From 36ea859345b6aa6bb64e553e7e86359405e602fb Mon Sep 17 00:00:00 2001 From: flavis Date: Mon, 20 Jul 2020 21:21:34 +0200 Subject: [PATCH] fix minor typos --- ana1.tex | 34 +++++++++++++++++----------------- ana14.tex | 11 +++++------ ana17.tex | 2 +- ana2.tex | 2 +- ana21.tex | 4 ++-- ana4.tex | 4 ++-- ana6.tex | 2 +- analysisII.pdf | Bin 964533 -> 963940 bytes 8 files changed, 29 insertions(+), 30 deletions(-) diff --git a/ana1.tex b/ana1.tex index 888aaab..d517db2 100644 --- a/ana1.tex +++ b/ana1.tex @@ -141,7 +141,7 @@ $f_n\colon [0,2] \to \R$, $f_n(x) \coloneqq \frac{1}{n} \sin \left( 2\pi n \cdot x \right) $ konvergiert gleichmäßig gegen $f(x) = 0$. \[ - | \sin(2 \pi n \cdot x) | \le 1 \quad \forall x \in R \implies + | \sin(2 \pi n \cdot x) | \le 1 \quad \forall x \in \R \implies f_n(x) \xrightarrow{n \to \infty} 0 \quad \forall x \in [0,2] \implies \text{punktweise Konvergenz} .\] Sei nun $\epsilon > 0$, dann $\exists N \in \N$ mit $\frac{1}{N} < \epsilon$. Damit folgt \[ @@ -197,10 +197,10 @@ Es sei $D = [a,b]$ und $f\colon [a,b] \to \R$ stetig. Dann gilt \begin{enumerate}[(i)] \item $(f_n)_{n\in\N}$ konvergiert gleichmäßig gegen $f \colon [a,b] \to \R - \iff \Vert f_n - f \Vert_\infty \to 0$, $n \to \infty$ + \iff \Vert f_n - f \Vert_\infty \xrightarrow{n \to \infty} 0$. \item Cauchy-Kriterium: $(f_n)_{n\in\N}$ konvergiert gleichmäßig auf $[a,b] \iff \forall \epsilon > 0$ $\exists N_0 \in \N$, s.d. $\forall n, m \ge N_0$ gilt - $\Vert f_n - f_m \Vert_{\infty} < \epsilon$ + $\Vert f_n - f_m \Vert_{\infty} < \epsilon$. \end{enumerate} \end{satz} @@ -280,7 +280,7 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^{b} f$? \begin{satz} - Seien $f_n \colon [a,b] \to R$ stetige Riemann-integrierbare Funktionen und $f\colon [a,b] \to \R$ + Seien $f_n \colon [a,b] \to \R$ stetige Riemann-integrierbare Funktionen und $f\colon [a,b] \to \R$ mit $\Vert f_n - f \Vert_\infty \xrightarrow{n \to \infty} 0$ (gleichmäßige Konvergenz). Dann gilt $f$ stetig und Riemann-integrierbar und \[ @@ -296,7 +296,7 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^ \begin{align*} \left| \int_{a}^{b} f_n(x) dx - \int_{a}^{b} f(x) dx\right| &= \left| \int_{a}^{b} (f_n(x) - f(x))dx\right|\\ &\le \int_{a}^{b} |f_n(x) - f(x)| dx\\ - &\le \max_{x \in [a,b]} |f_n(x) - f(x)| \cdot (a-b)\\ + &\le \max_{x \in [a,b]} |f_n(x) - f(x)| \cdot (b-a)\\ &=\underbrace{\norm{f_n - f}_\infty}_{\xrightarrow{n \to \infty} 0}\underbrace{(b-a)}_{\text{beschränkt}}. \end{align*} \end{proof} @@ -304,32 +304,32 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^ \begin{satz}\label{permutesumint} Seien $f_n \colon [a,b] \to \R$ stetige Funktionen $(n \in \N)$ und die Reihe $\sum_{n=0}^{\infty} f_n$ konvergiere gleichmäßig auf $[a,b]$, d.h. die Folge der Partialsummen - $(\sum_{n=0}^{N} f_n)_{n\in\N}$ sei gleichmäßig konvergent. Dann gilt: + $(\sum_{k=0}^{n} f_k)_{n\in\N}$ sei gleichmäßig konvergent. Dann gilt: \[ f(x) \coloneqq \sum_{n=0}^{\infty} f_n(x) \quad \forall x \in [a,b] .