update to current state of book

39 files changed, 3166 insertions, 3166 deletions

diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index 5aac429..1ab27fa 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -1,36 +1,36 @@
-
-#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-chapter4 =								\
-	../slides/4/ggt.tex						\
-	../slides/4/euklidmatrix.tex					\
-	../slides/4/euklidbeispiel.tex					\
-	../slides/4/euklidtabelle.tex					\
-	../slides/4/fp.tex						\
-	../slides/4/division.tex					\
-	../slides/4/gauss.tex						\
-	../slides/4/dh.tex						\
-	../slides/4/divisionpoly.tex					\
-	../slides/4/euklidpoly.tex					\
-	../slides/4/polynomefp.tex					\
-	../slides/4/schieberegister.tex					\
-	../slides/4/charakteristik.tex					\
-	../slides/4/char2.tex						\
-	../slides/4/frobenius.tex					\
-	../slides/4/qundr.tex						\
-	../slides/4/alpha.tex						\
-	../slides/4/galois/erweiterung.tex				\
-	../slides/4/galois/automorphismus.tex				\
-	../slides/4/galois/konstruktion.tex				\
-	../slides/4/galois/wuerfel.tex					\
-	../slides/4/galois/winkeldreiteilung.tex			\
-	../slides/4/galois/quadratur.tex				\
-	../slides/4/galois/radikale.tex					\
-	../slides/4/galois/aufloesbarkeit.tex				\
-	../slides/4/galois/sn.tex					\
-	../slides/4/chapter.tex
-
-
+

+#

+# Makefile.inc -- additional depencencies

+#

+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+#

+chapter4 =								\

+	../slides/4/ggt.tex						\

+	../slides/4/euklidmatrix.tex					\

+	../slides/4/euklidbeispiel.tex					\

+	../slides/4/euklidtabelle.tex					\

+	../slides/4/fp.tex						\

+	../slides/4/division.tex					\

+	../slides/4/gauss.tex						\

+	../slides/4/dh.tex						\

+	../slides/4/divisionpoly.tex					\

+	../slides/4/euklidpoly.tex					\

+	../slides/4/polynomefp.tex					\

+	../slides/4/schieberegister.tex					\

+	../slides/4/charakteristik.tex					\

+	../slides/4/char2.tex						\

+	../slides/4/frobenius.tex					\

+	../slides/4/qundr.tex						\

+	../slides/4/alpha.tex						\

+	../slides/4/galois/erweiterung.tex				\

+	../slides/4/galois/automorphismus.tex				\

+	../slides/4/galois/konstruktion.tex				\

+	../slides/4/galois/wuerfel.tex					\

+	../slides/4/galois/winkeldreiteilung.tex			\

+	../slides/4/galois/quadratur.tex				\

+	../slides/4/galois/radikale.tex					\

+	../slides/4/galois/aufloesbarkeit.tex				\

+	../slides/4/galois/sn.tex					\

+	../slides/4/chapter.tex

+

+

diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
index 0691e39..3015e7c 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -1,31 +1,31 @@
-%
-% chapter.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
-%
-\folie{4/ggt.tex}
-\folie{4/euklidmatrix.tex}
-\folie{4/euklidbeispiel.tex}
-\folie{4/euklidtabelle.tex}
-\folie{4/fp.tex}
-\folie{4/division.tex}
-\folie{4/gauss.tex}
-\folie{4/dh.tex}
-\folie{4/divisionpoly.tex}
-\folie{4/euklidpoly.tex}
-\folie{4/polynomefp.tex}
-\folie{4/alpha.tex}
-\folie{4/schieberegister.tex}
-\folie{4/charakteristik.tex}
-\folie{4/char2.tex}
-\folie{4/frobenius.tex}
-\folie{4/qundr.tex}
-\folie{4/galois/erweiterung.tex}
-\folie{4/galois/automorphismus.tex}
-\folie{4/galois/konstruktion.tex}
-\folie{4/galois/wuerfel.tex}
-\folie{4/galois/winkeldreiteilung.tex}
-\folie{4/galois/quadratur.tex}
-\folie{4/galois/radikale.tex}
-\folie{4/galois/aufloesbarkeit.tex}
-\folie{4/galois/sn.tex}
+%

