frac{\partial^{2}f}{\partial x_{3}^{2}}$ & $\frac{\partial^{2}f}{\partial x_{1} \partial x_{2}}$, & $\frac{\partial^{2}f}{\partial x_{1} \partial x_{3}}$ & $\frac{\partial^{2}f}{\partial x_{2} \partial x_{3}}$ + \end{tabular} + \end{table} \\ + Summennotation für partielle Ableitungen $2$-ter Ordnung: + $$ \sum_{\vert \alpha \vert = 2} \partial^{\alpha}f = \frac{\partial^{2}f}{\partial x_{1}^{2}} + \frac{\partial^{2}f}{\partial x_{2}^{2}} + \frac{\partial^{2}f}{\partial x_{3}^{2}} + \frac{\partial^{2}f}{\partial x_{1} \partial x_{2}} + \frac{\partial^{2}f}{\partial x_{1} \partial x_{3}} + \frac{\partial^{2}f}{\partial x_{2} \partial x_{3}}. $$ +\end{bsp} +\begin{bem} + Mit dieser Notation lässt sich die Taylor-Formel kompakt schreiben als: + \begin{salign*} + f(x+h) = \sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha} f(x)}{\alpha !} h^{\alpha} + \sum_{\vert \alpha \vert = r+1} \frac{\partial^{\alpha} f(x+\theta h)}{\alpha !} h^{\alpha}. + \end{salign*} +\end{bem} +\begin{proof} + Die Reihenfolge der Differenzierung kann vertauscht werden. Daraus folgt für alle $k$-Tupel $(i_{1},...,i_{k})$ mit $i_{j} \in \{1,...,n\}$ für $j \in \{1,...,k\}$: $$\partial_{i_{k}}...\partial_{i_{1}}f(y)\cdot h_{i_{1}} \cdot ... \cdot h_{i_{k}} = \partial_{1}^{\alpha_{1}}...\partial_{n}^{\alpha_{n}}f(y) \cdot h_{1}^{\alpha_{1}} \cdot ... \cdot h_{n}^{\alpha_{n}} = \partial^{\alpha} f(y) \cdot h^{\alpha} $$ + wobei $\alpha = (\alpha_{1},...,\alpha_{n})$ mit $\alpha_{s} \coloneqq \#\{ i_{j} \ | \ i_{j} = s \ \text{für} \ j \in \{1,...,k\} \}$ (Anzahl wie oft Index $s$ in $(i_{1},...,i_{k})$ vorkommt). \\ + Ohne Beweis (siehe S.72 Skript Ana2 Rolf Rannacher): für $\alpha\in \N_{0}^{n}$ mit $\vert \alpha \vert = k$ gilt: die Anzahl von $k$-Tupeln $(i_{1},...,i_{k})$ bei denen der Index $s$ genau $\alpha_{s}$-mal vorkommt ist $$\frac{k!}{\alpha_{1}! \cdot ... \cdot \alpha_{n}!} = \frac{k!}{\alpha!}.$$ + Daraus folgt + \begin{salign*} + \frac{\text{d}^{k} g}{\d{t^{k}}}(t) &= \sum_{i_{1},...,i_{k}=1}^{n} \frac{\partial^{k}f(x+th)}{\partial x_{i_{k}}...\partial x_{i_{1}}} \cdot h_{i_{1}} \cdot ... \cdot h_{i_{k}} = \sum_{\vert \alpha \vert = k} \frac{k!}{\alpha_{1}!\cdot ... \cdot \alpha_{n}!} \partial_{1}^{\alpha_{1}}...\partial_{n}^{\alpha_{n}}f(x+th) h_{1}^{\alpha_{1}} \cdot ... \cdot h_{n}^{\alpha_{n}} \\ + &= \sum_{\vert \alpha \vert = k} \frac{k!}{\alpha !} \partial^{\alpha}f(x+th)h^{\alpha}, + \end{salign*} + womit insbesondere auch gilt + \begin{salign*} + \sum_{k=0}^{r} \frac{1}{k!} \cdot \frac{\text{d}^{k} g}{\d{t^{k}}}(t) = \sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha}f(x+th)}{\alpha!}h^{\alpha}. + \end{salign*} + Und können ferner festhalten: + \begin{salign*} + f(x+h) &= g(1) \oldstackrel{\text{Taylor in } \R}{=} \sum_{k=0}^{r} \frac{1}{k!} \cdot \frac{\text{d}^{k} g}{\d{t^{k}}}(0)+ \frac{1}{(r+1)!} \frac{\text{d}^{r+1} g(\theta)}{\d{t^{r+1}}}\\ + &= \sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \frac{1}{(r+1)!}\sum_{\vert \alpha \vert = r+1} \frac{(r+1)! \cdot \partial^{\alpha}f(x+\theta h)}{\alpha!