\] ist stetig und Riemann-integrierbar und \begin{align*} - \int_{a}^{b} f(x) dx &= \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx \\ - \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx &= \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx + \int_{a}^{b} f(x) \d x &= \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) \d x \\ + \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) \d x &= \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) \d x ,\end{align*} d.h. die Reihe wird gliedweise integriert. \end{satz} \begin{proof} - $f_n$ sind stetig $\implies \sum_{n=0}^{N} f_n(x)$ stetig und Riemann-integrierbar. + $f_n$ sind stetig $\implies \sum_{k=0}^{n} f_k(x)$ stetig und Riemann-integrierbar. - Die Folge der Partialsummen $(\sum_{n=0}^{N} f_n)_{n\in\N}$ konvergiert gleichmäßig, d.h. + Die Folge der Partialsummen $(\sum_{k=0}^{n} f_k)_{n\in\N}$ konvergiert gleichmäßig, d.h. \[ - f(x) = \sum_{n=0}^{\infty} f_n(x) \text{ stetig } = \lim_{n \to \infty} \left( \sum_{n=0}^{N} f_n(x) \right) \text{ gleichmäßiger Limes} + f(x) = \sum_{n=0}^{\infty} f_n(x) \text{ stetig } = \lim_{n \to \infty} \left( \sum_{k=0}^{n} f_k(x) \right) \text{ gleichmäßiger Limes} .\] Es gilt \[ - \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx = - \sum_{n=0}^{N} \int_{a}^{b} f_n(x) dx + \int_{a}^{b} \sum_{n=N+1}^{\infty} f_n(x) dx + \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) \d x = + \sum_{n=0}^{N} \int_{a}^{b} f_n(x) \d x + \int_{a}^{b} \sum_{n=N+1}^{\infty} f_n(x) \d x .\] Die Reihe konvergiert gleichmäßig, d.h. $\forall \epsilon > 0$ $\exists N_{\epsilon} \in \N$, s.d. $\forall N \ge N_{\epsilon}$ gilt \begin{align*} - &\left| \int_{a}^{b} \sum_{n=N+1}^{\infty} f_n(x) dx \right| \le \epsilon \cdot (b-a) \\ - \implies &\left| \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx - \sum_{n=0}^{N} \int_{a}^{b} f_n(x) dx \right| \le \epsilon \\ - \implies& \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) dx = \lim_{N \to \infty} \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx = \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) dx + &\left| \int_{a}^{b} \sum_{n=N+1}^{\infty} f_n(x) \d x \right| \le \epsilon \cdot (b-a) \\ + \implies &\left| \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) \d x - \sum_{n=0}^{N} \int_{a}^{b} f_n(x) \d x \right| \le \epsilon \cdot (b-a)\\ + \implies& \int_{a}^{b} \sum_{n=0}^{\infty} f_n(x) \d x = \lim_{N \to \infty} \sum_{n=0}^{N} \int_{a}^{b} f_n(x) \d x = \sum_{n=0}^{\infty} \int_{a}^{b} f_n(x) \d x .\end{align*} \end{proof} @@ -338,7 +338,7 @@ Wichtige Frage: Wenn $f_n \to f$, gilt dann auch $\int_{a}^{b} f_n \to \int_{a}^ Dann konvergiert $\sum_{n=0}^{\infty} a_n (x - x_0)^{n}$ in jedem Intervall $[x_0 - r, x_0 + r]$ für $0 < r < \rho$ gleichmäßig und für $[a,b] \subset \;]x_0 - \rho, x_0 + \rho[$ gilt \[ - \int_{a}^{b} \sum_{n=0}^{\infty} a_n(x - x_0)^{n} dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1} + \int_{a}^{b} \sum_{n=0}^{\infty} a_n(x - x_0)^{n} \d x = \sum_{n=0}^{\infty} \frac{a_n}{n+1}(x-x_0)^{n+1} \Big|_{a}^{b} .