From a7a6dcb0d01f5200011a3c49d3783f34a8efed44 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Fri, 9 Apr 2021 08:05:22 +0200 Subject: new slides --- vorlesungen/06_msegalois/common.tex | 2 +- vorlesungen/06_msegalois/slides.tex | 15 +--- vorlesungen/slides/4/galois/aufloesbarkeit.tex | 106 ++++++++++++++++++++++++ vorlesungen/slides/4/galois/automorphismus.tex | 110 ++++++++++++++++++++++++- vorlesungen/slides/4/galois/erweiterung.tex | 51 ++++++++++++ vorlesungen/slides/4/galois/konstruktion.tex | 2 + vorlesungen/slides/4/galois/radikale.tex | 57 ++++++++++++- vorlesungen/slides/4/galois/sn.tex | 75 ++++++++++++++++- vorlesungen/slides/test.tex | 12 +-- 9 files changed, 406 insertions(+), 24 deletions(-) diff --git a/vorlesungen/06_msegalois/common.tex b/vorlesungen/06_msegalois/common.tex index 0700acf..50adc4f 100644 --- a/vorlesungen/06_msegalois/common.tex +++ b/vorlesungen/06_msegalois/common.tex @@ -9,7 +9,7 @@ \usetheme[hideothersubsections,hidetitle]{Hannover} } \beamertemplatenavigationsymbolsempty -\title[Titel]{Titel} +\title[Galois]{Galois-Theorie} \author[A.~Müller]{Prof. Dr. Andreas Müller} \date[]{} \newboolean{presentation} diff --git a/vorlesungen/06_msegalois/slides.tex b/vorlesungen/06_msegalois/slides.tex index 37739aa..95695c4 100644 --- a/vorlesungen/06_msegalois/slides.tex +++ b/vorlesungen/06_msegalois/slides.tex @@ -5,28 +5,19 @@ % \section{Körpererweiterungen} -% XXX Was ist eine Körpererweiterung \folie{4/galois/erweiterung.tex} -\section{Galois-Gruppe} -% XXX Übersetzung Körpererweiterungsstruktur in eine Gruppe -\folie{4/galois/automorphismus.tex} - \section{Geometrische Anwendungen} -% XXX Geometrische Konstruktionen \folie{4/galois/konstruktion.tex} -% XXX Verdoppelung des Würfels \folie{4/galois/wuerfel.tex} -% XXX Dreiteilung des Winkels \folie{4/galois/winkeldreiteilung.tex} -% XXX Quadratur des Kreises \folie{4/galois/quadratur.tex} +\section{Galois-Gruppe} +\folie{4/galois/automorphismus.tex} + \section{Lösbarkeit durch Radikale} -% XXX Wurzelformeln mit Radikalen \folie{4/galois/radikale.tex} -% XXX Auflösbarkeit einer Gruppe \folie{4/galois/aufloesbarkeit.tex} -% XXX S_n ist nicht auflösbar \folie{4/galois/sn.tex} diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex index 3215689..ef5902b 100644 --- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex +++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex @@ -4,11 +4,117 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \frametitle{Auflösbarkeit} +\vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Radikalerweiterung} +Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$ +einer Radikalerweiterung +\[ +\Bbbk \subset \Bbbk(\alpha) +\] +sind festgelegt durch Wahl von $f(\alpha)$. + +\begin{itemize} +\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt +\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein +\end{itemize} + +\end{block}} +\uncover<8->{% +\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$} +$\mathbb{Q}\subset \Bbbk$, +$n$ verschiedene Nullstellen $\mathbb{C}$: +\[ +\uncover<9->{ +\operatorname{Gal}(\Bbbk/\mathbb{Q}) +\cong +S_n} +\uncover<10->{ +\quad +\text{auflösbar?}} +\] +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\begin{block}{\uncover<5->{Galois-Gruppen}} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{1.2} + +\uncover<2->{ +\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3}); +\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen}; +} + +\node at (0,0) {$\mathbb{Q}$}; +\node at (0,{1*\s}) {$E_1$}; +\node at (0,{2*\s}) {$E_2$}; +\node at (0,{3*\s}) {$E_3$}; +\node at (0,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) {$\Bbbk$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s}); + +\begin{scope}[xshift=0.5cm] +\uncover<7->{ +\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3}); +\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe}; +} +\uncover<5->{ +\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$}; +\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$}; +\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$}; +\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$}; +\node at (1,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$}; +\node at (1,{0.5*\s}) {$\cap\mathstrut$}; +\node at (1,{1.5*\s}) {$\cap\mathstrut$}; +\node at (1,{2.5*\s}) {$\cap\mathstrut$}; +\node at (1,{3.5*\s}) {$\cap\mathstrut$}; +\node at (1,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<6->{ +\begin{scope}[xshift=2.5cm] +\node at (0,{0*\s}) {$G_n$}; +\node at (0,{1*\s}) {$G_{n-1}$}; +\node at (0,{2*\s}) {$G_{n-2}$}; +\node at (0,{3*\s}) {$G_{n-3}$}; +\node at (0,{5*\s}) {$G_0=\{e\}$}; +\node at (0,{0.5*\s}) {$\cap\mathstrut$}; +\node at (0,{1.5*\s}) {$\cap\mathstrut$}; +\node at (0,{2.5*\s}) {$\cap\mathstrut$}; +\node at (0,{3.5*\s}) {$\cap\mathstrut$}; +\node at (0,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<7->{ +\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$}; +\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$}; +\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$}; +\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch}; +} + +\end{scope} +\end{scope} + + + +\end{tikzpicture} +\end{center} +\end{block} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex index ab666cf..6051813 100644 --- a/vorlesungen/slides/4/galois/automorphismus.tex +++ b/vorlesungen/slides/4/galois/automorphismus.tex @@ -4,11 +4,115 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \begin{frame}[t] -\frametitle{Automorphismen} +\setlength{\abovedisplayskip}{4pt} +\setlength{\belowdisplayskip}{4pt} +\frametitle{Galois-Gruppe} +\vspace{-20pt} \begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} +\begin{column}{0.40\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{3.0} +\begin{scope}[xshift=-1.5cm] +\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut}; +\node at (0,{\s}) {$G$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0); +\node at (0,{-0.5*\s}) [left] {$[F:E]$}; +\node at (0,{0.5*\s}) [left] {$[G:F]$}; +\node at (0,0) {$F$}; +\node at (0,{-\s}) {$E$}; +\end{scope} +\uncover<3->{ +\begin{scope}[xshift=1.8cm] +\node at (0,{\s+0.1}) [above] {Gruppe\strut}; +\fill (0,{-\s}) circle[radius=0.06]; +\fill (0,0) circle[radius=0.06]; +\fill (0,{\s}) circle[radius=0.06]; +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=100,in=-100] (0,{\s}); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=80,in=-80] (0,0); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,0) to[out=80,in=-80] (0,{\s}); +\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$}; +\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$}; +\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$}; +\end{scope} +\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s}); +\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$}; +} +\uncover<4->{ +\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s}); +\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$}; +} +\end{tikzpicture} +\end{center} \end{column} -\begin{column}{0.48\textwidth} +\begin{column}{0.56\textwidth} +\uncover<2->{% +\begin{block}{Automorphismus} +\vspace{-10pt} +\[ +\operatorname{Aut}(F) += +\left\{ +f\colon F\to F +\left| +\begin{aligned} +f(x+y)&=f(x)+f(y)\\ +f(xy)&=f(x)f(y) +\end{aligned} +\right. +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Galois-Gruppe} +Automorphismen, die $E$ festlassen +\[ +{\color{red} +\operatorname{Gal}(F/E) +} += +\left\{ +\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Fixkörper} +$H\subset \operatorname{Aut}(F)$: +\begin{align*} +{\color{blue}F^H} +&= +\{x\in F\;|\; hx = x\forall h\in H\} +=\operatorname{Fix}(H) +\end{align*} +\end{block}} +\vspace{-13pt} +\uncover<5->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<6-> +\( +\operatorname{Gal}(\mathbb{C}/\mathbb{R}) += +\{ +\operatorname{id}_{\mathbb{C}}, +\operatorname{conj}\colon z\mapsto\overline{z} +\} +\) +\item<7-> +\( +\mathbb{C}^{\operatorname{conj}} += +\mathbb{R} +\) +\end{itemize} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex index 1cf0bec..6909849 100644 --- a/vorlesungen/slides/4/galois/erweiterung.tex +++ b/vorlesungen/slides/4/galois/erweiterung.tex @@ -4,11 +4,62 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \frametitle{Körpererweiterungen} +\vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\begin{block}{Körpererweiterung} +$E,F$ Körper: $E\subset F$ +\end{block} +\uncover<6->{% +\begin{block}{Vektorraum} +$F$ ist ein Vektorraum über $E$ +\end{block}} +\uncover<7->{% +\begin{block}{Endliche Körpererweiterung} +$\dim_E F < \infty$ +\end{block}} +\uncover<8->{% +\begin{block}{Adjunktion eines $\alpha$} +$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und +$\alpha$ enthält. +\end{block}} +\uncover<9->{% +\begin{block}{Algebraische Erweiterung} +$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von +$m(X)\in\Bbbk[X]$ +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiele} +\begin{enumerate} +\item<3-> +$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$ +\item<4-> +$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$ +\item<5-> +$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$ +\end{enumerate} +\end{block}} +\uncover<7->{% +\begin{block}{Grad} +$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls +\[ +\dim_E F = n =: [F:E] +\] +\uncover<8->{% +Gleichbedeutend: $\deg m(X) = n$} +\uncover<10->{% +\[ +E\subset F\subset G +\Rightarrow +[G:E] = [G:F]\cdot [F:E] +\] +(in unseren Fällen)} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex index 6afa359..094b570 100644 --- a/vorlesungen/slides/4/galois/konstruktion.tex +++ b/vorlesungen/slides/4/galois/konstruktion.tex @@ -142,4 +142,6 @@ $\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden} \end{center}} \end{column} \end{columns} +\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:} +Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$} \end{frame} diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex index 52fc4b9..e9e4ce8 100644 --- a/vorlesungen/slides/4/galois/radikale.tex +++ b/vorlesungen/slides/4/galois/radikale.tex @@ -4,11 +4,66 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \begin{frame}[t] -\frametitle{Radikale} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lösung durch Radikale} +\vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\begin{block}{Problemstellung} +Finde Nullstellen eines Polynomes +\[ +p(X) += +a_nX^n + a_{n-1}X^{n-1} ++\dots+ +a_1X+a_0 +\] +$p\in\mathbb{Q}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Radikale} +Geschachtelte Wurzelausdrücke +\[ +\sqrt[3]{ +-\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} ++ +\sqrt[3]{ +-\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} +\] +\uncover<3->{(Lösung von $x^3+px+q=0$)} +\end{block}} +\uncover<4->{% +\begin{block}{Lösbar durch Radikale} +Nullstelle von $p(X)$ ist ein Radikal +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Algebraische Formulierung} +Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$, +finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart, +dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{: +$\Bbbk = \mathbb{Q}[X]/(p)$} +\end{block}} +\uncover<7->{% +\begin{block}{Radikalerweiterung} +Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften +\begin{itemize} +\item<8-> $\alpha$ ist eine Einheitswurzel +\item<9-> $\alpha^k\in\Bbbk$ +\end{itemize} +\end{block}} +\vspace{-5pt} +\uncover<10->{% +\begin{block}{Lösbar durch Radikale} +Radikalerweiterungen +\[ +\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha +\] +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex index 0e3ebe2..1cae3fa 100644 --- a/vorlesungen/slides/4/galois/sn.tex +++ b/vorlesungen/slides/4/galois/sn.tex @@ -4,11 +4,84 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \begin{frame}[t] -\frametitle{Auflösbarkeit von $S_n$} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nichtauflösbarkeit von $S_n$} +\vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\begin{block}{Die symmetrische Gruppe $S_n$} +Permutationen auf $n$ Elementen +\[ +\sigma += +\begin{pmatrix} +1&2&3&\dots&n\\ +\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n) +\end{pmatrix} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Signum} +$t(\sigma)=\mathstrut$ Anzahl Transpositionen +\[ +\operatorname{sgn}(\sigma) += +(-1)^{t(\sigma)} += +\begin{cases} +\phantom{-}1&\text{$t(\sigma)$ gerade} +\\ +-1&\text{$t(\sigma)$ ungerade} +\end{cases} +\] +Homomorphismus! +\end{block}} +\uncover<3->{% +\begin{block}{Die alternierende Gruppe $A_n$} +\vspace{-12pt} +\[ +A_n = \ker \operatorname{sgn} += +\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\} +\] +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Normale Untergruppe} +\begin{itemize} +\item +$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$ +\item +$G/N$ ist wohldefiniert +\end{itemize} +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Einfache Gruppe} +$G$ einfach $\Leftrightarrow$ +\[ +H\triangleleft G +\; +\Rightarrow +\; +\text{$H=\{e\}$ oder $H=G$} +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$} +\begin{enumerate} +\item<7-> Zeigen, dass $A_5$ einfach ist +\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach +\end{enumerate} +\uncover<9->{% +$\Rightarrow$ i.~A.