}h^{\alpha} \\ + &= \sum_{\vert \alpha \vert\leq r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \underbrace{\sum_{\vert \alpha \vert = r+1} \frac{\partial^{\alpha}f(x+\theta h)}{\alpha!}h^{\alpha}}_{\text{Restglied in Lagrange-Form}} \\ + \intertext{und analog:} + f(x+h) &= g(1) \oldstackrel{\text{Taylor in } \R}{=}\sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \frac{1}{r!} \int_{0}^{1} \frac{\text{d}^{r+1}g(t)}{\d{t^{k+1}}}(1-t)^{r} \d{t} \\ + &= \sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \frac{1}{r!} \int_{0}^{1} \sum_{\vert \alpha \vert = r+1} \frac{(r+1)! \cdot \partial^{\alpha}f(x+t h)}{\alpha!}h^{\alpha}(1-t)^{r} \d{t} \\ + &= \sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \underbrace{(r+1) \int_{0}^{1} \sum_{\vert \alpha \vert = r+1} \frac{\partial^{\alpha}f(x+t h)}{\alpha!}h^{\alpha}(1-t)^{r} \d{t}}_{\text{Restglied in Integral-Form}} + \end{salign*} +\end{proof} +\begin{korollar} \label{kor::taylor-omega-form} + Seien $D \subset \R^{n}$ offen, $f\in C^{r+1}(D,\R)$. Dann gilt für alle $x \in D$ und $h \in \R^{n}$, für die $x+th \in D$ (für alle $t \in [0,1]$) $$f(x+h) = \sum_{\vert \alpha \vert \leq r+1} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \omega_{r+1}(x, h),$$ sodass $$\frac{\omega_{r+1}(x,h)}{\norm{h}^{r+1}} \oldstackrel{h \to \infty}{\longrightarrow} 0,$$ also $\omega_{r+1}(x,h) = o(\norm{h}^{r+1})$. +\end{korollar} +\begin{proof} + Da $D$ offen ist, existiert ein $\delta > 0$ mit $K_{\delta}(x) \subset D$. Nach Taylor-Formel existiert für alle $h \in \R^{n}$ mit $\norm{h} < \delta$ ein $\theta \in [0,1]$ sodass + \begin{salign*} + f(x+h) &= \sum_{\vert \alpha \vert \leq r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \sum_{\vert \alpha \vert = r+1} \frac{\partial^{\alpha}f(x+\theta h)}{\alpha!}h^{\alpha} \\ + &= \sum_{\vert \alpha \vert \leq r+1} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} + \underbrace{\sum_{\vert \alpha \vert = r+1} \frac{\partial^{\alpha}f(x+\theta h)-\partial^{\alpha}f(x)}{\alpha!}h^{\alpha}}_{\omega_{r+1}(x,h) \coloneqq} + \end{salign*} + Nun gilt noch zu zeigen, dass: $\frac{\omega_{r+1}(x,h)}{\norm{h}^{r+1}} \oldstackrel{h \to \infty}{\longrightarrow} 0$. \\ + Es gilt: + \begin{salign*} + \vert h^{\alpha} \vert = \vert h_{1}^{\alpha_{1}} \vert \cdot ... \cdot \vert h_{n}^{\alpha_{n}} \vert \leq \norm{h}^{\alpha_{1}} \cdot ... \cdot \norm{h}^{\alpha_{n}} = \norm{h}^{\vert \alpha \vert}. + \end{salign*} + Womit wir abschätzen können: + \begin{salign*} + \frac{\vert \omega_{r+1}(x,h) \vert}{\norm{h}^{r+1}} &= \frac{1}{\norm{h}^{r+1}} \sum_{\vert \alpha \vert = r+1} \frac{1}{\alpha !} \vert \partial^{\alpha}f(x+\theta h)-\partial^{\alpha}f(x) \vert \cdot \vert h_{1}^{\alpha_{1}} \vert \cdot ... \cdot \vert h_{n}^{\alpha_{n}} \vert \\ + &= \frac{\vert h^{r+1} \vert}{\norm{h}^{r+1}}\sum_{\vert \alpha \vert = r+1} \frac{1}{\alpha !} \vert \partial^{\alpha}f(x+\theta h)-\partial^{\alpha}f(x) \vert \\ + &\stackrel{\vert h^{r+1} \vert \leq \norm{h}^{r+1}}{\leq} \sum_{\vert \alpha \vert = r+1} \frac{1}{\alpha !