\] \end{korollar} diff --git a/ana14.tex b/ana14.tex index 192af02..e076d6a 100644 --- a/ana14.tex +++ b/ana14.tex @@ -150,13 +150,12 @@ Gesucht: Abbildung $f$, s.d. $y = f(x)$ mit $(x,f(x))$ löst \eqref{doublestar} \begin{minipage}[c]{0.35\textwidth} \begin{tikzpicture}[scale=0.8] - \draw[->] (0,-1.5) -- node[right,pos=.9] {$(x_0,y_0)$} (0,1.5); + \draw[->] (0,-1.5) -- node[right,pos=.9] {} (0,1.5); \draw[->] (-1.5,0) -- node[below,pos=1.3] {$F(x,y) = 0$} (1.5,0); \draw (0,0) circle (1cm); - \draw[fill] (0,1) circle (2pt); - \draw[fill=red,draw=red] (0,0) circle (2pt); - \node[color=red] at (.3,0) {)}; - \node[color=red] at (-.3,0) {(}; + \draw[fill=red,draw=red] (0,1) circle (2pt); + \node[color=red] at (.3,1) {)}; + \node[color=red] at (-.3,1) {(}; \draw[fill=blue,draw=blue] (1,0) circle (2pt); \node[color=blue] at (1.3,0) {)}; \node[color=blue] at (.7,0) {(}; @@ -172,7 +171,7 @@ Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung v \end{bsp} \begin{satz}[Satz über implizite Funktionen] \label{satz:sif} - Sei $D^x \subset \R^n$ offen, $D^y \subset \R^m$ offen, $F^1 \in C^1 (D^x\times D^y,\R^m)$ (stetig differenzierbar) und $(\hat{x}, \hat{y})\in D^x\times D^y$ mit $F(\hat x, \hat y) = 0$. Die $m\times m$ Matrix + Sei $D^x \subset \R^n$ offen, $D^y \subset \R^m$ offen, $F \in C^1 (D^x\times D^y,\R^m)$ (stetig differenzierbar) und $(\hat{x}, \hat{y})\in D^x\times D^y$ mit $F(\hat x, \hat y) = 0$. Die $m\times m$ Matrix \[ D_yF(x,y) =\begin{pmatrix} \pdv{F_1}{y_1} &\dots &\pdv{F_1}{y_m}\\ diff --git a/ana17.tex b/ana17.tex index 8814b5a..96121d6 100644 --- a/ana17.tex +++ b/ana17.tex @@ -43,7 +43,7 @@ Sei $f(t,x)$ stetig auf $D \subseteq \R \times \R^{n}$, $D$ abgeschlossen und $(t_0, y_0) \in D$. Sei weiter $y(t)$ Lösung der AWA für $t \in [t_0 - T, t_0 + T]$. - Dann ist $y$ nach rechts und links auf ein maximales Existenzintervall $I_{\text{max}} = (t_0 - T^{*}, t_0 + T^{*})$ + Dann ist $y$ nach rechts und links auf ein maximales Existenzintervall $I_{\text{max}} = (t_0 - T_{*}, t_0 + T^{*})$ bis zum Rand von $D$ (stetig diff'bar) fortsetzbar. \end{satz} diff --git a/ana2.tex b/ana2.tex index 1510ea9..76be396 100644 --- a/ana2.tex +++ b/ana2.tex @@ -21,7 +21,7 @@ Jetzt: Fourier Analysis! \begin{enumerate} \item Analog: Definitionen von uneigentlichen Riemann-integralen für komplexwertige Funktionen - \item Die Rechenregeln f+r das reelle Riemann-integral übertragen sich auf komplexwertige + \item Die Rechenregeln für das reelle Riemann-integral übertragen sich auf komplexwertige Integrale, insbesondere gilt: \begin{align*} \int_{a}^{b} \overline{f(x)} \d x &= \int_{a}^{b} \left( \text{Re}f(x) - i \cdot \text{Im}f(x) \right) \d x \\ diff --git a/ana21.tex b/ana21.tex index 2519cb4..9465eec 100644 --- a/ana21.tex +++ b/ana21.tex @@ -32,7 +32,7 @@ \item Es gilt die Abschätzung \begin{align*} \left|\int_\gamma f \d s\right| &= \left|\int_a^b f(\gamma(t)) \norm{\gamma'(t)} \d t\right| \\ - &\le \sup_{s\in [a,b]} \left|f(\gamma(s))\right| \cdot \int_a^b f(\gamma(t)) \norm{\gamma'(t)} \d t \\ + &\le \sup_{s\in [a,b]} \left|f(\gamma(s))\right| \cdot \int_a^b \norm{\gamma'(t)} \d t \\ &= \sup_{s\in [a,b]} \left|f(\gamma(s))\right| \cdot S(\gamma). \end{align*} \item Das Kurvenintegral ist invariant unter $C^1$-Parametertransformation. Sei $\varphi \colon [\alpha, \beta] \to [a,b]$ eine $C^1$ Parametertransformation. Dann gilt @@ -191,4 +191,4 @@ In diesem Fall erhalten wir eine Stammfunktion $\varphi_0\in