+% chapter.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi

+%

+\folie{4/ggt.tex}

+\folie{4/euklidmatrix.tex}

+\folie{4/euklidbeispiel.tex}

+\folie{4/euklidtabelle.tex}

+\folie{4/fp.tex}

+\folie{4/division.tex}

+\folie{4/gauss.tex}

+\folie{4/dh.tex}

+\folie{4/divisionpoly.tex}

+\folie{4/euklidpoly.tex}

+\folie{4/polynomefp.tex}

+\folie{4/alpha.tex}

+\folie{4/schieberegister.tex}

+\folie{4/charakteristik.tex}

+\folie{4/char2.tex}

+\folie{4/frobenius.tex}

+\folie{4/qundr.tex}

+\folie{4/galois/erweiterung.tex}

+\folie{4/galois/automorphismus.tex}

+\folie{4/galois/konstruktion.tex}

+\folie{4/galois/wuerfel.tex}

+\folie{4/galois/winkeldreiteilung.tex}

+\folie{4/galois/quadratur.tex}

+\folie{4/galois/radikale.tex}

+\folie{4/galois/aufloesbarkeit.tex}

+\folie{4/galois/sn.tex}

diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
index ef5902b..3d52b00 100644
--- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex
+++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex
@@ -1,120 +1,120 @@
-%
-% aufloesbarkeit.tex
-%
-% (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}
+%

+% aufloesbarkeit.tex

+%

+% (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 6051813..e59f9b9 100644
--- a/vorlesungen/slides/4/galois/automorphismus.tex
+++ b/vorlesungen/slides/4/galois/automorphismus.tex
@@ -1,118 +1,118 @@
-%
-% automorphismus.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{4pt}
-\setlength{\belowdisplayskip}{4pt}
-\frametitle{Galois-Gruppe}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\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}{0.56\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}
+%

+% automorphismus.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{4pt}

+\setlength{\belowdisplayskip}{4pt}

+\frametitle{Galois-Gruppe}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\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}{0.56\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 6909849..20b278e 100644
--- a/vorlesungen/slides/4/galois/erweiterung.tex
+++ b/vorlesungen/slides/4/galois/erweiterung.tex
@@ -1,65 +1,65 @@
-%
-% erweiterung.tex
-%
-% (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}
+%

+% erweiterung.tex

+%

+% (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/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile
index 444944e..fd197ce 100644
--- a/vorlesungen/slides/4/galois/images/Makefile
+++ b/vorlesungen/slides/4/galois/images/Makefile
@@ -1,12 +1,12 @@
-#
-# Makefile
-#
-# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-#
-all:	wuerfel2.png wuerfel.png
-
-wuerfel.png:	wuerfel.pov common.inc
-	povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov
-
-wuerfel2.png:	wuerfel2.pov common.inc
-	povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov
+#