~keine Lösung der +einer Polynomgleichung vom Grad $\ge 5$ durch Radikale +} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 5bdec4f..7e69436 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -25,13 +25,9 @@ % XXX Was ist eine Körpererweiterung %\folie{4/galois/erweiterung.tex} -\section{Galois-Gruppe} -% XXX Übersetzung Körpererweiterungsstruktur in eine Gruppe -%\folie{4/galois/automorphismus.tex} - \section{Geometrische Anwendungen} % XXX Geometrische Konstruktionen -\folie{4/galois/konstruktion.tex} +%\folie{4/galois/konstruktion.tex} % XXX Verdoppelung des Würfels %\folie{4/galois/wuerfel.tex} % XXX Dreiteilung des Winkels @@ -39,11 +35,15 @@ % XXX Quadratur des Kreises %\folie{4/galois/quadratur.tex} +\section{Galois-Gruppe} +% XXX Übersetzung Körpererweiterungsstruktur in eine Gruppe +%\folie{4/galois/automorphismus.tex} + \section{Lösbarkeit durch Radikale} % XXX Wurzelformeln mit Radikalen %\folie{4/galois/radikale.tex} % XXX Auflösbarkeit einer Gruppe %\folie{4/galois/aufloesbarkeit.tex} % XXX S_n ist nicht auflösbar -%\folie{4/galois/sn.tex} +\folie{4/galois/sn.tex} -- cgit v1.2.1 From 1fb743df08b0734932d510c6b11405d0a2dbbe47 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 19:57:13 +0200 Subject: new slides --- vorlesungen/common/template.tex | 19 +++ vorlesungen/slides/7/Makefile.inc | 20 +++ vorlesungen/slides/7/ableitung.tex | 62 +++++++++ vorlesungen/slides/7/algebraisch.tex | 105 +++++++++++++++ vorlesungen/slides/7/chapter.tex | 17 +++ vorlesungen/slides/7/dg.tex | 80 +++++++++++ vorlesungen/slides/7/drehung.tex | 113 ++++++++++++++++ vorlesungen/slides/7/einparameter.tex | 87 ++++++++++++ vorlesungen/slides/7/kurven.tex | 85 ++++++++++++ vorlesungen/slides/7/liealgebra.tex | 69 ++++++++++ vorlesungen/slides/7/mannigfaltigkeit.tex | 44 ++++++ vorlesungen/slides/7/parameter.tex | 58 ++++++++ vorlesungen/slides/7/semi.tex | 109 +++++++++++++++ vorlesungen/slides/7/sl2.tex | 216 ++++++++++++++++++++++++++++++ vorlesungen/slides/7/symmetrien.tex | 133 ++++++++++++++++++ vorlesungen/slides/Makefile.inc | 3 +- vorlesungen/slides/test.tex | 75 +++++------ 17 files changed, 1251 insertions(+), 44 deletions(-) create mode 100644 vorlesungen/common/template.tex create mode 100644 vorlesungen/slides/7/Makefile.inc create mode 100644 vorlesungen/slides/7/ableitung.tex create mode 100644 vorlesungen/slides/7/algebraisch.tex create mode 100644 vorlesungen/slides/7/chapter.tex create mode 100644 vorlesungen/slides/7/dg.tex create mode 100644 vorlesungen/slides/7/drehung.tex create mode 100644 vorlesungen/slides/7/einparameter.tex create mode 100644 vorlesungen/slides/7/kurven.tex create mode 100644 vorlesungen/slides/7/liealgebra.tex create mode 100644 vorlesungen/slides/7/mannigfaltigkeit.tex create mode 100644 vorlesungen/slides/7/parameter.tex create mode 100644 vorlesungen/slides/7/semi.tex create mode 100644 vorlesungen/slides/7/sl2.tex create mode 100644 vorlesungen/slides/7/symmetrien.tex diff --git a/vorlesungen/common/template.tex b/vorlesungen/common/template.tex new file mode 100644 index 0000000..a1343f8 --- /dev/null +++ b/vorlesungen/common/template.tex @@ -0,0 +1,19 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Template} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\end{column} +\begin{column}{0.48\textwidth} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc new file mode 100644 index 0000000..ef004ca --- /dev/null +++ b/vorlesungen/slides/7/Makefile.inc @@ -0,0 +1,20 @@ +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter5 = \ + ../slides/7/symmetrien.tex \ + ../slides/7/algebraisch.tex \ + ../slides/7/parameter.tex \ + ../slides/7/mannigfaltigkeit.tex \ + ../slides/7/sl2.tex \ + ../slides/7/drehung.tex \ + ../slides/7/semi.tex \ + ../slides/7/kurven.tex \ + ../slides/7/einparameter.tex \ + ../slides/7/ableitung.tex \ + ../slides/7/liealgebra.tex \ + ../slides/7/dg.tex \ + ../slides/7/chapter.tex + diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex new file mode 100644 index 0000000..b061b9a --- /dev/null +++ b/vorlesungen/slides/7/ableitung.tex @@ -0,0 +1,62 @@ +% +% ableitung.tex -- Ableitung in der Lie-Gruppe +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ableitung in der Matrix-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{O}(n)$} +$s \mapsto A(s)\in\operatorname{O}(n)$ +\begin{align*} +I +&= +A(s)^tA(s) +\\ +0 += +\frac{d}{ds} I +&= +\frac{d}{ds} (A(s)^t A(s)) +\\ +&= +\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s) +\intertext{An der Stelle $s=0$, d.~h.~$A(0)=I$} +0 +&= +\dot{A}(0)^t ++ +\dot{A}(0) +\\ +\Leftrightarrow +\qquad +\dot{A}(0)^t &= -\dot{A}(0) +\end{align*} +``Tangentialvektoren'' sind antisymmetrische Matrizen +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} +$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ +\begin{align*} +1 &= \det A(t) +\\ +0 += +\frac{d}{dt}1 +&= +\frac{d}{dt} \det A(t) +\intertext{mit dem Entwicklungssatz kann man nachrechnen:} +0&=\operatorname{Spur}\dot{A}(0) +\end{align*} +``Tangentialvektoren'' sind spurlose Matrizen +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex new file mode 100644 index 0000000..5b33566 --- /dev/null +++ b/vorlesungen/slides/7/algebraisch.tex @@ -0,0 +1,105 @@ +% +% algebraisch.tex -- algebraische Definition der Symmetrien +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Erhaltungsgrössen und Algebra} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Längen und Winkel} +Längenmessung mit Skalarprodukt +\begin{align*} +\|\vec{v}\|^2 +&= +\langle \vec{v},\vec{v}\rangle += +\vec{v}\cdot \vec{v} += +\vec{v}^t\vec{v} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Flächeninhalt/Volumen} +$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ +\\ +Volumen des Parallelepipeds: $\det V$ +\end{block} +\end{column} +\end{columns} +% +\vspace{-7pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Längenerhaltende Transformationen} +$A\in\operatorname{GL}_n(\mathbb{R})$ +\begin{align*} +\vec{x}^t\vec{y} +&= +(A\vec{x}) +\cdot +(A\vec{y}) += +(A\vec{x})^t +(A\vec{y}) +\\ +\vec{x}^tI\vec{y} +&= +\vec{x}^tA^tA\vec{y} +\Rightarrow I=A^tA +\end{align*} +Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$ +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Volumenerhaltende Transformationen} +$A\in\operatorname{GL}_n(\mathbb{R})$ +\begin{align*} +\det(V) +&= +\det(AV) += +\det(A)\det(V) +\\ +1&=\det(A) +\end{align*} +(Produktsatz für Determinante) +\end{block} +\end{column} +\end{columns} +% +\vspace{-3pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Orthogonale Matrizen} +Längentreue Abbildungen = orthogonale Matrizen: +\[ +O(n) += +\{ +A \in \operatorname{GL}_n(\mathbb{R}) +\;|\; +A^tA=I +\} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{``Spezielle'' Matrizen} +Volumen-/Orientierungserhaltende Transformationen: +\[ +\operatorname{SL}_n(\mathbb R) += +\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} +\] +\end{block} +\end{column} +\end{columns} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex new file mode 100644 index 0000000..44d46a6 --- /dev/null +++ b/vorlesungen/slides/7/chapter.tex @@ -0,0 +1,17 @@ +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\folie{7/semi.tex} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/dg.tex} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex new file mode 100644 index 0000000..36b1ade --- /dev/null +++ b/vorlesungen/slides/7/dg.tex @@ -0,0 +1,80 @@ +% +% dg.tex -- Differentialgleichung für die Exponentialabbildung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zurück zur Lie-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Tangentialvektor im Punkt $\gamma(t)$} +Ableitung von $\gamma(t)$ an der Stelle $t$: +\begin{align*} +\dot{\gamma}(t) +&= +\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} +\\ +&= +\frac{d}{ds} +\gamma(t+s) +\bigg|_{s=0} +\\ +&= +\frac{d}{ds} +\gamma(t)\gamma(s) +\bigg|_{s=0} +\\ +&= +\gamma(t) +\frac{d}{ds} +\gamma(s) +\bigg|_{s=0} += +\gamma(t) \dot{\gamma}(0) +\end{align*} +\end{block} +\vspace{-10pt} +\begin{block}{Differentialgleichung} +\vspace{-10pt} +\[ +\dot{\gamma}(t) = \gamma(t) A +\quad +\text{mit} +\quad +A=\dot{\gamma}(0)\in LG +\] +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\begin{block}{Lösung} +Exponentialfunktion +\[ +\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k +\] +\end{block} +\vspace{-5pt} +\begin{block}{Kontrolle: Tangentialvektor berechnen} +\vspace{-10pt} +\begin{align*} +\frac{d}{dt}e^{At} +&= +\sum_{k=1}^\infty A^k \frac{d}{dt} t^{k}{k!} +\\ +&= +\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A +\\ +&= +\sum_{k=0} A^k\frac{t^k}{k!} +A += +e^{At} A +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex new file mode 100644 index 0000000..ae0dbe3 --- /dev/null +++ b/vorlesungen/slides/7/drehung.tex @@ -0,0 +1,113 @@ +% +% drehung.tex -- Drehung aus streckungen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehung aus Streckungen und Scherungen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Drehung} +\[ +\operatorname{SO}(2) += +\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2) +\] +\end{block} +\begin{block}{Zusammensetzung} +Eine Drehung muss als Zusammensetzung geschrieben werden können: +\[ +D_{\alpha} += +\begin{pmatrix} +\cos\alpha & -\sin\alpha\\ +\sin\alpha &\phantom{-}\cos\alpha +\end{pmatrix} += +DST +\] +\end{block} +\begin{block}{Beispiel} +\vspace{-12pt} +\[ +D_{60^\circ} += +{\tiny +\begin{pmatrix}2&0\\0&\frac12\end{pmatrix} +\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} +} +\] +\end{block} +\end{column} +\begin{column}{0.58\textwidth} +\begin{block}{Ansatz} +\vspace{-12pt} +\begin{align*} +DST +&= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +1&-s\\ +0&1 +\end{pmatrix} +\begin{pmatrix} +1&0\\ +t&1 +\end{pmatrix} +\\ +&= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +-st&-s\\ + t& 1 +\end{pmatrix} +\\ +&= +\begin{pmatrix} +-stc^{-1}&{\color{darkgreen}sc^{-1}}\\ +{\color{blue}ct}&{\color{red}c} +\end{pmatrix} += +\begin{pmatrix} +\cos\alpha & {\color{darkgreen}- \sin\alpha} \\ +{\color{blue}\sin\alpha} & \phantom{-} {\color{red}\cos\alpha} +\end{pmatrix} +\end{align*} +\end{block} +\vspace{-10pt} +\begin{block}{Koeffizientenvergleich} +\vspace{-15pt} +\begin{align*} +{\color{red} c} +&= +{\color{red}\cos\alpha } +&& +& +{\color{blue} +t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\ +{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} +& +&\Rightarrow& +{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha +\\ +{\color{orange} -stc^{-t}} +&= +\rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex new file mode 100644 index 0000000..52924bf --- /dev/null +++ b/vorlesungen/slides/7/einparameter.tex @@ -0,0 +1,87 @@ +% +% einparameter.tex -- Einparameter Untergruppen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Einparameter Untergruppen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$, +die gleichzeitig eine Untergruppe von $G$ ist mit +\[ +\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R} +\] +\end{block} +\begin{block}{Drehungen} +Drehmatrizen bilden Einparameter- Untergruppen +\begin{align*} +t \mapsto D_{x,t} +&= +\begin{pmatrix} +1&0&0\\ +0&\cos t&-\sin t\\ +0&\sin t& \cos t +\end{pmatrix} +\\ +D_{x,t}D_{x,s} +&= +D_{x,t+s} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} +\vspace{-12pt} +\[ +\begin{pmatrix} +1&s\\ +0&1 +\end{pmatrix} +\begin{pmatrix} +1&t\\ +0&1 +\end{pmatrix} += +\begin{pmatrix} +1&s+t\\ +0&1 +\end{pmatrix} +\] +\end{block} +\vspace{-12pt} +\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} +\vspace{-12pt} +\[ +\begin{pmatrix} +e^s&0\\0&e^{-s} +\end{pmatrix} +\begin{pmatrix} +e^t&0\\0&e^{-t} +\end{pmatrix} += +\begin{pmatrix} +e^{t+s}&0\\0&e^{-(t+s)} +\end{pmatrix} +\] +\end{block} +\vspace{-12pt} +\begin{block}{Gemischt} +\vspace{-12pt} +\begin{gather*} +A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t +\\ +\text{dank}\quad +\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 +=I +\end{gather*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex new file mode 100644 index 0000000..196fa2a --- /dev/null +++ b/vorlesungen/slides/7/kurven.tex @@ -0,0 +1,85 @@ +% +% kurven.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kurven und Tangenten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Kurven} +Kurve in $\mathbb{R}^n$: +\vspace{-12pt} +\[ +\gamma +\colon +I=[a,b] \to \mathbb{R}^n +: +t\mapsto \gamma(t) += +\begin{pmatrix} +x_1(t)\\ +x_2(t)\\ +\vdots\\ +x_n(t) +\end{pmatrix} +\] +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (1,0.5); +\coordinate (B) at (4,0.5); +\coordinate (C) at (2,2.2); +\coordinate (D) at (5,2); +\coordinate (E) at ($(C)+(80:2)$); + +\draw[color=red,line width=1.4pt] + (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D); +\fill[color=red] (C) circle[radius=0.06]; +\node[color=red] at (C) [left] {$\gamma(t)$}; + +\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); +\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; + +\draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; +\draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Tangenten} +Ableitung +\[ +\frac{d}{dt}\gamma(t) += +\dot{\gamma}(t) += +\begin{pmatrix} +\dot{x}_1(t)\\ +\dot{x}_2(t)\\ +\vdots\\ +\dot{x}_n(t) +\end{pmatrix} +\] +Lineare Approximation: +\[ +\gamma(t+h) += +\gamma(t) ++ +\dot{\gamma}(t) \cdot h ++ +o(h) +\] +Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ +in $\mathbb{R}^n$ liegen +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex new file mode 100644 index 0000000..16a7aa0 --- /dev/null +++ b/vorlesungen/slides/7/liealgebra.tex @@ -0,0 +1,69 @@ +% +% liealgebra.tex -- Lie-Algebra +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra} +\vspace{-20pt} +\begin{block}{Vektorraum} +Tangentialvektoren im Punkt $I$: +\begin{center} +\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} +\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ +\hline +\operatorname{GL}_n(\mathbb{R}) +& beliebige Matrizen +& M_n(\mathbb{R}) +\\ +\operatorname{O(n)} +& antisymmetrische Matrizen +& \operatorname{o}(n) +\\ +\operatorname{SL}_n(\mathbb{R}) +& spurlose Matrizen +& \operatorname{sl}_2(\mathbb{R}) +\\ +\operatorname{U(n)} +& antihermitesche Matrizen +& \operatorname{u}(n) +\\ +\operatorname{SU(n)} +& spurlose, antihermitesche Matrizen +& \operatorname{su}(n) +\end{tabular} +\end{center} +\end{block} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Lie-Klammer} +Kommutator: $[A,B] = AB-BA$ +\end{block} +\begin{block}{Nachprüfen} +$[A,B]\in LG$ +für $A,B\in LG$ +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\begin{block}{Algebraische Eigenschaften} +\begin{itemize} +\item antisymmetrisch: $[A,B]=-[B,A]$ +\item Jacobi-Identität +\[ +[A,[B,C]]+ +[B,[C,A]]+ +[C,[A,B]] += 0 +\] +\end{itemize} +{\usebeamercolor[fg]{title} +Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex new file mode 100644 index 0000000..7809ea5 --- /dev/null +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -0,0 +1,44 @@ +% +% mannigfaltigkeit.tex -- Mannigfaltigkeit +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Mannigfaltigkeit} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ +\item differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet +\[ +\varphi_\beta\circ\varphi_\alpha^{-1} +\colon +\varphi_\alpha(U_\alpha\cap U_\beta) +\to +\varphi_\beta(U_\alpha\cap U_\beta) +\] +\item Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken +\end{itemize} +\end{block} +\vspace{-7pt} +\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$} +Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus +\end{block} +\vspace{-3pt} +\begin{block}{Lie-Gruppe} +Gruppe und Mannigfaltigkeit +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex new file mode 100644 index 0000000..ec129bb --- /dev/null +++ b/vorlesungen/slides/7/parameter.tex @@ -0,0 +1,58 @@ +% +% parameter.tex -- Parametrisierung der Matrizen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen Parametrisieren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.4\textwidth} +\begin{block}{Drehung um Achsen} +\begin{align*} +D_{x,\alpha} +&= +\begin{pmatrix} +1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha +\end{pmatrix} +\\ +D_{y,\beta} +&= +\begin{pmatrix} +\cos\beta&0&-\sin\beta\\0&1&0\\\sin\beta&0&\cos\beta +\end{pmatrix} +\\ +D_{z,\gamma} +&= +\begin{pmatrix} +\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 +\end{pmatrix} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\begin{block}{Drehung um $\vec{\omega}$} +$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel +\\ +$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse +\[ +\vec{x} +\mapsto +\cos\omega +\vec{x} ++ +(\vec{k}\times\vec{x})\sin\omega ++ +\vec{k}(\vec{k}\cdot\vec{x}) (1-\cos\omega) +\] +XXX TODO: Bild für Rodriguez Formel +\end{block} +\end{column} +\end{columns} +{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine +dreidimensionale Gruppe +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex new file mode 100644 index 0000000..46f6d03 --- /dev/null +++ b/vorlesungen/slides/7/semi.tex @@ -0,0 +1,109 @@ +% +% semi.tex -- Beispiele: semidirekte Produkte +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehung/Skalierung und Verschiebung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Skalierung und Verschiebung} +Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$ +\\ +Wirkung auf $\mathbb{R}$: +\[ +x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Drehung und Verschiebung} +Gruppe +$G= +\{ (\alpha,\vec{t}) +\;|\; +\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2 +\}$ +Wirkung auf $\mathbb{R}^2$: +\[ +\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} +\] +\end{block} +\end{column} +\end{columns} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(e^{s_1},t_1)(e^{s_2},t_2)x +&= +(e^{s_1},t_1)(e^{s_2}x+t_2) +\\ +&= +e^{s_1+s_2}x + e^{s_1}t_2+t_1 +\\ +(e^{s_1},t_1)(e^{s_2},t_2) +&= +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2) +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +\vec{x} +&= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2) +\\ +&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1 +\\ +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +&= +(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) +\end{align*} +\end{block} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Matrixschreibweise} +\vspace{-12pt} +\[ +g=(e^s,t) = +\begin{pmatrix} +e^s&t\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}x\\1\end{pmatrix} +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Matrixschreibweise} +\vspace{-12pt} +\[ +g=(\alpha,\vec{t}) = +\begin{pmatrix} +D_{\alpha}&\vec{t}\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}\vec{x}\\1\end{pmatrix} +\] +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex new file mode 100644 index 0000000..3480460 --- /dev/null +++ b/vorlesungen/slides/7/sl2.tex @@ -0,0 +1,216 @@ +% +% sl2.tex -- Beispiel: Parametrisierung von SL_2(R) +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t,fragile] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Determinante} +\[ +A=\begin{pmatrix} +a&b\\ +c&d +\end{pmatrix} +\;\Rightarrow\; +\det A = ad-bc +\] +\end{block} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Dimension} +\[ +4\; \text{Variablen} +- +1\; \text{Bedingung} += +3\; \text{Dimensionen} +\] +\end{block} +\end{column} +\end{columns} +\vspace{-10pt} +\uncover<3->{% +\begin{columns}[t,onlytextwidth] +\def\s{0.94} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \fill[color=blue!20] (-1,-1) rectangle (1,1); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \foreach \d in {4,...,10}{ + \only<\d>{ + \pgfmathparse{1+(\d-4)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + ({-\t},{-1/\t}) rectangle (\t,{1/\t}); + \foreach \x in {-2,...,2}{ + \draw[color=red,line width=0.3pt] + ({\x*\t},-3) -- ({\x*\t},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{\y/\t}) -- (3,{\y/\t}); + } + } + } + \uncover<11->{ + \xdef\t{1.6} + \fill[color=red!40,opacity=0.5] + ({-\t},{-1/\t}) rectangle (\t,{1/\t}); + \foreach \x in {-2,...,2}{ + \draw[color=red,line width=0.3pt] + ({\x*\t},-3) -- ({\x*\t},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{\y/\t}) -- (3,{\y/\t}); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<3->{% + \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + D + = + \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\fill[color=blue!20] (-1,-1) rectangle (1,1); +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \foreach \d in {11,...,17}{ + \only<\d>{ + \pgfmathparse{(\d-11)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + ({-1+\t*(-1)},{-1}) + -- + ({1+\t*(-1)},{-1}) + -- + ({1+\t},{1}) + -- + ({-1+\t},{1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({-3+\t*\y},\y) -- ({3+\t*\y},\y); + } + } + } + \uncover<18->{ + \xdef\t{0.6} + \fill[color=red!40,opacity=0.5] + ({-1+\t*(-1)},{-1}) + -- + ({1+\t*(-1)},{-1}) + -- + ({1+\t},{1}) + -- + ({-1+\t},{1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({-3+\t*\y},\y) -- ({3+\t*\y},\y); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<11->{ + \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + S + = + \begin{pmatrix} 1&s\\ 0&1\end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\fill[color=blue!20] (-1,-1) rectangle (1,1); +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \foreach \d in {18,...,24}{ + \only<\d>{ + \pgfmathparse{(\d-18)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + (-1,{\t*(-1)-1}) + -- + (1,{\t*1-1}) + -- + (1,{\t*1+1}) + -- + (-1,{\t*(-1)+1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); + } + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<18->{% +\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + T + = + \begin{pmatrix} 1&0\\t&1\end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\end{columns}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex new file mode 100644 index 0000000..79f9ef7 --- /dev/null +++ b/vorlesungen/slides/7/symmetrien.tex @@ -0,0 +1,133 @@ +% +% symmetrien.tex -- Symmetrien +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Symmetrien} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Diskrete Symmetrien} +\begin{itemize} +\item +Ebenen-Spiegelung: +\[ +{\tiny +\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} +} +\mapsto +{\tiny +\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}, +} +\; +\vec{x} +\mapsto +\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} +\] +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{10} +\def\b{50} +\def\r{2} +\coordinate (O) at (0,0); +\coordinate (A) at (\b:\r); +\coordinate (B) at ({180+2*\a-\b}:\r); +\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); +\coordinate (N) at (\a:2); +\coordinate (D) at (\a:{\r*cos(\b-\a)}); +\clip (-2.5,-0.45) rectangle (2.5,1.95); + +\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) -- cycle; +\draw[->,color=darkgreen] (O) -- (N); +\node[color=darkgreen] at (N) [above] {$\vec{n}$}; + + +\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) -- cycle; +\fill[color=red] (O) circle[radius=0.06]; +\draw[color=red] ({\a-90}:2) -- ({\a+90}:2); +\fill[color=blue] (C) circle[radius=0.06]; +\draw[color=blue,line width=0.1pt] (A) -- (D); +\node[color=darkgreen] at (D) [below,rotate=\a] {$(\vec{n}\cdot\vec{x})\vec{n}$}; +\draw[color=blue,line width=0.5pt] (A)--(B); + +\node[color=blue] at (A) [above right] {$\vec{x}$}; +\node[color=blue] at (B) [above left] {$\vec{x}'$}; + +\node[color=red] at (O) [below left] {$O$}; + +\draw[->,color=blue,shorten <= 0.06cm] (O) -- (A); +\draw[->,color=blue,shorten <= 0.06cm] (O) -- (B); + +\end{tikzpicture} +\end{center} +\vspace{-5pt} +$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ +\item +Punkt-Spiegelung: +\[ +{\tiny +\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} +} +\mapsto +- +{\tiny +\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*} +} +\] +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Kontinuierliche Symmetrien} +\begin{itemize} +\item Translation: +\( +\vec{x} \mapsto \vec{x} + \vec{t} +\) +\item Drehung: +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{25} +\def\r{1.3} +\coordinate (O) at (0,0); +\begin{scope} +\clip (-1.1,-0.1) rectangle (2.3,2.3); +\draw[color=red] (O) circle[radius=2]; +\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle; +\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; +\node at ({0.5*\a}:1) {$\alpha$}; +\node at ({90+0.5*\a}:1) {$\alpha$}; +\draw[->,color=blue] (O) -- (\a:2); +\draw[->,color=darkgreen] (O) -- ({90+\a}:2); +\end{scope} +\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; +\end{tikzpicture} +\end{center} +\[ +\begin{pmatrix}x\\y\end{pmatrix} +\mapsto +\begin{pmatrix} +{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\ +{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} +\end{pmatrix} +\begin{pmatrix}x\\y\end{pmatrix} +\] +\end{itemize} +\end{block} +\vspace{-10pt} +\begin{block}{Definition} +Längen/Winkel bleiben erhalten +\\ +$\Rightarrow$ $\exists$ Erhaltungsgrösse +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index 4bf9348..e2271b8 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -9,9 +9,10 @@ include ../slides/2/Makefile.inc include ../slides/3/Makefile.inc include ../slides/4/Makefile.inc include ../slides/5/Makefile.inc +include ../slides/7/Makefile.inc include ../slides/8/Makefile.inc include ../slides/9/Makefile.inc slides = \ $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ - $(chapter5) $(chapter8) $(chapter9) + $(chapter5) $(chapter7) $(chapter8) $(chapter9) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 7e69436..7062509 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -1,49 +1,38 @@ % % test.tex collection of all slides % -% (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil % -%\folie{5/verzerrung.tex} -%\folie{5/plan.tex} -%\folie{5/planbeispiele.tex} -%\folie{5/approximation.tex} - -% XXX Visualisierung Cayley-Hamilton-Produkte -% XXX \folie{5/chvisual.tex} -% XXX stone weierstrass incomplete -%\folie{5/stoneweierstrass.tex} -%\folie{5/swbeweis.tex} - -% XXX polynome auf dem spektrum -% XXX Motiviation für *-Operation -%\folie{5/normal.tex} -%\folie{5/normalbeispiel34.tex} - - -\section{Körpererweiterungen} -% XXX Was ist eine Körpererweiterung -%\folie{4/galois/erweiterung.tex} - -\section{Geometrische Anwendungen} -% XXX Geometrische Konstruktionen -%\folie{4/galois/konstruktion.tex} -% XXX Verdoppelung des Würfels -%\folie{4/galois/wuerfel.tex} -% XXX Dreiteilung des Winkels -%\folie{4/galois/winkeldreiteilung.tex} -% XXX Quadratur des Kreises -%\folie{4/galois/quadratur.tex} - -\section{Galois-Gruppe} -% XXX Übersetzung Körpererweiterungsstruktur in eine Gruppe -%\folie{4/galois/automorphismus.tex} - -\section{Lösbarkeit durch Radikale} -% XXX Wurzelformeln mit Radikalen -%\folie{4/galois/radikale.tex} -% XXX Auflösbarkeit einer Gruppe -%\folie{4/galois/aufloesbarkeit.tex} -% XXX S_n ist nicht auflösbar -\folie{4/galois/sn.tex} +\section{Matrizen-Gruppen} +% Was sind Symmetrien +\folie{7/symmetrien.tex} +% Algebraische Bedingungen für Matrixgruppen +\folie{7/algebraisch.tex} +% Parametrisierung, Beispiel SO(3) +\folie{7/parameter.tex} +% Mannigfaltigkeiten +\folie{7/mannigfaltigkeit.tex} +% Weitere Beispiele +% SL_2(R) +%\folie{7/sl2.tex} +%\folie{7/drehung.tex} +% Semidirekte Produkte SO(2) x R^2, R^+ x R +\folie{7/semi.tex} + +\section{Ableitungen} +% Kurven in einer Gruppe +\folie{7/kurven.tex} +% Einparameter-Gruppen +\folie{7/einparameter.tex} +% Ableitung einer Einparameter-Gruppe +\folie{7/ableitung.tex} +% Lie-Algebra +\folie{7/liealgebra.tex} + +\section{Exponentialabbildung} +% Differentialgleichung für die Exponentialabbildung +\folie{7/dg.tex} +% Exponentialabbilung +%\folie{7/exponential.tex} -- cgit v1.2.1 From 7242d266e3e348e40209ac51e703a0acb6d78ca3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 21:13:29 +0200 Subject: add rodriguez image --- vorlesungen/slides/7/images/Makefile | 12 +++ vorlesungen/slides/7/images/rodriguez.jpg | Bin 0 -> 168776 bytes vorlesungen/slides/7/images/rodriguez.png | Bin 0 -> 261798 bytes vorlesungen/slides/7/images/rodriguez.pov | 118 ++++++++++++++++++++++++++++++ vorlesungen/slides/7/parameter.tex | 17 ++++- 5 files changed, 144 insertions(+), 3 deletions(-) create mode 100644 vorlesungen/slides/7/images/Makefile create mode 100644 vorlesungen/slides/7/images/rodriguez.jpg create mode 100644 vorlesungen/slides/7/images/rodriguez.png create mode 100644 vorlesungen/slides/7/images/rodriguez.pov diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile new file mode 100644 index 0000000..1b9590d --- /dev/null +++ b/vorlesungen/slides/7/images/Makefile @@ -0,0 +1,12 @@ +# +# Makefile -- Illustrationen zu +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +#  +all: rodriguez.jpg + +rodriguez.png: rodriguez.pov + povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov + +rodriguez.jpg: rodriguez.png + convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg diff --git a/vorlesungen/slides/7/images/rodriguez.jpg b/vorlesungen/slides/7/images/rodriguez.jpg new file mode 100644 index 0000000..5c49700 Binary files /dev/null and b/vorlesungen/slides/7/images/rodriguez.jpg differ diff --git a/vorlesungen/slides/7/images/rodriguez.png b/vorlesungen/slides/7/images/rodriguez.png new file mode 100644 index 0000000..6d9e9e4 Binary files /dev/null and b/vorlesungen/slides/7/images/rodriguez.png differ diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov new file mode 100644 index 0000000..07aec19 --- /dev/null +++ b/vorlesungen/slides/7/images/rodriguez.pov @@ -0,0 +1,118 @@ +// +// rodriguez.