} \underbrace{\vert \partial^{\alpha}f(x+\theta h)-\partial^{\alpha}f(x) \vert}_{\oldstackrel{h \to 0}{\longrightarrow} 0, \ \text{weil stetig sind alle} \ \partial^{\alpha}f} \oldstackrel{h \to 0}{\longrightarrow} 0. + \end{salign*} +\end{proof} +\begin{korollar} + Seien $D \subset \R^{n}$ offen, $f \in C^{2}(D,\R)$. Dann gilt für alle $x \in D$ und $h \in \R^{n}$, sodass $x+th \in D$ (für alle $t \in [0,1]$): + \begin{salign*} + f(x+h) = f(x) + \left( \nabla f(x), h \right)_{2} + \frac{1}{2} \left( H_{f}(x)h, h\right)_{2} + \omega_{2}(x,h), && \text{mit} \ \frac{\omega_{2}(x,h)}{\norm{h}^{2}} \oldstackrel{h \to \infty}{\longrightarrow} 0. + \end{salign*} + Dabei ist $H_{f}(x)$ die Hesse-Matrix von $f$. Das heißt das Taylor-Polynom 2. Ordnung ist eine quadratische Approximation von $f$. Für $f \in C^{1}(D,\R)$ gilt: + \begin{salign*} + f(x+h) = f(x) + \left( \nabla f(x), h \right)_{2} + \omega_{1}(x,h), && \text{mit} \ \frac{\omega_{1}(x,h)}{\norm{h}} \oldstackrel{h \to \infty}{\longrightarrow} 0. + \end{salign*} + Das heißt das Taylor-Polynom 1. Ordnung ist eine lineare Approximation von $f$. +\end{korollar} +\begin{proof} + Folgt aus \ref{kor::taylor-omega-form} mir $r+1=2$ bzw. $r+1 = 1$ sowie: + \begin{salign*} + \sum_{\vert \alpha \vert = 0} \frac{\partial^{\alpha}f(x)}{\alpha!} h^{\alpha} &= f(x), \\ + \sum_{\vert \alpha \vert = 1} \frac{\partial^{\alpha}f(x)}{\alpha!} h^{\alpha} &= \frac{1}{1!} \cdot \sum_{i=1}^{n} \pdv{f(x)}{x_{i}} \cdot h_{i} = \left( \nabla f(x), h \right)_{2}, \\ + \sum_{\vert \alpha \vert = 2} \frac{\partial^{\alpha}f(x)}{\alpha!} h^{\alpha} &= \frac{1}{2!} \sum_{i,j=1}^{n} \frac{\partial^{2} f(x)}{\partial x_{i} \partial x_{j}} h_{i} \cdot h_{j} = \frac{1}{2} \left( H_{f}(x)h, h\right)_{2}. + \end{salign*} +\end{proof} +\begin{definition} + Seien $D \subset \R^{n}$ offen und $f\colon D \to \R$ beliebig oft partiell differenzierbar, $x \in D$. Dann heißt die \underline{Taylor-Reihe von $f$ in $x$}: + $$ T_{\infty}^{f}(x+h) = \sum_{\vert \alpha \vert = 0}^{\infty} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha}.$$ +\end{definition} +\begin{korollar} + Seien $D \subset \R^{n}$ offen und $f\colon D \to \R$ eine beliebig oft differenzierbare Funktion. Dann gilt $$ T_{r}^{f}(x+h) = \sum_{\vert \alpha \vert = 0}^{r} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha} \oldstackrel{r \to \infty}{\longrightarrow} f(x+h)$$ wenn + \begin{equation} \label{eqq::taylor-rest-gegen-null} + R_{r+1}^{f}(x,h) \oldstackrel{r \to \infty}{\longrightarrow} 0, \ \ \ \ \ x \in D. + \end{equation} + Eine hinreichende Bedingung für (\ref{eqq::taylor-rest-gegen-null}) ist $$\sup_{\vert \alpha \vert \geq 0} \ \sup_{x \in D} \vert \partial^{\alpha} f(x) \vert \leq M_{f} < \infty,$$ das heißt alle partiellen Ableitungen von $f$ sind gleichmäßig beschränkt. +\end{korollar} +\begin{proof} + Siehe Skript Ana2 Rolf Rannacher, Seite 74, Übungsaufgabe 3.23. +\end{proof} + +\begin{bem} + Die Taylor Reihe von $f$ konvergiert gegen $f$, wenn die Funktion $f$ durch eine Potenzreihe beschrieben wird (also wenn $f$ analytisch ist). +\end{bem} + + + \end{document}