+# Makefile

+#

+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+#

+all:	wuerfel2.png wuerfel.png

+

+wuerfel.png:	wuerfel.pov common.inc

+	povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov

+

+wuerfel2.png:	wuerfel2.pov common.inc

+	povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov

diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc
index 6cfcabe..44ee4c8 100644
--- a/vorlesungen/slides/4/galois/images/common.inc
+++ b/vorlesungen/slides/4/galois/images/common.inc
@@ -1,89 +1,89 @@
-//
-// common.inc
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#version 3.7;
-#include "colors.inc"
-#include "textures.inc"
-#include "stones.inc"
-
-global_settings {
-	assumed_gamma 1
-}
-
-#declare imagescale = 0.133;
-#declare O = <0, 0, 0>;
-#declare E = <1, 1, 1>;
-#declare a = pow(2, 1/3);
-#declare at = 0.02;
-
-camera {
-	location <3, 2, 12>
-	look_at E * (a / 2) * 0.93
-	right x * imagescale
-	up y * imagescale
-}
-
-light_source {
-	<11, 20, 16> color White
-	area_light <1,0,0> <0,0,1>, 10, 10
-	adaptive 1
-	jitter
-}
-
-sky_sphere {
-	pigment {
-		color rgb<1,1,1>
-	}
-}
-
-#macro wuerfelgitter(A, AT)
-	cylinder { O, <A, 0, 0>, AT }
-	cylinder { O, <0, A, 0>, AT }
-	cylinder { O, <0, 0, A>, AT }
-	cylinder { <A, 0, 0>, <A, A, 0>, AT }
-	cylinder { <A, 0, 0>, <A, 0, A>, AT }
-	cylinder { <0, A, 0>, <A, A, 0>, AT }
-	cylinder { <0, A, 0>, <0, A, A>, AT }
-	cylinder { <0, 0, A>, <A, 0, A>, AT }
-	cylinder { <0, 0, A>, <0, A, A>, AT }
-	cylinder { <A, A, 0>, <A, A, A>, AT }
-	cylinder { <A, 0, A>, <A, A, A>, AT }
-	cylinder { <0, A, A>, <A, A, A>, AT }
-	sphere { <0, 0, 0>, AT }
-	sphere { <A, 0, 0>, AT }
-	sphere { <0, A, 0>, AT }
-	sphere { <0, 0, A>, AT }
-	sphere { <A, A, 0>, AT }
-	sphere { <A, 0, A>, AT }
-	sphere { <0, A, A>, AT }
-	sphere { <A, A, A>, AT }
-#end
-
-#macro wuerfel()
-	union {
-		box { O, E }
-		wuerfelgitter(1, 0.5*at)
-		texture {
-			T_Grnt24
-		}
-		finish {
-			specular 0.9
-			metallic
-		}
-	}
-#end
-
-#macro wuerfel2()
-	union {
-		wuerfelgitter(a, at)
-		pigment {
-			color rgb<0.8,0.4,0.4>
-		}
-		finish {
-			specular 0.9
-			metallic
-		}
-	}
-#end
+//

+// common.inc

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#version 3.7;

+#include "colors.inc"

+#include "textures.inc"

+#include "stones.inc"

+

+global_settings {

+	assumed_gamma 1

+}

+

+#declare imagescale = 0.133;

+#declare O = <0, 0, 0>;

+#declare E = <1, 1, 1>;

+#declare a = pow(2, 1/3);

+#declare at = 0.02;

+

+camera {

+	location <3, 2, 12>

+	look_at E * (a / 2) * 0.93

+	right x * imagescale

+	up y * imagescale

+}

+

+light_source {

+	<11, 20, 16> color White

+	area_light <1,0,0> <0,0,1>, 10, 10

+	adaptive 1

+	jitter

+}

+

+sky_sphere {

+	pigment {

+		color rgb<1,1,1>

+	}

+}

+

+#macro wuerfelgitter(A, AT)

+	cylinder { O, <A, 0, 0>, AT }

+	cylinder { O, <0, A, 0>, AT }

+	cylinder { O, <0, 0, A>, AT }

+	cylinder { <A, 0, 0>, <A, A, 0>, AT }

+	cylinder { <A, 0, 0>, <A, 0, A>, AT }

+	cylinder { <0, A, 0>, <A, A, 0>, AT }

+	cylinder { <0, A, 0>, <0, A, A>, AT }

+	cylinder { <0, 0, A>, <A, 0, A>, AT }

+	cylinder { <0, 0, A>, <0, A, A>, AT }

+	cylinder { <A, A, 0>, <A, A, A>, AT }

+	cylinder { <A, 0, A>, <A, A, A>, AT }

+	cylinder { <0, A, A>, <A, A, A>, AT }

+	sphere { <0, 0, 0>, AT }

+	sphere { <A, 0, 0>, AT }

+	sphere { <0, A, 0>, AT }

+	sphere { <0, 0, A>, AT }

+	sphere { <A, A, 0>, AT }

+	sphere { <A, 0, A>, AT }

+	sphere { <0, A, A>, AT }

+	sphere { <A, A, A>, AT }

+#end

+

+#macro wuerfel()

+	union {

+		box { O, E }

+		wuerfelgitter(1, 0.5*at)

+		texture {

+			T_Grnt24

+		}

+		finish {

+			specular 0.9

+			metallic

+		}

+	}

+#end

+

+#macro wuerfel2()

+	union {

+		wuerfelgitter(a, at)

+		pigment {

+			color rgb<0.8,0.4,0.4>

+		}

+		finish {

+			specular 0.9

+			metallic

+		}

+	}

+#end

diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov
index a5db465..a0466f3 100644
--- a/vorlesungen/slides/4/galois/images/wuerfel.pov
+++ b/vorlesungen/slides/4/galois/images/wuerfel.pov
@@ -1,9 +1,9 @@
-//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-
+//