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.020; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <8, 15, -50> + look_at <0.1, 0.475, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-4, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#declare K = vnormalize(<0.2,1,0.1>); +#declare X = vnormalize(<1.1,1,-1.2>); +#declare O = <0,0,0>; + +#declare r = vlength(vcross(K, X)) / vlength(K); + +#declare l = 1.0; + +arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) +arrow(< 0, 0, -l >, < 0, 0, l >, at, White) +arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) + +arrow(O, X, at, Red) +arrow(O, K, at, Blue) + +#macro punkt(H,phi) + ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K) +#end + +cone { vdot(K, X) * K, r, O, 0 + pigment { + color rgbt<0.6,0.6,0.6,0.5> + } + finish { + specular 0.9 + metallic + } +} + + +union { + #declare phistep = pi / 100; + #declare phi = 0; + #while (phi < 2 * pi - phistep/2) + sphere { punkt(K, phi), at/2 } + cylinder { + punkt(X, phi), + punkt(X, phi + phistep), + at/2 + } + #declare phi = phi + phistep; + #end + pigment { + color Orange + } + finish { + specular 0.9 + metallic + } +} + +arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow) +#declare Darkgreen = rgb<0,0.5,0>; +arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen) diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index ec129bb..b719207 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -4,6 +4,7 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} \begin{frame}[t] \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} @@ -41,17 +42,27 @@ $\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse \[ \vec{x} \mapsto +(\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}) \cos\omega -\vec{x} + (\vec{k}\times\vec{x})\sin\omega + -\vec{k}(\vec{k}\cdot\vec{x}) (1-\cos\omega) +\vec{k}(\vec{k}\cdot\vec{x}) \] -XXX TODO: Bild für Rodriguez Formel +\vspace{-40pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; +\node[color=red] at (1.6,-0.9) {$\vec{x}$}; +\node[color=blue] at (0.5,2) {$\vec{k}$}; +\node[color=darkgreen] at (-3,1.1) {$\vec{k}\times\vec{x}$}; +\node[color=yellow] at (2.2,-0.2) {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +\end{tikzpicture} +\end{center} \end{block} \end{column} \end{columns} +\vspace{-15pt} {\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine dreidimensionale Gruppe \end{frame} -- cgit v1.2.1 From e8fcbfb46f9d7badd043b9daca9659bbd45f84ee Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 22:56:28 +0200 Subject: new images --- vorlesungen/slides/7/Makefile.inc | 1 + vorlesungen/slides/7/chapter.tex | 1 + vorlesungen/slides/7/images/Makefile | 9 +- vorlesungen/slides/7/images/c/c01.jpg | Bin 0 -> 89652 bytes vorlesungen/slides/7/images/c/c02.jpg | Bin 0 -> 91977 bytes 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+++ b/vorlesungen/slides/7/Makefile.inc @@ -15,6 +15,7 @@ chapter5 = \ ../slides/7/einparameter.tex \ ../slides/7/ableitung.tex \ ../slides/7/liealgebra.tex \ + ../slides/7/kommutator.tex \ ../slides/7/dg.tex \ ../slides/7/chapter.tex diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 44d46a6..5878164 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -14,4 +14,5 @@ \folie{7/einparameter.tex} \folie{7/ableitung.tex} \folie{7/liealgebra.tex} +\folie{7/kommutator.tex} \folie{7/dg.tex} diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile index 1b9590d..cc67c8a 100644 --- a/vorlesungen/slides/7/images/Makefile +++ b/vorlesungen/slides/7/images/Makefile @@ -3,10 +3,17 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule #  -all: rodriguez.jpg +all: rodriguez.jpg rodriguez.png: rodriguez.pov povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov rodriguez.jpg: rodriguez.png convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg + +commutator: commutator.ini commutator.pov common.inc + povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini +jpg: + for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done + + diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg new file mode 100644 index 0000000..b2dbdb2 Binary files /dev/null and b/vorlesungen/slides/7/images/c/c01.jpg differ diff --git a/vorlesungen/slides/7/images/c/c02.jpg b/vorlesungen/slides/7/images/c/c02.jpg new file mode 100644 index 0000000..9b45ba3 Binary files /dev/null and b/vorlesungen/slides/7/images/c/c02.jpg differ diff --git a/vorlesungen/slides/7/images/c/c03.jpg b/vorlesungen/slides/7/images/c/c03.jpg new file mode 100644 index 0000000..882be40 Binary files /dev/null and b/vorlesungen/slides/7/images/c/c03.jpg differ diff --git a/vorlesungen/slides/7/images/c/c04.jpg b/vorlesungen/slides/7/images/c/c04.jpg new file mode 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index 0000000..0e27c9a --- /dev/null +++ b/vorlesungen/slides/7/images/common.inc @@ -0,0 +1,70 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.025; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <18, 15, -50> + look_at <0.0, 0.5, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-40, 30, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#declare l = 1.2; + +arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) +arrow(< 0, 0, -l >, < 0, 0, l >, at, White) +arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) + diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini new file mode 100644 index 0000000..8c2211e --- /dev/null +++ b/vorlesungen/slides/7/images/commutator.ini @@ -0,0 +1,8 @@ +Input_File_Name=commutator.pov +Initial_Frame=1 +Final_Frame=60 +Initial_Clock=1 +Final_Clock=60 +Cyclic_Animation=off +Pause_when_Done=off + diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m new file mode 100644 index 0000000..5a448db --- /dev/null +++ b/vorlesungen/slides/7/images/commutator.m @@ -0,0 +1,111 @@ +# +# commutator.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +X = [ + 0, 0, 0; + 0, 0, -1; + 0, 1, 0 +]; + +Y = [ + 0, 0, 1; + 0, 0, 0; + -1, 0, 0 +]; + +Z = [ + 0, -1, 0; + 1, 0, 0; + 0, 0, 0 +]; + +function retval = Dx(alpha) + retval = [ + 1, 0, 0 ; + 0, cos(alpha), -sin(alpha); + 0, sin(alpha), cos(alpha) + ]; +end + +function retval = Dy(beta) + retval = [ + cos(beta), 0, sin(beta); + 0, 1, 0 ; + -sin(beta), 0, cos(beta) + ]; +end + +t = 0.9; +P = Dx(t) * Dy(t) +Q = Dy(t) * Dx(t) +P - Q +(P - Q) * [0;0;1] + +function retval = kurven(filename, t) + retval = -1; + N = 20; + fn = fopen(filename, "w"); + fprintf(fn, "//\n"); + fprintf(fn, "// %s\n", filename); + fprintf(fn, "//\n"); + fprintf(fn, "#macro XYkurve()\n"); + for i = (0:N-1) + v1 = Dx(t * i / N) * [0;0;1]; + v2 = Dx(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + for i = (0:N-1) + v1 = Dx(t) * Dy(t * i / N) * [0;0;1]; + v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#end\n"); + fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#macro YXkurve()\n"); + for i = (0:N-1) + v1 = Dy(t * i / N) * [0;0;1]; + v2 = Dy(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + for i = (0:N-1) + v1 = Dy(t) * Dx(t * i / N) * [0;0;1]; + v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#end\n"); + fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n", + v2(1,1), v2(3,1), v2(2,1)); + + fclose(fn); + retval = 0; +end + +function retval = kurve(i) + n = pi / 180; + filename = sprintf("f/%04d.inc", i); + kurven(filename, n * i); +end + +for i = (1:60) + kurve(i); +end diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov new file mode 100644 index 0000000..9ae11b9 --- /dev/null +++ b/vorlesungen/slides/7/images/commutator.pov @@ -0,0 +1,59 @@ +// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +sphere { O, 0.99 + pigment { + color rgbt<1,1,1,0.5> + } + finish { + specular 0.9 + metallic + } +} + +#declare filename = concat("f/", str(clock, -4, 0), ".inc"); + +#include filename + +#declare n1 = vcross(<0,1,0>, finalXY); +#declare n2 = vcross(<0,1,0>, finalYX); + +intersection { + sphere { O, 1 } + plane { -n1, 0 } + plane { n2, 0 } + pigment { + color rgb<0,0.4,0.1> + } + finish { + specular 0.9 + metallic + } +} + +union { + XYkurve() + pigment { + color Red + } + finish { + specular 0.9 + metallic + } +} + +union { + YXkurve() + pigment { + color Blue + } + finish { + specular 0.9 + metallic + } +} + diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex new file mode 100644 index 0000000..f01debe --- /dev/null +++ b/vorlesungen/slides/7/kommutator.tex @@ -0,0 +1,139 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kommutator in $\operatorname{SO}(3)$} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] 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+\includegraphics[width=\textwidth]{../slides/7/images/c/c51.jpg}};} +\only<52>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c52.jpg}};} +\only<53>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c53.jpg}};} +\only<54>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c54.jpg}};} +\only<55>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c55.jpg}};} +\only<56>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c56.jpg}};} +\only<57>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c57.jpg}};} +\only<58>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c58.jpg}};} +\only<59>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c59.jpg}};} +\only<60>{\node at (0,0) { +\includegraphics[width=\textwidth]{../slides/7/images/c/c60.jpg}};} +}{} +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex index 3480460..a65b4f6 100644 --- a/vorlesungen/slides/7/sl2.tex +++ b/vorlesungen/slides/7/sl2.tex @@ -50,6 +50,7 @@ c&d \foreach \y in {-2,...,2}{ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); } + \ifthenelse{\boolean{presentation}}{ \foreach \d in {4,...,10}{ \only<\d>{ \pgfmathparse{1+(\d-4)/10} @@ -66,6 +67,7 @@ c&d } } } + }{} \uncover<11->{ \xdef\t{1.6} \fill[color=red!40,opacity=0.5] @@ -105,6 +107,7 @@ c&d \foreach \y in {-2,...,2}{ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); } + \ifthenelse{\boolean{presentation}}{ \foreach \d in {11,...,17}{ \only<\d>{ \pgfmathparse{(\d-11)/10} @@ -128,6 +131,7 @@ c&d } } } + }{} \uncover<18->{ \xdef\t{0.6} \fill[color=red!40,opacity=0.5] @@ -174,6 +178,7 @@ c&d \foreach \y in {-2,...,2}{ \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); } + \ifthenelse{\boolean{presentation}}{ \foreach \d in {18,...,24}{ \only<\d>{ \pgfmathparse{(\d-18)/10} @@ -197,6 +202,27 @@ c&d } } } + }{} + \uncover<25->{ + \xdef\t{0.6} + \fill[color=red!40,opacity=0.5] + (-1,{\t*(-1)-1}) + -- + (1,{\t*1-1}) + -- + (1,{\t*1+1}) + -- + (-1,{\t*(-1)+1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); + } + } \end{scope} \draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; \draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 7062509..ee6829e 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -15,8 +15,8 @@ \folie{7/mannigfaltigkeit.tex} % Weitere Beispiele % SL_2(R) -%\folie{7/sl2.tex} -%\folie{7/drehung.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} % Semidirekte Produkte SO(2) x R^2, R^+ x R \folie{7/semi.tex} @@ -29,10 +29,10 @@ \folie{7/ableitung.tex} % Lie-Algebra \folie{7/liealgebra.tex} +% Kommutator +\folie{7/kommutator.tex} \section{Exponentialabbildung} % Differentialgleichung für die Exponentialabbildung \folie{7/dg.tex} -% Exponentialabbilung -%\folie{7/exponential.tex} -- cgit v1.2.1 From 4d96632a4cb5dce985c92bc5daebe272dcbc1410 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 23:01:39 +0200 Subject: typos --- vorlesungen/slides/7/kommutator.tex | 2 +- vorlesungen/slides/7/liealgebra.tex | 1 - 2 files changed, 1 insertion(+), 2 deletions(-) diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index f01debe..f9004df 100644 --- a/vorlesungen/slides/7/kommutator.tex +++ b/vorlesungen/slides/7/kommutator.tex @@ -130,9 +130,9 @@ \includegraphics[width=\textwidth]{../slides/7/images/c/c58.jpg}};} \only<59>{\node at (0,0) { \includegraphics[width=\textwidth]{../slides/7/images/c/c59.jpg}};} +}{} \only<60>{\node at (0,0) { \includegraphics[width=\textwidth]{../slides/7/images/c/c60.jpg}};} -}{} \end{tikzpicture} \end{center} \end{frame} diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 16a7aa0..892216e 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -8,7 +8,6 @@ \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} \frametitle{Lie-Algebra} -\vspace{-20pt} \begin{block}{Vektorraum} Tangentialvektoren im Punkt $I$: \begin{center} -- cgit v1.2.1 From ac447649b63709b23842c821bba91134d545b9dc Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sat, 10 Apr 2021 23:05:32 +0200 Subject: add presentation 7 --- vorlesungen/07_lie/Makefile | 33 +++++++++++++++++++++++++++++++++ vorlesungen/07_lie/MathSem-07-lie.tex | 14 ++++++++++++++ vorlesungen/07_lie/common.tex | 16 ++++++++++++++++ vorlesungen/07_lie/lie-handout.tex | 11 +++++++++++ vorlesungen/07_lie/slides.tex | 23 +++++++++++++++++++++++ 5 files changed, 97 insertions(+) create mode 100644 vorlesungen/07_lie/Makefile create mode 100644 vorlesungen/07_lie/MathSem-07-lie.tex create mode 100644 vorlesungen/07_lie/common.tex create mode 100644 vorlesungen/07_lie/lie-handout.tex create mode 100644 vorlesungen/07_lie/slides.tex diff --git a/vorlesungen/07_lie/Makefile b/vorlesungen/07_lie/Makefile new file mode 100644 index 0000000..ceb0e44 --- /dev/null +++ b/vorlesungen/07_lie/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- lie +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: lie-handout.pdf MathSem-07-lie.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-07-lie.pdf: MathSem-07-lie.tex $(SOURCES) + pdflatex MathSem-07-lie.tex + +lie-handout.pdf: lie-handout.tex $(SOURCES) + pdflatex lie-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSem-07-lie.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-07-lie.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-07-lie.