C^1(D,\R)$ durch $\varphi_0(x)\coloneqq \int_\gamma F$, wobei $\gamma$ ein Integrationsweg von einem gewählten Punkt $x_0\in D$ zu $x\in D$ ist. Die Menge aller Potentiale von $F$ ist gegeben durch $\{\varphi_0+c\,|\,c\in \R\}$. \end{satz} -\end{document} \ No newline at end of file +\end{document} diff --git a/ana4.tex b/ana4.tex index ae7a5de..dac1497 100644 --- a/ana4.tex +++ b/ana4.tex @@ -74,7 +74,7 @@ \item beschränkt, falls $\forall k\in \N:\; x^{(k)} \in K_{R}(0)$, $K_{R}(0)$ eine Kugelumgebung von $0$ mit Radius $R$. $$K_{r}(a) \coloneqq \{ x \in \K^{n} \ | \ \norm{x-a}_{\infty} < r\}.$$ \item Cauchy-Folge, wenn $\forall \varepsilon > 0, \exists N_{\varepsilon} \in \N$ sodass $\forall k,l \geq N_{\varepsilon}$ gilt: $\norm{x^{(k)} - x^{(l)}}_{\infty} < \varepsilon$. - \item konvergent gegen ein $x \in \K^{n}$, wenn $\norm{x^{(k)} - x}_{\infty} \to 0$ für $k \to \infty$. \\ + \item konvergent gegen ein $x \in \K^{n}$, wenn $\norm{x^{(k)} - x}_{\infty} \xrightarrow{k \to \infty} 0$. \\ geometrisch: jede Kugelumgebung $K_{\varepsilon}(x)$ enthält fast alle Folgenelemente $x^{(k)}$ (d.h. alle bis auf endlich viele). \end{enumerate} \end{definition} @@ -154,7 +154,7 @@ Bezeichnung: $\norm{\cdot}$ irgendeine Norm. \end{definition} \begin{definition}[offene Menge] - Eine Menge $O \in \K^{n}$ heißt offen, falls $O$ eine Umgebung jedes Punktes aus $O$ ($ x \in O$) ist, das heißt $\forall x \in O, \exists \varepsilon > 0$ mit $K_{\varepsilon}(x) \subseteq O$. + Eine Menge $O \subseteq \K^{n}$ heißt offen, falls $O$ eine Umgebung jedes Punktes aus $O$ ($ x \in O$) ist, das heißt $\forall x \in O, \exists \varepsilon > 0$ mit $K_{\varepsilon}(x) \subseteq O$. \end{definition} \begin{bsp} diff --git a/ana6.tex b/ana6.tex index cc14991..c7ad046 100644 --- a/ana6.tex +++ b/ana6.tex @@ -185,7 +185,7 @@ Wichtige Ungleichungen \end{bem} \begin{satz}[Gram-Schmidt-Verfahren] - Sei $\{a^{(1)},\dots ,a^{(n)}\}$ eine Basis des $\K^n$. Dann ist $\{b^{(1)},\dots ,b^{(n)}\}$ konstruirt durch das \underline{Orthogonalisierungsverfahren} von Gram und Schmidt eine \underline{Orhonormalbasis}. + Sei $\{a^{(1)},\dots ,a^{(n)}\}$ eine Basis des $\K^n$. Dann ist $\{b^{(1)},\dots ,b^{(n)}\}$, konstruiert durch das \underline{Orthogonalisierungsverfahren} von Gram und Schmidt, eine \underline{Orhonormalbasis}. \begin{align*} b^{(1)}&\coloneqq\frac{a^{(1)}}{\norm{a^{(1)}}_2}\\ \Tilde{b}^{(k)}&\coloneqq a^{(k)}-\sum_{j=1}^{k-1}(a^{(k)},b^{(j)})_2\cdot b^{(j)}\\ diff --git a/analysisII.pdf b/analysisII.pdf index 2642bc7cc89378ea15bdf0aa0a4752ffb0ca5fb3..cdc639402a663eb4e380840b64a0c76f099aba0f 100644 GIT binary patch delta 542789 zcmV)#K##w*?>gk_I*=uofh!3re~pvNPUJ8UMfd#`UM33{DLamn9k1po5Q1z*Gb}>4 zlQ#4?QXWYBJ!KmSS}+Wo#IDr6r+ky_)wegux^a=Qj%DohzxTs>v3DYJh8hH_oYgH8 zsshX_rj&#VD^{=B@8rNW+n3RfDVHdd^OTE%uJ`o?sTD(nN-5#hTnZ>?f60{qDP>gO z41EhECDHK^7CKozrAj5uWCrV+Sh`F_ne^jiQ;vyCF=;%3$sEc;GA>F$fzk2GPU9*i z&1SIe6s@2AzIQ>xF{2aF!8jJ6OA%E^)ICmp8~aZUZM%s6lc{Xw@oydAN9aPs{y5ER z8YYxxKUSEJZ;V}=?E^)je-krwur_3Zfi5&CT@jlS%*)8WwyQpJ&zh!vr#h;`hDw10 zv$Nkhb5x*xyun&W>x+Jzo9KNxcUv>OyFdCz@D!>)0|*4E)$Z&uH`}-fxiZ6K+*iR- 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