+// wuerfel.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#include "common.inc"

+

+wuerfel()

+

diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov
index ac32b2f..a11bab0 100644
--- a/vorlesungen/slides/4/galois/images/wuerfel2.pov
+++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov
@@ -1,9 +1,9 @@
-//
-// wuerfel.pov
-//
-// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-//
-#include "common.inc"
-
-wuerfel()
-wuerfel2()
+//

+// wuerfel.pov

+//

+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+//

+#include "common.inc"

+

+wuerfel()

+wuerfel2()

diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex
index 094b570..b461d44 100644
--- a/vorlesungen/slides/4/galois/konstruktion.tex
+++ b/vorlesungen/slides/4/galois/konstruktion.tex
@@ -1,147 +1,147 @@
-%
-% konstruktion.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Konstruktion mit Zirkel und Lineal}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Strahlensatz}
-\uncover<6->{%
-Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$
-kann mit Zirkel und Lineal konstruiert werden.}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<7->{%
-\begin{block}{Kreis--Gerade}
-Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$
-\uncover<13->{%
-$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}
-\end{block}}
-\end{column}
-\end{columns}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\s{0.5}
-\def\t{0.45}
-
-\coordinate (A) at (0,0);
-\coordinate (B) at ({10*\t},0);
-
-\uncover<2->{
-	\draw (0,0) -- (30:{10.5*\s});
-}
-
-\uncover<3->{
-	\foreach \x in {0,...,10}{
-		\fill (30:{\x*\s}) circle[radius=0.03];
-	}
-	\foreach \x in {0,1,2,3,4,7,8,9}{
-		\node at (30:{\x*\s}) [above] {\tiny $\x$};
-	}
-	\node at (30:{10*\s}) [above right] {$q=10$};
-}
-
-\uncover<4->{
-	\foreach \x in {1,...,10}{
-		\fill (0:{\x*\t}) circle[radius=0.03];
-		\draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});
-	}
-}
-
-\draw (A) -- (0:{10.5*\t});
-\node at (A) [below left] {$A$};
-\node at (B) [below right] {$B$};
-\fill (A) circle[radius=0.05];
-\fill (B) circle[radius=0.05];
-
-\uncover<5->{
-	\node at (30:{6*\s}) [above left] {$p=6$};
-	\draw[line width=0.2pt] (0,0) -- (0,-0.4);
-	\draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);
-	\draw[<->] (0,-0.3) -- ({6*\t},-0.3);
-	\node at ({3*\t},-0.4) [below]
-		{$\displaystyle\frac{p}{q}\cdot\overline{AB}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<8->{%
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-
-%\foreach \x in {8,...,14}{
-%	\only<\x>{\node at (4,4) {$\x$};}
-%}
-
-\def\r{4}
-\def\a{50}
-
-\coordinate (A) at ({\r*cos(\a)},0);
-
-\uncover<10->{
-	\fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;
-	\fill[color=gray]
-		(95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;
-}
-
-\draw[->] (0,0) -- (95:\r);
-\node at (95:{0.5*\r}) [left] {$c$};
-
-\begin{scope}
-	\clip (-1,-0.3) rectangle (4.5,4.1);
-	\uncover<10->{
-		\draw (-1,0) -- (5,0);
-		\draw[->] (0,0) -- (\r,0);
-		\draw (0,0) circle[radius=\r];
-		\draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);
-	}
-\end{scope}
-
-\uncover<11->{
-	\fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;
-}
-
-\uncover<9->{
-	\fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;
-	\fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];
-	\draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);
-}
-
-\uncover<11->{
-	\draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);
-	\node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};
-}
-
-\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);
-\fill[color=blue] (0,0) circle[radius=0.04];
-\fill[color=blue] (A) circle[radius=0.04];
-\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};
-
-\uncover<12->{
-	\fill[color=white,opacity=0.8]
-		({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})
-		rectangle
-		({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});
-
-	\node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]
-		{$b=\sqrt{c^2-a^2}$};
-	\draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);
-	\fill[color=red] (\a:\r) circle[radius=0.05];
-	\fill[color=red] (A) circle[radius=0.05];
-}
-
-\end{tikzpicture}
-\end{center}}
-\end{column}
-\end{columns}
-\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:}
-Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$}
-\end{frame}
+%