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-07-lie.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/07_lie/MathSem-07-lie.tex b/vorlesungen/07_lie/MathSem-07-lie.tex new file mode 100644 index 0000000..4eeca41 --- /dev/null +++ b/vorlesungen/07_lie/MathSem-07-lie.tex @@ -0,0 +1,14 @@ +% +% MathSem-07-lie.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/07_lie/common.tex b/vorlesungen/07_lie/common.tex new file mode 100644 index 0000000..8472b93 --- /dev/null +++ b/vorlesungen/07_lie/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Lie]{Lie-Gruppen und Lie-Algebren} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/07_lie/lie-handout.tex b/vorlesungen/07_lie/lie-handout.tex new file mode 100644 index 0000000..dbdb386 --- /dev/null +++ b/vorlesungen/07_lie/lie-handout.tex @@ -0,0 +1,11 @@ +% +% lie-handout.tex -- Handout XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/07_lie/slides.tex b/vorlesungen/07_lie/slides.tex new file mode 100644 index 0000000..db4a3fc --- /dev/null +++ b/vorlesungen/07_lie/slides.tex @@ -0,0 +1,23 @@ +% +% slides.tex -- Vorlesung über Lie-Theorie +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Matrizen-Gruppen} +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\folie{7/semi.tex} + +\section{Ableitungen} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/kommutator.tex} + +\section{Exponentialabbildung} +\folie{7/dg.tex} -- cgit v1.2.1 From 9248e1b412136d41d2ad7b302f682bfcc05e5174 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 09:07:21 +0200 Subject: phasen --- vorlesungen/slides/7/algebraisch.tex | 40 +++++++++++++--------- vorlesungen/slides/7/symmetrien.tex | 64 +++++++++++++++++++++--------------- vorlesungen/slides/test.tex | 24 +++++++------- 3 files changed, 75 insertions(+), 53 deletions(-) diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex index 5b33566..31d209a 100644 --- a/vorlesungen/slides/7/algebraisch.tex +++ b/vorlesungen/slides/7/algebraisch.tex @@ -19,23 +19,25 @@ Längenmessung mit Skalarprodukt \langle \vec{v},\vec{v}\rangle = \vec{v}\cdot \vec{v} -= -\vec{v}^t\vec{v} +\uncover<2->{= +\vec{v}^t\vec{v}} \end{align*} \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<3->{% \begin{block}{Flächeninhalt/Volumen} $n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ \\ Volumen des Parallelepipeds: $\det V$ -\end{block} +\end{block}} \end{column} \end{columns} % \vspace{-7pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<4->{% \begin{block}{Längenerhaltende Transformationen} $A\in\operatorname{GL}_n(\mathbb{R})$ \begin{align*} @@ -44,38 +46,45 @@ $A\in\operatorname{GL}_n(\mathbb{R})$ (A\vec{x}) \cdot (A\vec{y}) -= +\uncover<5->{= (A\vec{x})^t -(A\vec{y}) +(A\vec{y})} \\ +\uncover<6->{ \vec{x}^tI\vec{y} &= -\vec{x}^tA^tA\vec{y} -\Rightarrow I=A^tA +\vec{x}^tA^tA\vec{y}} +\uncover<7->{ +\Rightarrow I=A^tA} \end{align*} -Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$ -\end{block} +\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<9->{% \begin{block}{Volumenerhaltende Transformationen} $A\in\operatorname{GL}_n(\mathbb{R})$ \begin{align*} \det(V) &= \det(AV) -= -\det(A)\det(V) +\uncover<10->{= +\det(A)\det(V)} \\ -1&=\det(A) +\uncover<11->{ +1&=\det(A)} \end{align*} +\uncover<10->{ (Produktsatz für Determinante) -\end{block} +} +\end{block}} \end{column} \end{columns} % \vspace{-3pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<12->{% \begin{block}{Orthogonale Matrizen} Längentreue Abbildungen = orthogonale Matrizen: \[ @@ -87,9 +96,10 @@ A \in \operatorname{GL}_n(\mathbb{R}) A^tA=I \} \] -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<13->{% \begin{block}{``Spezielle'' Matrizen} Volumen-/Orientierungserhaltende Transformationen: \[ @@ -97,7 +107,7 @@ Volumen-/Orientierungserhaltende Transformationen: = \{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} \] -\end{block} +\end{block}} \end{column} \end{columns} diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex index 79f9ef7..35d62d8 100644 --- a/vorlesungen/slides/7/symmetrien.tex +++ b/vorlesungen/slides/7/symmetrien.tex @@ -14,7 +14,7 @@ \begin{column}{0.48\textwidth} \begin{block}{Diskrete Symmetrien} \begin{itemize} -\item +\item<2-> Ebenen-Spiegelung: \[ {\tiny @@ -22,12 +22,13 @@ Ebenen-Spiegelung: } \mapsto {\tiny -\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}, +\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*} } -\; +\uncover<4->{\!,\; \vec{x} \mapsto \vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} +} \] \vspace{-10pt} \begin{center} @@ -41,34 +42,39 @@ Ebenen-Spiegelung: \coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); \coordinate (N) at (\a:2); \coordinate (D) at (\a:{\r*cos(\b-\a)}); +\uncover<3->{ \clip (-2.5,-0.45) rectangle (2.5,1.95); -\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) -- cycle; -\draw[->,color=darkgreen] (O) -- (N); -\node[color=darkgreen] at (N) [above] {$\vec{n}$}; + \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) + -- cycle; + \draw[->,color=darkgreen] (O) -- (N); + \node[color=darkgreen] at (N) [above] {$\vec{n}$}; -\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) -- cycle; -\fill[color=red] (O) circle[radius=0.06]; -\draw[color=red] ({\a-90}:2) -- ({\a+90}:2); -\fill[color=blue] (C) circle[radius=0.06]; -\draw[color=blue,line width=0.1pt] (A) -- (D); -\node[color=darkgreen] at (D) [below,rotate=\a] {$(\vec{n}\cdot\vec{x})\vec{n}$}; -\draw[color=blue,line width=0.5pt] (A)--(B); + \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) + -- cycle; + \fill[color=red] (O) circle[radius=0.06]; + \draw[color=red] ({\a-90}:2) -- ({\a+90}:2); + \fill[color=blue] (C) circle[radius=0.06]; + \draw[color=blue,line width=0.1pt] (A) -- (D); + \node[color=darkgreen] at (D) [below,rotate=\a] + {$(\vec{n}\cdot\vec{x})\vec{n}$}; + \draw[color=blue,line width=0.5pt] (A)--(B); -\node[color=blue] at (A) [above right] {$\vec{x}$}; -\node[color=blue] at (B) [above left] {$\vec{x}'$}; + \node[color=blue] at (A) [above right] {$\vec{x}$}; + \node[color=blue] at (B) [above left] {$\vec{x}'$}; -\node[color=red] at (O) [below left] {$O$}; + \node[color=red] at (O) [below left] {$O$}; -\draw[->,color=blue,shorten <= 0.06cm] (O) -- (A); -\draw[->,color=blue,shorten <= 0.06cm] (O) -- (B); + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A); + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B); +} \end{tikzpicture} \end{center} \vspace{-5pt} $\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ -\item +\item<5-> Punkt-Spiegelung: \[ {\tiny @@ -84,13 +90,15 @@ Punkt-Spiegelung: \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<6->{% \begin{block}{Kontinuierliche Symmetrien} \begin{itemize} -\item Translation: +\item<7-> Translation: \( \vec{x} \mapsto \vec{x} + \vec{t} \) -\item Drehung: +\item<8-> Drehung: +\vspace{-3pt} \begin{center} \begin{tikzpicture}[>=latex,thick] \def\a{25} @@ -103,14 +111,15 @@ Punkt-Spiegelung: \fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; \node at ({0.5*\a}:1) {$\alpha$}; \node at ({90+0.5*\a}:1) {$\alpha$}; -\draw[->,color=blue] (O) -- (\a:2); -\draw[->,color=darkgreen] (O) -- ({90+\a}:2); +\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2); +\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2); \end{scope} \draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; \draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; \end{tikzpicture} \end{center} \[ +\uncover<9->{% \begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix} @@ -118,15 +127,18 @@ Punkt-Spiegelung: {\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} \end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} +} \] \end{itemize} -\end{block} +\end{block}} \vspace{-10pt} +\uncover<10->{% \begin{block}{Definition} Längen/Winkel bleiben erhalten \\ -$\Rightarrow$ $\exists$ Erhaltungsgrösse -\end{block} +\uncover<11->{% +$\Rightarrow$ $\exists$ Erhaltungsgrösse} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index ee6829e..45b16d1 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -6,33 +6,33 @@ \section{Matrizen-Gruppen} % Was sind Symmetrien -\folie{7/symmetrien.tex} +%\folie{7/symmetrien.tex} % Algebraische Bedingungen für Matrixgruppen \folie{7/algebraisch.tex} % Parametrisierung, Beispiel SO(3) -\folie{7/parameter.tex} +%\folie{7/parameter.tex} % Mannigfaltigkeiten -\folie{7/mannigfaltigkeit.tex} +%\folie{7/mannigfaltigkeit.tex} % Weitere Beispiele % SL_2(R) -\folie{7/sl2.tex} -\folie{7/drehung.tex} +%\folie{7/sl2.tex} +%\folie{7/drehung.tex} % Semidirekte Produkte SO(2) x R^2, R^+ x R -\folie{7/semi.tex} +%\folie{7/semi.tex} \section{Ableitungen} % Kurven in einer Gruppe -\folie{7/kurven.tex} +%\folie{7/kurven.tex} % Einparameter-Gruppen -\folie{7/einparameter.tex} +%\folie{7/einparameter.tex} % Ableitung einer Einparameter-Gruppe -\folie{7/ableitung.tex} +%\folie{7/ableitung.tex} % Lie-Algebra -\folie{7/liealgebra.tex} +%\folie{7/liealgebra.tex} % Kommutator -\folie{7/kommutator.tex} +%\folie{7/kommutator.tex} \section{Exponentialabbildung} % Differentialgleichung für die Exponentialabbildung -\folie{7/dg.tex} +%\folie{7/dg.tex} -- cgit v1.2.1 From 15881729aa3f1293d546a1692a02094ed3f24e2b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 10:30:05 +0200 Subject: phases --- vorlesungen/slides/7/ableitung.tex | 40 ++++---- vorlesungen/slides/7/dg.tex | 36 +++++--- vorlesungen/slides/7/drehung.tex | 53 +++++++---- vorlesungen/slides/7/einparameter.tex | 18 ++-- vorlesungen/slides/7/kommutator.tex | 147 ++++++++++++++++++------------ vorlesungen/slides/7/kurven.tex | 31 +++++-- vorlesungen/slides/7/liealgebra.tex | 27 +++++- vorlesungen/slides/7/mannigfaltigkeit.tex | 12 ++- vorlesungen/slides/7/parameter.tex | 51 +++++++++-- vorlesungen/slides/7/semi.tex | 34 ++++--- vorlesungen/slides/test.tex | 4 +- 11 files changed, 301 insertions(+), 152 deletions(-) diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex index b061b9a..12f9084 100644 --- a/vorlesungen/slides/7/ableitung.tex +++ b/vorlesungen/slides/7/ableitung.tex @@ -12,49 +12,55 @@ \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Ableitung in $\operatorname{O}(n)$} +\uncover<2->{% $s \mapsto A(s)\in\operatorname{O}(n)$ +} \begin{align*} -I +\uncover<3->{I &= -A(s)^tA(s) +A(s)^tA(s)} \\ -0 +\uncover<4->{0 = \frac{d}{ds} I &= -\frac{d}{ds} (A(s)^t A(s)) +\frac{d}{ds} (A(s)^t A(s))} \\ -&= -\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s) -\intertext{An der Stelle $s=0$, d.~h.~$A(0)=I$} -0 +&\uncover<5->{= +\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)} +\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}} +\uncover<7->{0 &= \dot{A}(0)^t + -\dot{A}(0) +\dot{A}(0)} \\ -\Leftrightarrow +\uncover<8->{\Leftrightarrow \qquad -\dot{A}(0)^t &= -\dot{A}(0) +\dot{A}(0)^t &= -\dot{A}(0)} \end{align*} +\uncover<9->{% ``Tangentialvektoren'' sind antisymmetrische Matrizen +} \end{block} \end{column} \begin{column}{0.48\textwidth} \begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} +\uncover<2->{% $s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ +} \begin{align*} -1 &= \det A(t) +\uncover<3->{1 &= \det A(t)} \\ -0 +\uncover<10->{0 = \frac{d}{dt}1 &= -\frac{d}{dt} \det A(t) -\intertext{mit dem Entwicklungssatz kann man nachrechnen:} -0&=\operatorname{Spur}\dot{A}(0) +\frac{d}{dt} \det A(t)} +\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}} +\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)} \end{align*} -``Tangentialvektoren'' sind spurlose Matrizen +\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} \end{block} \end{column} \end{columns} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 36b1ade..a99cebb 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -15,29 +15,35 @@ Ableitung von $\gamma(t)$ an der Stelle $t$: \begin{align*} \dot{\gamma}(t) -&= +&\uncover<2->{= \frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} +} \\ -&= +&\uncover<3->{= \frac{d}{ds} \gamma(t+s) \bigg|_{s=0} +} \\ -&= +&\uncover<4->{= \frac{d}{ds} \gamma(t)\gamma(s) \bigg|_{s=0} +} \\ -&= +&\uncover<5->{= \gamma(t) \frac{d}{ds} \gamma(s) \bigg|_{s=0} -= +} +\uncover<6->{= \gamma(t) \dot{\gamma}(0) +} \end{align*} \end{block} \vspace{-10pt} +\uncover<7->{% \begin{block}{Differentialgleichung} \vspace{-10pt} \[ @@ -47,33 +53,39 @@ Ableitung von $\gamma(t)$ an der Stelle $t$: \quad A=\dot{\gamma}(0)\in LG \] -\end{block} +\end{block}} \end{column} \begin{column}{0.50\textwidth} +\uncover<8->{% \begin{block}{Lösung} Exponentialfunktion \[ \exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k \] -\end{block} +\end{block}} \vspace{-5pt} +\uncover<9->{% \begin{block}{Kontrolle: Tangentialvektor berechnen} \vspace{-10pt} \begin{align*} \frac{d}{dt}e^{At} -&= +&\uncover<10->{= \sum_{k=1}^\infty A^k \frac{d}{dt} t^{k}{k!} +} \\ -&= +&\uncover<11->{= \sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A +} \\ -&= +&\uncover<12->{= \sum_{k=0} A^k\frac{t^k}{k!