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zoqF8_6JRzOnq)~Y8{aph_r|%TD?izVGPPv1*(y?IV3aaD`*SlVGxf}ENnRog!J8Ye zk<l}(hz$JCh1J_g6VeT*CaJK^zBSdB=Ncvu?a zVpJm)7G13)Wf|XM8km6OYJ!RN(!fYE2&r%tjGh|iak5`hG8F8Q zgmE_ZL3j|ZD?Z_l5%mn@sGHLR_v98;6C6G`{B_ph9I~=U-L`2<4jrp!mX&%}Koj&2 z^8fhuC-0TNac7;>=>PouD?jr8`sMXCo{+PWxcc||zkk2-i)rMq&+otf_R`OvUQSOB zm(%5ZS)QIhKfRpl<hM_<1SuaFko)2w|5^tJpb|X?my(+ Bu+;zn diff --git a/ana13.tex b/ana13.tex index 1049477..e9e5ff3 100644 --- a/ana13.tex +++ b/ana13.tex @@ -95,7 +95,7 @@ \norm{\int_{0}^{1} J_{f}(x+sh) h \d{s} }_{2} &\leq \int_{0}^{1} \norm{J_{f}(x+sh) h}_{2} \d{s} \\ & \leq \int_{0}^{1} \norm{J_{f}(x+sh)}_{2} \norm{h}_{2} \d{s} \\ &\leq \sup_{0 0$ 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} + &= \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}}_{\eqqcolon \, \omega_{r+1}(x,h)} \end{salign*} - Nun gilt noch zu zeigen, dass: $\frac{\omega_{r+1}(x,h)}{\norm{h}^{r+1}} \oldstackrel{h \to \infty}{\longrightarrow} 0$. \\ + Nun gilt noch zu zeigen, dass: $\frac{\omega_{r+1}(x,h)}{\norm{h}^{r+1}} \oldstackrel{h \to 0}{\longrightarrow} 0$. \\ 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}. + \vert h^{\alpha} \vert = \vert h_{1}^{\alpha_{1}} \vert \dotsm \vert h_{n}^{\alpha_{n}} \vert \leq \norm{h}^{\alpha_{1}} \dotsm \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 \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 \dotsm \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. + &\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 alle} \ \partial^{\alpha}f \text{ stetig sind}} \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. + 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 0}{\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. + 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 0}{\longrightarrow} 0. \end{salign*} Das heißt das Taylor-Polynom 1. Ordnung ist eine lineare Approximation von $f$. \end{korollar} @@ -277,7 +277,7 @@ $$ T_{\infty}^{f}(x+h) = \sum_{\vert \alpha \vert = 0}^{\infty} \frac{\partial^{\alpha}f(x)}{\alpha!}h^{\alpha}.$$ \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 + 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} diff --git a/analysisII.pdf b/analysisII.pdf index 99b3de90e85fd66368817217151cabce169bcb64..4d06f8a33518bb4f53c51d4efe6d7fde1b2baeaf 100644 GIT binary patch delta 90461 zcmZsiQ*_=>-0l0swr$&J)Y!IdtKlzc%%(w~*tYF7cGB35ZR`D?b1vSSb2s~P*36pq z-FrSeqKsw#jRuukMM8?1orN2LdgkxoG6LAO0)$nkx()(yfnCXAu^N8B4nTo;!KQZ5 zm<_~h#hQV2aN^A$;HD7Hr@DB*$bk-^8j|LG4uSlE_udeA9y5;Ft_ zm9m+gx`!hIs}dPI8yflY&|qR|Z3Cu%<-1s!?Gk;G{sm`d9^Bfv(2z)Fq4s^&{k*Eu5MuSNVKP-y zG*)KP=m_pEP;KRdD}aiLYiN&7enDtn&~+?1xpb?dq%;&ZG%C5LhPnq5wqA_b9-O5E zAWO{ofi(l2u;L;(1s#G{9)u!OsV4O#_sudw8cx!`b`6O-dM&|FRtaj$ynOGAAQmn3 zx9^R~>89|s*d}(pO*5s@n9#za7>Wy;c#J05Fb0yDQU=%5u|OI{uf3bB0&81xt&R@Y 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