+% konstruktion.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\frametitle{Konstruktion mit Zirkel und Lineal}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{block}{Strahlensatz}

+\uncover<6->{%

+Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$

+kann mit Zirkel und Lineal konstruiert werden.}

+\end{block}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<7->{%

+\begin{block}{Kreis--Gerade}

+Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$

+\uncover<13->{%

+$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden}

+\end{block}}

+\end{column}

+\end{columns}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\s{0.5}

+\def\t{0.45}

+

+\coordinate (A) at (0,0);

+\coordinate (B) at ({10*\t},0);

+

+\uncover<2->{

+	\draw (0,0) -- (30:{10.5*\s});

+}

+

+\uncover<3->{

+	\foreach \x in {0,...,10}{

+		\fill (30:{\x*\s}) circle[radius=0.03];

+	}

+	\foreach \x in {0,1,2,3,4,7,8,9}{

+		\node at (30:{\x*\s}) [above] {\tiny $\x$};

+	}

+	\node at (30:{10*\s}) [above right] {$q=10$};

+}

+

+\uncover<4->{

+	\foreach \x in {1,...,10}{

+		\fill (0:{\x*\t}) circle[radius=0.03];

+		\draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t});

+	}

+}

+

+\draw (A) -- (0:{10.5*\t});

+\node at (A) [below left] {$A$};

+\node at (B) [below right] {$B$};

+\fill (A) circle[radius=0.05];

+\fill (B) circle[radius=0.05];

+

+\uncover<5->{

+	\node at (30:{6*\s}) [above left] {$p=6$};

+	\draw[line width=0.2pt] (0,0) -- (0,-0.4);

+	\draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4);

+	\draw[<->] (0,-0.3) -- ({6*\t},-0.3);

+	\node at ({3*\t},-0.4) [below]

+		{$\displaystyle\frac{p}{q}\cdot\overline{AB}$};

+}

+

+\end{tikzpicture}

+\end{center}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<8->{%

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+

+%\foreach \x in {8,...,14}{

+%	\only<\x>{\node at (4,4) {$\x$};}

+%}

+

+\def\r{4}

+\def\a{50}

+

+\coordinate (A) at ({\r*cos(\a)},0);

+

+\uncover<10->{

+	\fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle;

+	\fill[color=gray]

+		(95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle;

+}

+

+\draw[->] (0,0) -- (95:\r);

+\node at (95:{0.5*\r}) [left] {$c$};

+

+\begin{scope}

+	\clip (-1,-0.3) rectangle (4.5,4.1);

+	\uncover<10->{

+		\draw (-1,0) -- (5,0);

+		\draw[->] (0,0) -- (\r,0);

+		\draw (0,0) circle[radius=\r];

+		\draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5);

+	}

+\end{scope}

+

+\uncover<11->{

+	\fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle;

+}

+

+\uncover<9->{

+	\fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle;

+	\fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07];

+	\draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1);

+}

+

+\uncover<11->{

+	\draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r);

+	\node[color=blue] at (\a:{0.5*\r}) [above left] {$c$};

+}

+

+\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0);

+\fill[color=blue] (0,0) circle[radius=0.04];

+\fill[color=blue] (A) circle[radius=0.04];

+\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$};

+

+\uncover<12->{

+	\fill[color=white,opacity=0.8]

+		({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25})

+		rectangle

+		({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25});

+

+	\node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right]

+		{$b=\sqrt{c^2-a^2}$};

+	\draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r);

+	\fill[color=red] (\a:\r) circle[radius=0.05];

+	\fill[color=red] (A) circle[radius=0.05];