} A -= +} +\uncover<13->{= e^{At} A +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index ae0dbe3..7744e99 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -13,12 +13,20 @@ \begin{columns}[t,onlytextwidth] \begin{column}{0.38\textwidth} \begin{block}{Drehung} +{\color{blue}Längen}, {\color<2->{blue}Winkel}, +{\color<2->{darkgreen}Orientierung} +erhalten +\uncover<2->{ \[ \operatorname{SO}(2) = -\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2) -\] +{\color{blue}\operatorname{O}(2)} +\cap +{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})} +\]} +\vspace{-20pt} \end{block} +\uncover<3->{% \begin{block}{Zusammensetzung} Eine Drehung muss als Zusammensetzung geschrieben werden können: \[ @@ -31,7 +39,9 @@ D_{\alpha} = DST \] -\end{block} +\end{block}} +\vspace{-10pt} +\uncover<12->{% \begin{block}{Beispiel} \vspace{-12pt} \[ @@ -43,9 +53,10 @@ D_{60^\circ} \begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} } \] -\end{block} +\end{block}} \end{column} \begin{column}{0.58\textwidth} +\uncover<4->{% \begin{block}{Ansatz} \vspace{-12pt} \begin{align*} @@ -64,7 +75,7 @@ c^{-1}&0\\ t&1 \end{pmatrix} \\ -&= +&\uncover<5->{= \begin{pmatrix} c^{-1}&0\\ 0 &c @@ -73,40 +84,48 @@ c^{-1}&0\\ -st&-s\\ t& 1 \end{pmatrix} +} \\ -&= +&\uncover<6->{= \begin{pmatrix} --stc^{-1}&{\color{darkgreen}sc^{-1}}\\ -{\color{blue}ct}&{\color{red}c} -\end{pmatrix} -= +{\color<11->{orange}-stc^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<9->{blue}ct}&{\color<8->{red}c} +\end{pmatrix}} +\uncover<7->{= \begin{pmatrix} -\cos\alpha & {\color{darkgreen}- \sin\alpha} \\ -{\color{blue}\sin\alpha} & \phantom{-} {\color{red}\cos\alpha} -\end{pmatrix} +{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\ +{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha} +\end{pmatrix}} \end{align*} -\end{block} +\end{block}} \vspace{-10pt} +\uncover<7->{% \begin{block}{Koeffizientenvergleich} \vspace{-15pt} \begin{align*} +\uncover<8->{ {\color{red} c} &= -{\color{red}\cos\alpha } +{\color{red}\cos\alpha }} && & +\uncover<9->{ {\color{blue} -t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\ +t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ +\uncover<10->{ {\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} & &\Rightarrow& {\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha +} \\ +\uncover<11->{ {\color{orange} -stc^{-t}} &= \rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex index 52924bf..5171085 100644 --- a/vorlesungen/slides/7/einparameter.tex +++ b/vorlesungen/slides/7/einparameter.tex @@ -7,17 +7,20 @@ \begin{frame}[t] \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} -\frametitle{Einparameter Untergruppen} +\frametitle{Einparameter-Untergruppen} \vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} \begin{block}{Definition} Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$, -die gleichzeitig eine Untergruppe von $G$ ist mit +die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit} \[ +\uncover<3->{ \gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R} +} \] \end{block} +\uncover<4->{% \begin{block}{Drehungen} Drehmatrizen bilden Einparameter- Untergruppen \begin{align*} @@ -33,9 +36,10 @@ D_{x,t}D_{x,s} &= D_{x,t+s} \end{align*} -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<5->{% \begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} \vspace{-12pt} \[ @@ -53,8 +57,9 @@ D_{x,t+s} 0&1 \end{pmatrix} \] -\end{block} +\end{block}} \vspace{-12pt} +\uncover<6->{% \begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} \vspace{-12pt} \[ @@ -69,8 +74,9 @@ e^t&0\\0&e^{-t} e^{t+s}&0\\0&e^{-(t+s)} \end{pmatrix} \] -\end{block} +\end{block}} \vspace{-12pt} +\uncover<7->{% \begin{block}{Gemischt} \vspace{-12pt} \begin{gather*} @@ -80,7 +86,7 @@ A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t \begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 =I \end{gather*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index f9004df..84bf034 100644 --- a/vorlesungen/slides/7/kommutator.tex +++ b/vorlesungen/slides/7/kommutator.tex @@ -4,6 +4,7 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} \begin{frame}[t] \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} @@ -11,128 +12,154 @@ \vspace{-20pt} \begin{center} \begin{tikzpicture}[>=latex,thick] +\def\t{14.0cm} \ifthenelse{\boolean{presentation}}{ \only<1>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c01.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};} \only<2>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c02.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};} 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-\includegraphics[width=\textwidth]{../slides/7/images/c/c51.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};} \only<52>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c52.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};} \only<53>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c53.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};} \only<54>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c54.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};} \only<55>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c55.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};} \only<56>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c56.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};} \only<57>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c57.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};} \only<58>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c58.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};} \only<59>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c59.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};} }{} \only<60>{\node at (0,0) { -\includegraphics[width=\textwidth]{../slides/7/images/c/c60.jpg}};} +\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};} +\coordinate (A) at (-0.3,3); +\coordinate (B) at (-1.1,2); +\coordinate (C) at (-2.1,-1.2); +\draw[->,color=red,line width=1.4pt] + (A) + to[out=-143,in=60] + (B) + to[out=-120,in=80] + (C); +%\fill[color=red] (B) circle[radius=0.08]; +\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$}; +\coordinate (D) at (0.3,3.2); +\coordinate (E) at (1.8,2.8); +\coordinate (F) at (5.2,-0.3); +\draw[->,color=blue,line width=1.4pt] + (D) + to[out=-10,in=157] + (E) + to[out=-23,in=120] + (F); +\fill[color=blue] (E) circle[radius=0.08]; +\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$}; +\draw[->,color=darkgreen,line width=1.4pt] + (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6); +\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$}; \end{tikzpicture} \end{center} \end{frame} diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex index 196fa2a..e0690eb 100644 --- a/vorlesungen/slides/7/kurven.tex +++ b/vorlesungen/slides/7/kurven.tex @@ -20,6 +20,7 @@ Kurve in $\mathbb{R}^n$: I=[a,b] \to \mathbb{R}^n : t\mapsto \gamma(t) +\uncover<2->{ = \begin{pmatrix} x_1(t)\\ @@ -27,6 +28,7 @@ x_2(t)\\ \vdots\\ x_n(t) \end{pmatrix} +} \] \vspace{-15pt} \begin{center} @@ -42,16 +44,21 @@ x_n(t) \fill[color=red] (C) circle[radius=0.06]; \node[color=red] at (C) [left] {$\gamma(t)$}; -\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); -\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; +\uncover<4->{ + \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); + \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; +} -\draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; -\draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; +\uncover<2->{ + \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; + \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; +} \end{tikzpicture} \end{center} \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<4->{% \begin{block}{Tangenten} Ableitung \[ @@ -66,6 +73,7 @@ Ableitung \dot{x}_n(t) \end{pmatrix} \] +\uncover<5->{% Lineare Approximation: \[ \gamma(t+h) @@ -75,10 +83,21 @@ Lineare Approximation: \dot{\gamma}(t) \cdot h + o(h) -\] +\]}% +\vspace{-10pt} +\begin{itemize} +\item<6-> Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ in $\mathbb{R}^n$ liegen -\end{block} +\item<7-> +Gilt auch für +\[ +\operatorname{GL}_n(\mathbb{R}) +\uncover<8->{\subset M_n(\mathbb{R})} +\uncover<9->{ = \mathbb{R}^{n\times n}} +\] +\end{itemize} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 892216e..574467b 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -8,50 +8,64 @@ \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} \frametitle{Lie-Algebra} +\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}} \begin{block}{Vektorraum} Tangentialvektoren im Punkt $I$: \begin{center} \begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} \text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ \hline +\uncover<2->{ \operatorname{GL}_n(\mathbb{R}) & beliebige Matrizen & M_n(\mathbb{R}) +} \\ +\uncover<3->{ \operatorname{O(n)} & antisymmetrische Matrizen & \operatorname{o}(n) +} \\ +\uncover<4->{ \operatorname{SL}_n(\mathbb{R}) & spurlose Matrizen & \operatorname{sl}_2(\mathbb{R}) +} \\ +\uncover<5->{ \operatorname{U(n)} & antihermitesche Matrizen & \operatorname{u}(n) +} \\ +\uncover<6->{ \operatorname{SU(n)} & spurlose, antihermitesche Matrizen & \operatorname{su}(n) +} \end{tabular} \end{center} \end{block} \vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.40\textwidth} +\uncover<7->{% \begin{block}{Lie-Klammer} Kommutator: $[A,B] = AB-BA$ -\end{block} +\end{block}} +\uncover<8->{% \begin{block}{Nachprüfen} $[A,B]\in LG$ für $A,B\in LG$ -\end{block} +\end{block}} \end{column} \begin{column}{0.56\textwidth} +\uncover<9->{% \begin{block}{Algebraische Eigenschaften} \begin{itemize} -\item antisymmetrisch: $[A,B]=-[B,A]$ -\item Jacobi-Identität +\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ +\item<11-> Jacobi-Identität \[ [A,[B,C]]+ [B,[C,A]]+ @@ -59,9 +73,12 @@ für $A,B\in LG$ = 0 \] \end{itemize} +\vspace{-13pt} +\uncover<12->{% {\usebeamercolor[fg]{title} Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ -\end{block} +} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex index 7809ea5..077dc9d 100644 --- a/vorlesungen/slides/7/mannigfaltigkeit.tex +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -18,8 +18,8 @@ \begin{column}{0.48\textwidth} \begin{block}{Definition} \begin{itemize} -\item Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ -\item differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet +\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ +\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet \[ \varphi_\beta\circ\varphi_\alpha^{-1} \colon @@ -27,17 +27,19 @@ \to \varphi_\beta(U_\alpha\cap U_\beta) \] -\item Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken +\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken \end{itemize} \end{block} \vspace{-7pt} +\uncover<5->{% \begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$} Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus -\end{block} +\end{block}} \vspace{-3pt} +\uncover<6->{% \begin{block}{Lie-Gruppe} Gruppe und Mannigfaltigkeit -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index b719207..1e8549c 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -5,6 +5,7 @@ % \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkyellow}{rgb}{1,0.8,0} \begin{frame}[t] \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} @@ -13,53 +14,85 @@ \begin{columns}[t,onlytextwidth] \begin{column}{0.4\textwidth} \begin{block}{Drehung um Achsen} +\vspace{-12pt} \begin{align*} +\uncover<2->{ D_{x,\alpha} &= \begin{pmatrix} 1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha \end{pmatrix} +} \\ +\uncover<3->{ D_{y,\beta} &= \begin{pmatrix} \cos\beta&0&-\sin\beta\\0&1&0\\\sin\beta&0&\cos\beta \end{pmatrix} +} \\ +\uncover<4->{ D_{z,\gamma} &= \begin{pmatrix} \cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 \end{pmatrix} +} +\intertext{\uncover<5->{beliebige Drehung:}} +\uncover<5->{ +D +&= +D_{x,\alpha} +D_{y,\beta} +D_{z,\gamma} +} \end{align*} \end{block} \end{column} \begin{column}{0.56\textwidth} -\begin{block}{Drehung um $\vec{\omega}$} +\uncover<6->{% +\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional} +\uncover<7->{% $\omega=|\vec{\omega}|=\mathstrut$Drehwinkel +} \\ +\uncover<8->{% $\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse +} \[ -\vec{x} +{\color{red}\vec{x}} \mapsto -(\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}) +\uncover<10->{ +({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) \cos\omega + -(\vec{k}\times\vec{x})\sin\omega +} +\uncover<11->{ +({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega + -\vec{k}(\vec{k}\cdot\vec{x}) +} +\uncover<9->{ +{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) +} \] \vspace{-40pt} \begin{center} \begin{tikzpicture}[>=latex,thick] -\node at (0,0) {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; +\node at (0,0) + {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; \node[color=red] at (1.6,-0.9) {$\vec{x}$}; \node[color=blue] at (0.5,2) {$\vec{k}$}; -\node[color=darkgreen] at (-3,1.1) {$\vec{k}\times\vec{x}$}; -\node[color=yellow] at (2.2,-0.2) {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +\uncover<11->{ + \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; +} +\uncover<10->{ + \node[color=yellow] at (2.2,-0.2) + {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +} \end{tikzpicture} \end{center} -\end{block} +\end{block}} \end{column} \end{columns} \vspace{-15pt} diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 46f6d03..66b8d27 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -21,6 +21,7 @@ x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<2->{% \begin{block}{Drehung und Verschiebung} Gruppe $G= @@ -32,51 +33,57 @@ Wirkung auf $\mathbb{R}^2$: \[ \vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} \] -\end{block} +\end{block}} \end{column} \end{columns} \vspace{-15pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<3->{% \begin{block}{Verknüpfung} \vspace{-15pt} \begin{align*} (e^{s_1},t_1)(e^{s_2},t_2)x -&= -(e^{s_1},t_1)(e^{s_2}x+t_2) +&\uncover<4->{= +(e^{s_1},t_1)(e^{s_2}x+t_2)} \\ -&= -e^{s_1+s_2}x + e^{s_1}t_2+t_1 +&\uncover<5->{= +e^{s_1+s_2}x + e^{s_1}t_2+t_1} \\ +\uncover<6->{ (e^{s_1},t_1)(e^{s_2},t_2) &= -(e^{s_1}e^{s_2},t_1+e^{s_1}t_2) +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} \end{align*} -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<7->{% \begin{block}{Verknüpfung} \vspace{-15pt} \begin{align*} (\alpha_1,\vec{t}_1) (\alpha_2,\vec{t}_2) \vec{x} -&= -(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2) +&\uncover<8->{= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} \\ -&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1 +&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1} \\ +\uncover<10->{ (\alpha_1,\vec{t}_1) (\alpha_2,\vec{t}_2) &= (\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \vspace{-10pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<11->{% \begin{block}{Matrixschreibweise} \vspace{-12pt} \[ @@ -88,9 +95,10 @@ e^s&t\\ \quad\text{auf}\quad \begin{pmatrix}x\\1\end{pmatrix} \] -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<12->{% \begin{block}{Matrixschreibweise} \vspace{-12pt} \[ @@ -102,7 +110,7 @@ D_{\alpha}&\vec{t}\\ \quad\text{auf}\quad \begin{pmatrix}\vec{x}\\1\end{pmatrix} \] -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 45b16d1..0cab79f 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -8,7 +8,7 @@ % Was sind Symmetrien %\folie{7/symmetrien.tex} % Algebraische Bedingungen für Matrixgruppen -\folie{7/algebraisch.tex} +%\folie{7/algebraisch.tex} % Parametrisierung, Beispiel SO(3) %\folie{7/parameter.tex} % Mannigfaltigkeiten @@ -34,5 +34,5 @@ \section{Exponentialabbildung} % Differentialgleichung für die Exponentialabbildung -%\folie{7/dg.tex} +\folie{7/dg.tex} -- cgit v1.2.1 From 78a4379a40a594fd89a74bbb0d5e6cc52f8213f6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 11:03:04 +0200 Subject: typos --- vorlesungen/slides/7/parameter.tex | 15 ++++++++++----- 1 file changed, 10 insertions(+), 5 deletions(-) diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index 1e8549c..12b5221 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -61,8 +61,10 @@ $\omega=|\vec{\omega}|=\mathstrut$Drehwinkel $\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse } \[ +\uncover<9->{ {\color{red}\vec{x}} \mapsto +} \uncover<10->{ ({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) \cos\omega @@ -79,10 +81,12 @@ $\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse \vspace{-40pt} \begin{center} \begin{tikzpicture}[>=latex,thick] -\node at (0,0) - {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; -\node[color=red] at (1.6,-0.9) {$\vec{x}$}; -\node[color=blue] at (0.5,2) {$\vec{k}$}; +\uncover<9->{ + \node at (0,0) + {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; + \node[color=red] at (1.6,-0.9) {$\vec{x}$}; + \node[color=blue] at (0.5,2) {$\vec{k}$}; +} \uncover<11->{ \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; } @@ -96,7 +100,8 @@ $\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse \end{column} \end{columns} \vspace{-15pt} +\uncover<5->{% {\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine -dreidimensionale Gruppe +dreidimensionale Gruppe} \end{frame} \egroup -- cgit v1.2.1 From 9b47f234c30f63c41bfff952a9be32bab79302cd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Sun, 11 Apr 2021 19:12:02 +0200 Subject: fix some typos --- buch/test3.tex | 91 +++++++++++++++++++++++++++++++++++ vorlesungen/07_lie/Makefile | 4 +- vorlesungen/07_lie/MathSem-07-lie.tex | 4 ++ vorlesungen/slides/7/dg.tex | 2 +- 4 files changed, 98 insertions(+), 3 deletions(-) create mode 100644 buch/test3.tex diff --git a/buch/test3.tex b/buch/test3.tex new file mode 100644 index 0000000..71b1529 --- /dev/null +++ b/buch/test3.tex @@ -0,0 +1,91 @@ +% +% test3.tex -- Test 3 +% +% (c) 2021 Prof. Dr. Andreas Mueller, OST +% +%\documentclass[a4paper,12pt]{book} +\documentclass[a4paper,12pt]{article} +\usepackage{geometry} +\geometry{papersize={210mm,297mm},total={165mm,260mm}} +\usepackage{ngerman} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage[all]{xy} +\usepackage{txfonts} +\usepackage{alltt} +\usepackage{verbatim} +\usepackage{paralist} +\usepackage{makeidx} +\usepackage{array} +\usepackage{hyperref} +\usepackage{caption} +\usepackage{subcaption} +\usepackage{standalone} +\usepackage{environ} +\usepackage{tikz} +\input{../common/linsys.tex} +\newcounter{beispiel} +\newenvironment{beispiele}{ +\bgroup\smallskip\parindent0pt\bf Beispiele\egroup + +\begin{list}{\arabic{beispiel}.} + {\usecounter{beispiel} + \setlength{\labelsep}{5mm} + \setlength{\rightmargin}{0pt} +}}{\end{list}} +\newcounter{uebungsaufgabe} +% environment fuer uebungsaufgaben +\newenvironment{uebungsaufgaben}{ +\begin{list}{\arabic{uebungsaufgabe}.} + {\usecounter{uebungsaufgabe} + \setlength{\labelwidth}{2cm} + \setlength{\leftmargin}{0pt} + \setlength{\labelsep}{5mm} + \setlength{\rightmargin}{0pt} + \setlength{\itemindent}{0pt} +}}{\end{list}\vfill\pagebreak} +\newenvironment{teilaufgaben}{ +\begin{enumerate} +\renewcommand{\labelenumi}{\alph{enumi})} +}{\end{enumerate}} +% Loesung +\NewEnviron{loesung}{% +\begin{proof}[Lösung]% +\renewcommand{\qedsymbol}{$\bigcirc$} +\BODY +\end{proof}} +\NewEnviron{bewertung}{\relax} +\NewEnviron{diskussion}{ +\BODY +} +\RenewEnviron{loesung}{\relax} +\RenewEnviron{diskussion}{\relax} +\newenvironment{hinweis}{% +\renewcommand{\qedsymbol}{} +\begin{proof}[Hinweis]}{\end{proof}} + +\begin{document} +{\parindent0pt\hbox to\hsize{% +Name: \hbox to7cm{\dotfill} Vorname: \dotfill}} +\vspace{0.5cm} + +\section*{Kurztest 3} + +\begin{uebungsaufgaben} + +\item +\input chapters/60-gruppen/uebungsaufgaben/6001.tex +%\item +%\input chapters/60-gruppen/uebungsaufgaben/6002.tex + +\end{uebungsaufgaben} + +\end{document} diff --git a/vorlesungen/07_lie/Makefile b/vorlesungen/07_lie/Makefile index ceb0e44..1788301 100644 --- a/vorlesungen/07_lie/Makefile +++ b/vorlesungen/07_lie/Makefile @@ -15,7 +15,7 @@ MathSem-07-lie.pdf: MathSem-07-lie.tex $(SOURCES) lie-handout.pdf: lie-handout.tex $(SOURCES) pdflatex lie-handout.tex -thumbnail: thumbnail.jpg # fix1.jpg +thumbnail: thumbnail.jpg fix1.jpg thumbnail.pdf: MathSem-07-lie.pdf pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ @@ -26,7 +26,7 @@ thumbnail.jpg: thumbnail.pdf fix1.pdf: MathSem-07-lie.pdf pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ - MathSem-07-lie.pdf 1 + MathSem-07-lie.pdf 205 fix1.jpg: fix1.pdf convert -density 300 fix1.pdf \ -resize 1920x1080 -units PixelsPerInch fix1.jpg diff --git a/vorlesungen/07_lie/MathSem-07-lie.tex b/vorlesungen/07_lie/MathSem-07-lie.tex index 4eeca41..8a5557d 100644 --- a/vorlesungen/07_lie/MathSem-07-lie.tex +++ b/vorlesungen/07_lie/MathSem-07-lie.tex @@ -9,6 +9,10 @@ \begin{document} \begin{frame} \titlepage +\vspace{-1.5cm} +\begin{center} +\includegraphics[width=10cm]{../slides/7/images/rodriguez.jpg} +\end{center} \end{frame} \input{slides.tex} \end{document} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index a99cebb..4447bac 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -70,7 +70,7 @@ Exponentialfunktion \begin{align*} \frac{d}{dt}e^{At} &\uncover<10->{= -\sum_{k=1}^\infty A^k \frac{d}{dt} t^{k}{k!} +\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!} } \\ &\uncover<11->{= -- cgit v1.2.1 From eb8f7d58101bd19006019dfbfafca92dd080fe41 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 12 Apr 2021 13:30:14 +0200 Subject: typos, new slides --- vorlesungen/07_lie/slides.tex | 3 + vorlesungen/common/slide-template.tex | 19 +++++ vorlesungen/common/template.tex | 19 ----- vorlesungen/slides/7/Makefile.inc | 1 + vorlesungen/slides/7/chapter.tex | 1 + vorlesungen/slides/7/drehanim.tex | 155 ++++++++++++++++++++++++++++++++++ vorlesungen/slides/7/drehung.tex | 12 +-- vorlesungen/slides/test.tex | 5 +- 8 files changed, 188 insertions(+), 27 deletions(-) create mode 100644 vorlesungen/common/slide-template.tex delete mode 100644 vorlesungen/common/template.tex create mode 100644 vorlesungen/slides/7/drehanim.tex diff --git a/vorlesungen/07_lie/slides.tex b/vorlesungen/07_lie/slides.tex index db4a3fc..19131d8 100644 --- a/vorlesungen/07_lie/slides.tex +++ b/vorlesungen/07_lie/slides.tex @@ -10,6 +10,9 @@ \folie{7/mannigfaltigkeit.tex} \folie{7/sl2.tex} \folie{7/drehung.tex} +\ifthenelse{\boolean{presentation}}{ +\folie{7/drehanim.tex} +}{} \folie{7/semi.tex} \section{Ableitungen} diff --git a/vorlesungen/common/slide-template.tex b/vorlesungen/common/slide-template.tex new file mode 100644 index 0000000..a1343f8 --- /dev/null +++ b/vorlesungen/common/slide-template.tex @@ -0,0 +1,19 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Template} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\end{column} +\begin{column}{0.48\textwidth} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/common/template.tex b/vorlesungen/common/template.tex deleted file mode 100644 index a1343f8..0000000 --- a/vorlesungen/common/template.tex +++ /dev/null @@ -1,19 +0,0 @@ -% -% template.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Template} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\end{column} -\begin{column}{0.48\textwidth} -\end{column} -\end{columns} -\end{frame} -\egroup diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 540b080..7afeea1 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -10,6 +10,7 @@ chapter5 = \ ../slides/7/mannigfaltigkeit.tex \ ../slides/7/sl2.tex \ ../slides/7/drehung.tex \ + ../slides/7/drehanim.tex \ ../slides/7/semi.tex \ ../slides/7/kurven.tex \ ../slides/7/einparameter.tex \ diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 5878164..079cf16 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -9,6 +9,7 @@ \folie{7/mannigfaltigkeit.tex} \folie{7/sl2.tex} \folie{7/drehung.tex} +\folie{7/drehanim.tex} \folie{7/semi.tex} \folie{7/kurven.tex} \folie{7/einparameter.tex} diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex new file mode 100644 index 0000000..ac136f1 --- /dev/null +++ b/vorlesungen/slides/7/drehanim.tex @@ -0,0 +1,155 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) } + +\makeatletter +\hoffset=-2cm +\advance\textwidth2cm +\hsize\textwidth +\columnwidth\textwidth +\makeatother + +\begin{frame}[t,plain] +\vspace{-5pt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=white] (-4,-4) rectangle (9,4.5); + +\def\a{60} + +\pgfmathparse{tan(\a)} +\xdef\T{\pgfmathresult} + +\pgfmathparse{-sin(\a)*cos(\a)} +\xdef\S{\pgfmathresult} + +\pgfmathparse{1/cos(\a)} +\xdef\E{\pgfmathresult} + +\def\N{20} +\pgfmathparse{2*\N} +\xdef\Nzwei{\pgfmathresult} +\pgfmathparse{3*\N} +\xdef\Ndrei{\pgfmathresult} + +\node at (4.2,4.2) [below right] {\begin{minipage}{7cm} +\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$} +\begin{itemize} +\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized +as shear mappings and axis rescalings +\[ +A= +\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix} +\begin{pmatrix}1&s\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\t&1\end{pmatrix} +\] +\item Most rotations can be decomposed into a product of +shear mappings and axis rescalings +\end{itemize} +\end{block} +\end{minipage}}; + +\foreach \d in {1,2,...,\Ndrei}{ + % Scherung in Y-Richtung + \ifnum \d>\N + \pgfmathparse{\T} + \else + \pgfmathparse{\T*(\d-1)/(\N-1)} + \fi + \xdef\t{\pgfmathresult} + + % Scherung in X-Richtung + \ifnum \d>\Nzwei + \xdef\s{\S} + \else + \ifnum \d<\N + \xdef\s{0} + \else + \ifnum \d=\N + \xdef\s{0} + \else + \pgfmathparse{\S*(\d-\N-1)/(\N-1)} + \xdef\s{\pgfmathresult} + \fi + \fi + \fi + + % Reskalierung der Achsen + \ifnum \d>\Nzwei + \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))} + \else + \pgfmathparse{1} + \fi + \xdef\e{\pgfmathresult} + + % Matrixelemente + \pgfmathparse{(\e)*((\s)*(\t)+1)} + \xdef\A{\pgfmathresult} + + \pgfmathparse{(\e)*(\s)} + \xdef\B{\pgfmathresult} + + \pgfmathparse{(\t)/(\e)} + \xdef\C{\pgfmathresult} + + \pgfmathparse{1/(\e)} + \xdef\D{\pgfmathresult} + + \only<\d>{ + \node at (5.0,-0.9) [below right] {$ + \begin{aligned} + t &= \t \\ + s &= \s \\ + d &= \e \\ + D &= \begin{pmatrix} + \A&\B\\ + \C&\D + \end{pmatrix} + \qquad + \only<60>{\checkmark} + \end{aligned} + $}; + } + + \begin{scope} + \clip (-4.05,-4.05) rectangle (4.05,4.05); + \only<\d>{ + \foreach \x in {-6,...,6}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{-12} -- \punkt{\x}{12}; + } + \foreach \y in {-12,...,12}{ + \draw[color=darkgreen,line width=0.5pt] + \punkt{-6}{\y} -- \punkt{6}{\y}; + } + + \foreach \r in {1,2,3,4}{ + \draw[color=red] plot[domain=0:359,samples=360] + ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))}) + -- + cycle; + } + } + \end{scope} + +% \uncover<\d>{ +% \node at (5,4) {\d}; +% } +} + +\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}]; +\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}]; + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index 7744e99..2d7b317 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -50,7 +50,7 @@ D_{60^\circ} {\tiny \begin{pmatrix}2&0\\0&\frac12\end{pmatrix} \begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} -\begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} +\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix} } \] \end{block}} @@ -81,14 +81,14 @@ c^{-1}&0\\ 0 &c \end{pmatrix} \begin{pmatrix} --st&-s\\ - t& 1 +1-st&-s\\ + t& 1 \end{pmatrix} } \\ &\uncover<6->{= \begin{pmatrix} -{\color<11->{orange}-stc^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ {\color<9->{blue}ct}&{\color<8->{red}c} \end{pmatrix}} \uncover<7->{= @@ -120,9 +120,9 @@ t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ } \\ \uncover<11->{ -{\color{orange} -stc^{-t}} +{\color{orange} (1-st)c^{-t}} &= -\rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $} } \end{align*} \end{block}} diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 0cab79f..4673f76 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -16,7 +16,8 @@ % Weitere Beispiele % SL_2(R) %\folie{7/sl2.tex} -%\folie{7/drehung.tex} +\folie{7/drehung.tex} +%\folie{7/drehanim.tex} % Semidirekte Produkte SO(2) x R^2, R^+ x R %\folie{7/semi.tex} @@ -34,5 +35,5 @@ \section{Exponentialabbildung} % Differentialgleichung für die Exponentialabbildung -\folie{7/dg.tex} +%\folie{7/dg.tex} -- cgit v1.2.1 From 9642e623fe93b8da49e7b5f0b60a2e37fc940803 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 12 Apr 2021 13:54:02 +0200 Subject: add new template --- vorlesungen/common/README | 28 ++++++++++++++++ vorlesungen/common/presentation-template.tex | 49 ++++++++++++++++++++++++++++ 2 files changed, 77 insertions(+) create mode 100644 vorlesungen/common/README create mode 100644 vorlesungen/common/presentation-template.tex diff --git a/vorlesungen/common/README b/vorlesungen/common/README new file mode 100644 index 0000000..1ed40aa --- /dev/null +++ b/vorlesungen/common/README @@ -0,0 +1,28 @@ +Die beiden Files + + presentation-template.tex + slide-template.tex + +können als Basis für die eigene Präsentation verwendet werden. +Dazu geht man wie folgt vor: + +1. In einem Arbeitsverzeichnis eine Kopie von presentation-template.tex +anlegen und im file Author und Titel anpassen. Im Folgenden wird diese +Kopie als beispiel-praesentation.tex bezeichnet. + +2. Für jede Folie der Präsentation im Arbeitsverzeichnis eine Kopie von +slide-template.tex anlegen und den Inhalt anpassen. + +3. Die Slides mit Hilfe von Input-Befehlen, die in presentation-template.tex +eingetragen werden, in die Präsentation importieren. + +4. Die Präsentation mit dem Befehl + + pdflatex beispiel-praesentation.tex + +erzeugen, es entsteht das File beispile-praesentation.pdf + +Diese Vorgehen erlaubt, die Reihenfolge der Folien während der Vorbereitung +zu ändern oder zwecks Beschleunigung des pdflatex-Laufs während der +Entwicklung auszukommentieren. + diff --git a/vorlesungen/common/presentation-template.tex b/vorlesungen/common/presentation-template.tex new file mode 100644 index 0000000..9f92489 --- /dev/null +++ b/vorlesungen/common/presentation-template.tex @@ -0,0 +1,49 @@ +% +% presentation-template.tex -- Präsentation +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{ifthen} +\usepackage{lmodern} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{adjustbox} +\usepackage{multimedia} +\usepackage{verbatim} +\usepackage{wasysym} +\usepackage{stmaryrd} +\usepackage{tikz} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{decorations.pathreplacing} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Titel]{Titel} +\author[A. Uthor]{A. Uthor} +\date[]{} +\newboolean{presentation} +\setboolean{presentation}{true} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\input{slide.tex} + +\end{document} -- cgit v1.2.1 From 2a2f981422ab9c2ab11c69c19b87ffda6d0f855a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 12 Apr 2021 14:03:27 +0200 Subject: typo --- vorlesungen/slides/7/parameter.