+}

+

+\end{tikzpicture}

+\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/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex
index f5763b9..f9510ba 100644
--- a/vorlesungen/slides/4/galois/quadratur.tex
+++ b/vorlesungen/slides/4/galois/quadratur.tex
@@ -1,66 +1,66 @@
-%
-% quadratur.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Quadratur des Kreises}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.44\textwidth}
-\begin{center}
-\uncover<2->{%
-\begin{tikzpicture}[>=latex,thick]
-
-\def\r{2.8}
-\pgfmathparse{sqrt(3.14159)*\r/2}
-\xdef\s{\pgfmathresult}
-
-\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);
-\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];
-
-\uncover<3->{
-	\draw[->,color=red] (0,0) -- (50:\r);
-	\fill[color=red] (0,0) circle[radius=0.04];
-	\node[color=red] at (50:{0.5*\r}) [below right] {$r$};
-}
-
-\uncover<4->{
-	\draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});
-	\draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});
-	\draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});
-	\node[color=blue] at (0,{-\s-0.6}) [below] {$l$};
-}
-
-\uncover<5->{
-	\node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}
-		\;\Rightarrow\;
-		{\color{blue}l}={\color{red}\sqrt{\pi}r}$};
-}
-
-\end{tikzpicture}}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere ein zu einem Kreis flächengleiches Quadrat
-\end{block}
-\uncover<6->{%
-\begin{block}{Modifizierte Aufgabe}
-Konstruiere eine Strecke, deren Länge Lösung der Gleichung
-$x^2-\pi=0$ ist.
-\end{block}}
-\uncover<7->{%
-\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]
-\begin{itemize}
-\item<8-> Lösung in einem Erweiterungskörper
-\item<9-> Lösung ist Nullstelle eines Polynoms 
-\item<10-> Lösung ist algebraisch
-\item<11-> $\pi$ ist {\bf nicht} algebraisch
-\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% quadratur.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\frametitle{Quadratur des Kreises}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.44\textwidth}

+\begin{center}

+\uncover<2->{%

+\begin{tikzpicture}[>=latex,thick]

+

+\def\r{2.8}

+\pgfmathparse{sqrt(3.14159)*\r/2}

+\xdef\s{\pgfmathresult}

+

+\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s);

+\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r];

+

+\uncover<3->{

+	\draw[->,color=red] (0,0) -- (50:\r);

+	\fill[color=red] (0,0) circle[radius=0.04];

+	\node[color=red] at (50:{0.5*\r}) [below right] {$r$};

+}

+

+\uncover<4->{

+	\draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7});

+	\draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7});

+	\draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6});

+	\node[color=blue] at (0,{-\s-0.6}) [below] {$l$};

+}

+

+\uncover<5->{

+	\node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2}

+		\;\Rightarrow\;

+		{\color{blue}l}={\color{red}\sqrt{\pi}r}$};

+}

+

+\end{tikzpicture}}

+\end{center}

+\end{column}

+\begin{column}{0.52\textwidth}

+\begin{block}{Aufgabe}

+Konstruiere ein zu einem Kreis flächengleiches Quadrat

+\end{block}

+\uncover<6->{%

+\begin{block}{Modifizierte Aufgabe}

+Konstruiere eine Strecke, deren Länge Lösung der Gleichung

+$x^2-\pi=0$ ist.

+\end{block}}

+\uncover<7->{%

+\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch]

+\begin{itemize}

+\item<8-> Lösung in einem Erweiterungskörper

+\item<9-> Lösung ist Nullstelle eines Polynoms 

+\item<10-> Lösung ist algebraisch

+\item<11-> $\pi$ ist {\bf nicht} algebraisch

+\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})}

+\qedhere

+\end{itemize}

+\end{proof}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex
index e9e4ce8..cb08dca 100644
--- a/vorlesungen/slides/4/galois/radikale.tex
+++ b/vorlesungen/slides/4/galois/radikale.tex
@@ -1,69 +1,69 @@
-%
-% radikale.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\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}
+%

+% radikale.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\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 1cae3fa..f340825 100644
--- a/vorlesungen/slides/4/galois/sn.tex
+++ b/vorlesungen/slides/4/galois/sn.tex
@@ -1,87 +1,87 @@
-%
-% sn.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\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}
+%