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index 12b5221..52c8e4a 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -28,7 +28,7 @@ D_{x,\alpha} D_{y,\beta} &= \begin{pmatrix} -\cos\beta&0&-\sin\beta\\0&1&0\\\sin\beta&0&\cos\beta +\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta \end{pmatrix} } \\ -- cgit v1.2.1 From 9c6f818be2619504d9394d20a7e503f881cf3371 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Mon, 12 Apr 2021 16:30:23 +0200 Subject: add new files --- vorlesungen/00_template/mathman2.png | Bin 0 -> 46259 bytes vorlesungen/stream/background.png | Bin 0 -> 33587 bytes vorlesungen/stream/background2.png | Bin 0 -> 46601 bytes 3 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 vorlesungen/00_template/mathman2.png create mode 100644 vorlesungen/stream/background.png create mode 100644 vorlesungen/stream/background2.png diff --git a/vorlesungen/00_template/mathman2.png b/vorlesungen/00_template/mathman2.png new file mode 100644 index 0000000..70b2059 Binary files /dev/null and b/vorlesungen/00_template/mathman2.png differ diff --git a/vorlesungen/stream/background.png b/vorlesungen/stream/background.png new file mode 100644 index 0000000..6cd215d Binary files /dev/null and b/vorlesungen/stream/background.png differ diff --git a/vorlesungen/stream/background2.png b/vorlesungen/stream/background2.png new file mode 100644 index 0000000..43f8484 Binary files /dev/null and b/vorlesungen/stream/background2.png differ -- cgit v1.2.1 From 96c61feb021015cb3e5a2ae74ed7fd64236da8cd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 13 Apr 2021 08:56:40 +0200 Subject: add experiment to reedsolomon --- buch/papers/reedsolomon/experiments/f.m | 56 +++++++++++++++++++++++++++++++++ 1 file changed, 56 insertions(+) create mode 100644 buch/papers/reedsolomon/experiments/f.m diff --git a/buch/papers/reedsolomon/experiments/f.m b/buch/papers/reedsolomon/experiments/f.m new file mode 100644 index 0000000..ba58825 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/f.m @@ -0,0 +1,56 @@ +# +# f.m -- Reed-Solomon-Visualisierung mit FFT +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +N = 64; +b = 32; +l = N + b; + +signal = zeros(l,1); +signal(1:N,1) = round(10 * rand(N,1)); +signal + +codiert = fft(signal) + +plot(abs(signal)); +xlim([1, l]); +title("Signal"); +pause() + +fehler = zeros(l,1); +fehler(21,1) = 2; +fehler(75,1) = 1; +fehler(7,1) = 2; + +plot(fehler); +xlim([1, l]); +title("Fehler"); +pause() + +empfangen = codiert + fehler; + +plot(abs(empfangen)); +xlim([1, l]); +title("Empfangen"); +pause() + +decodiert = ifft(empfangen) +plot(abs(decodiert)); +xlim([1, l]); +title("Decodiert"); +pause() + +syndrom = decodiert; +syndrom(1:N,1) = zeros(N,1) +plot(abs(syndrom)); +xlim([1, l]); +title("Syndrom"); +pause() + +locator = abs(fft(syndrom)) + +plot(locator); +xlim([1, l]); +title("Locator"); +pause() -- cgit v1.2.1 From 516b6c8a4d7672a13847d1fb71be2df213459d3a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 13 Apr 2021 09:47:35 +0200 Subject: update fs-fft --- buch/papers/reedsolomon/experiments/f.m | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/buch/papers/reedsolomon/experiments/f.m b/buch/papers/reedsolomon/experiments/f.m index ba58825..6bdc741 100644 --- a/buch/papers/reedsolomon/experiments/f.m +++ b/buch/papers/reedsolomon/experiments/f.m @@ -11,13 +11,18 @@ signal = zeros(l,1); signal(1:N,1) = round(10 * rand(N,1)); signal -codiert = fft(signal) - plot(abs(signal)); xlim([1, l]); title("Signal"); pause() +codiert = fft(signal) + +plot(abs(codiert)); +xlim([1, l]); +title("Codiert"); +pause() + fehler = zeros(l,1); fehler(21,1) = 2; fehler(75,1) = 1; -- cgit v1.2.1 From ee33b6de909df12cdd757abcb5db04fc9d2b5a56 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 13 Apr 2021 15:46:41 +0200 Subject: kgV --- buch/chapters/30-endlichekoerper/euklid.tex | 236 ++++++++++++++++++++++++++++ 1 file changed, 236 insertions(+) diff --git a/buch/chapters/30-endlichekoerper/euklid.tex b/buch/chapters/30-endlichekoerper/euklid.tex index db326f8..9bc36a6 100644 --- a/buch/chapters/30-endlichekoerper/euklid.tex +++ b/buch/chapters/30-endlichekoerper/euklid.tex @@ -431,6 +431,7 @@ zur Bestimmung des grössten gemeinsamen Teilers von $76415$ und $23205$ zur Berechnung der Koeffizienten $c_k$ und $d_k$ Wir schreiben die gefundenen Zahlen in eine Tabelle: \begin{center} +\label{buch:endlichekoerper:beispiel1erweitert} \renewcommand{\arraystretch}{1.1} \begin{tabular}{|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|} \hline @@ -614,4 +615,239 @@ Aus den letzten zwei Zeilen folgt $ua-vb = ab/g - ab/g = 0$, wie erwartet. \end{beispiel} +% +% Das kleinste gemeinsame Vielfache +% +\subsection{Das kleinste gemeinsame Vielfache +\label{buch:subsection:daskgv}} +Das kleinste gemeinsame Vielfache zweier Zahlen $a$ und $b$ ist +\[ +\operatorname{kgV}(a,b) += +\frac{ab}{\operatorname{ggT}(a,b)}. +\] +Wir suchen nach einen Algorithmus, mit dem man das kleinste gemeinsame +Vielfache effizient berechnen kann. + +Die Zahlen $a$ und $b$ sind beide Vielfache des grössten gemeinsamen +Teilers $g=\operatorname{ggT}(a,b)$, es gibt also Zahlen $u$ und $v$ derart, +dass $a=ug$ und $b=vg$. +Wenn $t$ ein gemeinsamer Teiler von $u$ und $v$ ist, dann ist $tg$ ein +grösserer gemeinsamer Teiler von $a$ und $b$. +Dies kann nicht sein, also müssen $u$ und $v$ teilerfremd sein. +Das kleinste gemeinsame Vielfache von $a$ und $b$ ist dann $ugv=av=ub$. +Die Bestimmung des kleinsten gemeinsamen Vielfachen ist also gleichbedeutend +mit der Bestimmung der Zahle $u$ und $v$. + +Die definierende Eigenschaften von $u$ und $v$ kann man in Matrixform als +\begin{equation} +\begin{pmatrix} +a\\b +\end{pmatrix} += +\underbrace{ +\begin{pmatrix} +u&?\\ +v&? +\end{pmatrix}}_{\displaystyle =K} +\begin{pmatrix} +\operatorname{ggT}(a,b)\\ 0 +\end{pmatrix} +\label{buch:eindlichekoerper:eqn:uvmatrix} +\end{equation} +geschrieben werden, wobei wir die Matrixelemente $?$ nicht kennen. +Diese Elemente müssen wir auch nicht kennen, um $u$ und $v$ zu bestimmen. + +Bei der Bestimmung des grössten gemeinsamen Teilers wurde der Vektor auf +der rechten Seite von~\eqref{buch:eindlichekoerper:eqn:uvmatrix} bereits +gefunden. +Die Matrizen $Q(q_i)$, die die einzelne Schritte des euklidischen +Algorithmus beschreiben, ergeben ihn als +\[ +\begin{pmatrix} +\operatorname{ggT}(a,b)\\0 +\end{pmatrix} += +Q(q_n)Q(q_{n-1}) \dots Q(q_1)Q(q_0) +\begin{pmatrix}a\\b\end{pmatrix}. +\] +Indem wir die Matrizen $Q(q_n)$ bis $Q(q_0)$ auf die linke Seite der +Gleichung schaffen, erhalten wir +\[ +\begin{pmatrix}a\\b\end{pmatrix} += +Q(q_0)^{-1} +Q(q_1)^{-1} +\dots +Q(q_{n-1})^{-1} +Q(q_n) +\begin{pmatrix}\operatorname{ggT}(a,b)\\0\end{pmatrix}. +\] +Eine mögliche Lösung für die Matrix $K$ in +\eqref{buch:eindlichekoerper:eqn:uvmatrix} +ist der die Matrix +\[ +K += +Q(q_0)^{-1} +Q(q_1)^{-1} +\dots +Q(q_{n-1})^{-1} +Q(q_n). +\] +Insbesondere ist die Matrix $K$ die Inverse der früher gefundenen +Matrix $Q$. + +Die Berechnung der Matrix $K$ als Inverse von $Q$ ist nicht sehr +effizient. +Genauso wie es möglich war, das Produkt $Q$ der Matrizen +$Q(q_k)$ iterativ zu bestimmen, muss es auch eine Rekursionsformel +für das Produkt der inversen Matrizen $Q(q_k)^{-1}$ geben. + +Schreiben wir die die gesuchte Matrix +\[ +K_k += +Q(q_0)^{-1}\dots Q(q_{k-1})^{-1} += +\begin{pmatrix} +e_k & e_{k-1}\\ +f_k & f_{k-1} +\end{pmatrix}, +\] +dann kann, kann $K_k$ durch die Rekursion +\[ +K_{k+1} += +K_{k} Q(q_k)^{-1} += +K_k K(q_k) +\qquad\text{mit}\qquad +K_0 = \begin{pmatrix}1&0\\0&1\end{pmatrix} = I +\] +berechnen. +Die Inverse von $Q(q)$ ist +\[ +K(q) += +Q(q)^{-1} += +\frac{1}{\det Q(q)} +\begin{pmatrix} +q&1\\ +1&0 +\end{pmatrix} +\quad\text{denn}\quad +K(q)Q(q) += +\begin{pmatrix} +q&1\\ +1&0 +\end{pmatrix} +\begin{pmatrix} +0&1\\ +1&-q +\end{pmatrix} += +\begin{pmatrix} +1&0\\ +0&1 +\end{pmatrix}. +\] +Da die zweite Spalte von $K(q)$ die erste Spalte einer Einheitsmatrix +ist, wird die zweite Spalte des Produktes $AK(q)$ immer die erste Spalte +von $A$ sein. +In $K_{k+1}$ ist daher nur die erste Spalte neu, die zweite Spalte ist +die erste Spalte von $K_k$. + +Wenn $K_k$ die Matrixelemente +\[ +K_k += +\begin{pmatrix} +e_k & e_{k-1} \\ +f_k & f_{k-1} +\end{pmatrix} +\qquad\text{und}\qquad +K_0 = +\begin{pmatrix} +1&0\\ +0&1 +\end{pmatrix} +\Rightarrow +\left\{ +\begin{aligned} +e_0 &= 1 & e_{-1} &= 0\\ +f_0 &= 0 & f_{-1} &= 1 +\end{aligned} +\right. +\] +Daraus kann man Rekursionsformeln für die Folgen $e_k$ und $f_k$ +ablesen, es gilt +\begin{align*} +e_{k+1} &= q_ke_k + e_{k-1} \\ +f_{k+1} &= q_kf_k + f_{k-1} +\end{align*} +für $k=0,1,\dots ,n$. +Damit können $e_k$ und $f_k$ gleichzeitig mit den Zahlen $c_k$ und $d_k$ +in einer Tabelle berechnen. + +\begin{beispiel} +Wir erweitern das Beispiel von +Seite~\pageref{buch:endlichekoerper:beispiel1erweitert} +um die beiden Spalten zur Berechnung von $e_k$ und $f_k$: +\begin{center} +\renewcommand{\arraystretch}{1.1} +\begin{tabular}{|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|>{$}r<{$}>{$}r<{$}|} +\hline +k& a_k& b_k& q_k& r_k& c_k& d_k& e_k& f_k\\ +\hline + & & & & & 1& 0& 0& 1\\ +0& 76415& 23205& 3& 6800& 0& 1& 1& 0\\ +1& 23205& 6800& 3& 2805& 1& -3& 3& 1\\ +2& 6800& 2805& 2& 1190& -3& 10& 10& 3\\ +3& 2805& 1190& 2& 425& 7& -23& 23& 7\\ +4& 1190& 425& 2& 340& -17& 56& 56& 17\\ +5& 425& 340& 1& 85& 41& -135& 135& 41\\ +6& 340& 85& 4& 0& -58& 191& 191& 58\\ +7& 85& 0& & & 273& -899& 899& 273\\ +\hline +\end{tabular} +\end{center} +Der grösste gemeinsame Teiler ist $\operatorname{ggT}(a,b)=85$. +Aus der letzten Zeile der Tabelle kann man jetzt die Zahlen $u=e_7=899$ +und $v=f_7=273$ ablesen, und tatsächlich ist +\[ +a=76415 = 899\cdot 85 +\qquad\text{und}\qquad +b=23205 = 273 \cdot 85. +\] +Daraus kann man dann auch das kleinste gemeinsame Vielfache ablesen, es ist +\[ +\operatorname{kgV}(a,b) += +\operatorname{kgV}(76415,23205) += +\left\{ +\begin{aligned} +ub +&= +899\cdot 23205\\ +va +&= +273\cdot 76415 +\end{aligned} +\right\} += +20861295. +\qedhere +\] +\end{beispiel} + +Der erweiterte Algorithmus kann auch dazu verwendet werden, +das kleinste gemeinsame Vielfache zweier Polynome zu berechnen. +Dies wird zum Beispiel bei der Decodierung des Reed-Solomon-Codes in +Kapitel~\ref{chapter:reedsolomon} verwendet. + + -- cgit v1.2.1 From b94b4240a20b40871b914ddd7ae5df14f020e112 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 13 Apr 2021 15:58:59 +0200 Subject: typos --- buch/chapters/30-endlichekoerper/euklid.tex | 38 +++++++---------------------- 1 file changed, 9 insertions(+), 29 deletions(-) diff --git a/buch/chapters/30-endlichekoerper/euklid.tex b/buch/chapters/30-endlichekoerper/euklid.tex index 9bc36a6..8aa2f71 100644 --- a/buch/chapters/30-endlichekoerper/euklid.tex +++ b/buch/chapters/30-endlichekoerper/euklid.tex @@ -637,7 +637,7 @@ grösserer gemeinsamer Teiler von $a$ und $b$. Dies kann nicht sein, also müssen $u$ und $v$ teilerfremd sein. Das kleinste gemeinsame Vielfache von $a$ und $b$ ist dann $ugv=av=ub$. Die Bestimmung des kleinsten gemeinsamen Vielfachen ist also gleichbedeutend -mit der Bestimmung der Zahle $u$ und $v$. +mit der Bestimmung der Zahlen $u$ und $v$. Die definierende Eigenschaften von $u$ und $v$ kann man in Matrixform als \begin{equation} @@ -704,7 +704,7 @@ Genauso wie es möglich war, das Produkt $Q$ der Matrizen $Q(q_k)$ iterativ zu bestimmen, muss es auch eine Rekursionsformel für das Produkt der inversen Matrizen $Q(q_k)^{-1}$ geben. -Schreiben wir die die gesuchte Matrix +Schreiben wir die gesuchte Matrix \[ K_k = @@ -715,8 +715,8 @@ e_k & e_{k-1}\\ f_k & f_{k-1} \end{pmatrix}, \] -dann kann, kann $K_k$ durch die Rekursion -\[ +dann kann man $K_k$ durch die Rekursion +\begin{equation} K_{k+1} = K_{k} Q(q_k)^{-1} @@ -724,7 +724,8 @@ K_{k} Q(q_k)^{-1} K_k K(q_k) \qquad\text{mit}\qquad K_0 = \begin{pmatrix}1&0\\0&1\end{pmatrix} = I -\] +\label{buch:endlichekoerper:eqn:kgvrekursion} +\end{equation} berechnen. Die Inverse von $Q(q)$ ist \[ @@ -760,30 +761,9 @@ von $A$ sein. In $K_{k+1}$ ist daher nur die erste Spalte neu, die zweite Spalte ist die erste Spalte von $K_k$. -Wenn $K_k$ die Matrixelemente -\[ -K_k -= -\begin{pmatrix} -e_k & e_{k-1} \\ -f_k & f_{k-1} -\end{pmatrix} -\qquad\text{und}\qquad -K_0 = -\begin{pmatrix} -1&0\\ -0&1 -\end{pmatrix} -\Rightarrow -\left\{ -\begin{aligned} -e_0 &= 1 & e_{-1} &= 0\\ -f_0 &= 0 & f_{-1} &= 1 -\end{aligned} -\right. -\] -Daraus kann man Rekursionsformeln für die Folgen $e_k$ und $f_k$ -ablesen, es gilt +Aus der Rekursionsformel \eqref{buch:endlichekoerper:eqn:kgvrekursion} +für die Matrizen $K_k$ kann man jetzt eine Rekursionsbeziehung +für die Folgen $e_k$ und $f_k$ ablesen, es gilt \begin{align*} e_{k+1} &= q_ke_k + e_{k-1} \\ f_{k+1} &= q_kf_k + f_{k-1} -- cgit v1.2.1