+% sn.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\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/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
index 54b941b..28c07fe 100644
--- a/vorlesungen/slides/4/galois/winkeldreiteilung.tex
+++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex
@@ -1,94 +1,94 @@
-%
-% winkeldreiteilung.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Winkeldreiteilung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.43\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\r{5}
-\def\a{25}
-
-\uncover<3->{
-	\draw[line width=0.7pt] (\r,0) arc (0:90:\r);
-}
-
-\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;
-\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};
-
-\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);
-
-\uncover<2->{
-	\fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;
-	\draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);
-	\node[color=red] at ({0.5*\a}:{0.7*\r})
-		{$\displaystyle\frac{\alpha}{3}$};
-}
-
-\uncover<3->{
-	\fill[color=blue] ({3*\a}:\r) circle[radius=0.05];
-	\draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);
-
-	\fill[color=red] ({\a}:\r) circle[radius=0.05];
-	\draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);
-
-	\draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];
-	\draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];
-}
-
-
-\uncover<4->{
-\node at ({0.5*\r},-0.5) [below] {$\displaystyle
-\cos{\color{blue}\alpha}
-=
-4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}
-$};
-}
-
-\uncover<5->{
-	\node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};
-	\node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.53\textwidth}
-\begin{block}{Aufgabe}
-Teile einen Winkel in drei gleiche Teile
-\end{block}
-\vspace{-2pt}
-\uncover<6->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere $x$ aus $a$ derart, dass
-\[
-p(x)
-=
-x^3-\frac34 x -a = 0
-\]
-\uncover<7->{%
-$a=0$:}
-\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}
-\end{block}}
-\vspace{-2pt}
-\uncover<10->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<11->
-$a\ne 0$ $\Rightarrow$  $p(x)$ irreduzibel
-\item<12->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<13->
-Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% winkeldreiteilung.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{5pt}

+\setlength{\belowdisplayskip}{5pt}

+\frametitle{Winkeldreiteilung}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.43\textwidth}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\r{5}

+\def\a{25}

+

+\uncover<3->{

+	\draw[line width=0.7pt] (\r,0) arc (0:90:\r);

+}

+

+\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle;

+\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$};

+

+\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r);

+

+\uncover<2->{

+	\fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle;

+	\draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r);

+	\node[color=red] at ({0.5*\a}:{0.7*\r})

+		{$\displaystyle\frac{\alpha}{3}$};

+}

+

+\uncover<3->{

+	\fill[color=blue] ({3*\a}:\r) circle[radius=0.05];

+	\draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1);

+

+	\fill[color=red] ({\a}:\r) circle[radius=0.05];

+	\draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1);

+

+	\draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}];

+	\draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}];

+}

+

+

+\uncover<4->{

+\node at ({0.5*\r},-0.5) [below] {$\displaystyle

+\cos{\color{blue}\alpha}

+=

+4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3}

+$};

+}

+

+\uncover<5->{

+	\node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$};

+	\node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$};

+}

+

+\end{tikzpicture}

+\end{center}

+\end{column}

+\begin{column}{0.53\textwidth}

+\begin{block}{Aufgabe}

+Teile einen Winkel in drei gleiche Teile

+\end{block}

+\vspace{-2pt}

+\uncover<6->{%

+\begin{block}{Algebraisierte Aufgabe}

+Konstruiere $x$ aus $a$ derart, dass

+\[

+p(x)

+=

+x^3-\frac34 x -a = 0

+\]

+\uncover<7->{%

+$a=0$:}

+\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$}

+\end{block}}

+\vspace{-2pt}

+\uncover<10->{%

+\begin{proof}[Unmöglichkeitsbeweis]

+\begin{itemize}

+\item<11->

+$a\ne 0$ $\Rightarrow$  $p(x)$ irreduzibel

+\item<12->

+$p(x)$ definiert eine Körpererweiterung vom Grad $3$

+\item<13->

+Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$

+\qedhere

+\end{itemize}

+\end{proof}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex
index ada6079..907d60a 100644
--- a/vorlesungen/slides/4/galois/wuerfel.tex
+++ b/vorlesungen/slides/4/galois/wuerfel.tex
@@ -1,64 +1,64 @@
-%
-% wuerfel.tex
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\begin{frame}[t]
-\frametitle{Würfelverdoppelung}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};
-\uncover<2->{
-\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};
-}
-
-\uncover<3->{
-	\draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);
-	\node[color=blue] at (0.75,-2.3) [above] {$a$};
-}
-
-\uncover<4->{
-	\begin{scope}[yshift=0.03cm]
-		\draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);
-		\draw[color=red] (2.85,-2.7) -- (2.85,-3.0);
-		\draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);
-	\end{scope}
-	\node[color=red] at (0.36,-2.9) [below] {$b$};
-}
-
-\uncover<5->{
-\node at (0,-4) {$
-	2{\color{blue}a}^3={\color{red}b}^3
-	\uncover<6->{\;\Rightarrow\;
-	\frac{b}{a} = \sqrt[3]{2}}$};
-}
-
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.52\textwidth}
-\begin{block}{Aufgabe}
-Konstruiere einen Würfel mit doppeltem Volumen
-\end{block}
-\uncover<7->{%
-\begin{block}{Algebraisierte Aufgabe}
-Konstruiere eine Nullstelle von $p(x)=x^3-2$
-\end{block}}
-\uncover<8->{%
-\begin{proof}[Unmöglichkeitsbeweis]
-\begin{itemize}
-\item<9->
-$p(x)$ irreduzibel
-\item<10->
-$p(x)$ definiert eine Körpererweiterung vom Grad $3$
-\item<11->
-Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar
-\qedhere
-\end{itemize}
-\end{proof}}
-\end{column}
-\end{columns}
-\end{frame}
+%

+% wuerfel.tex

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\begin{frame}[t]

+\frametitle{Würfelverdoppelung}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}};

+\uncover<2->{

+\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}};

+}

+

+\uncover<3->{

+	\draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25);

+	\node[color=blue] at (0.75,-2.3) [above] {$a$};

+}

+

+\uncover<4->{

+	\begin{scope}[yshift=0.03cm]

+		\draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19);

+		\draw[color=red] (2.85,-2.7) -- (2.85,-3.0);

+		\draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9);

+	\end{scope}

+	\node[color=red] at (0.36,-2.9) [below] {$b$};

+}

+

+\uncover<5->{

+\node at (0,-4) {$

+	2{\color{blue}a}^3={\color{red}b}^3

+	\uncover<6->{\;\Rightarrow\;

+	\frac{b}{a} = \sqrt[3]{2}}$};

+}

+

+\end{tikzpicture}

+\end{center}

+\end{column}

+\begin{column}{0.52\textwidth}

+\begin{block}{Aufgabe}

+Konstruiere einen Würfel mit doppeltem Volumen

+\end{block}

+\uncover<7->{%

+\begin{block}{Algebraisierte Aufgabe}

+Konstruiere eine Nullstelle von $p(x)=x^3-2$

+\end{block}}

+\uncover<8->{%

+\begin{proof}[Unmöglichkeitsbeweis]

+\begin{itemize}

+\item<9->

+$p(x)$ irreduzibel

+\item<10->

+$p(x)$ definiert eine Körpererweiterung vom Grad $3$

+\item<11->

+Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar

+\qedhere

+\end{itemize}

+\end{proof}}

+\end{column}

+\end{columns}

+\end{frame}

diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index 7afeea1..2391099 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -1,22 +1,22 @@
-#
-# 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/drehanim.tex					\
-	../slides/7/semi.tex						\
-	../slides/7/kurven.tex						\
-	../slides/7/einparameter.tex					\
-	../slides/7/ableitung.tex					\
-	../slides/7/liealgebra.tex					\
-	../slides/7/kommutator.tex					\
-	../slides/7/dg.tex						\
-	../slides/7/chapter.tex
-
+#

+# 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/drehanim.tex					\

+	../slides/7/semi.tex						\

+	../slides/7/kurven.tex						\

+	../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/ableitung.tex b/vorlesungen/slides/7/ableitung.tex
index 12f9084..5a4b94e 100644
--- a/vorlesungen/slides/7/ableitung.tex
+++ b/vorlesungen/slides/7/ableitung.tex
@@ -1,68 +1,68 @@
-%
-% 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 $\operatorname{O}(n)$}
-\uncover<2->{%
-$s \mapsto A(s)\in\operatorname{O}(n)$
-}
-\begin{align*}
-\uncover<3->{I
-&=
-A(s)^tA(s)}
-\\
-\uncover<4->{0
-=
-\frac{d}{ds} I
-&=
-\frac{d}{ds} (A(s)^t A(s))}
-\\
-&\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)}
-\\
-\uncover<8->{\Leftrightarrow
-\qquad
-\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*}
-\uncover<3->{1 &= \det A(t)}
-\\
-\uncover<10->{0
-=
-\frac{d}{dt}1
-&=
-\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*}
-\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen}
-\end{block}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% 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 $\operatorname{O}(n)$}

+\uncover<2->{%

+$s \mapsto A(s)\in\operatorname{O}(n)$

+}

+\begin{align*}

+\uncover<3->{I

+&=

+A(s)^tA(s)}

+\\

+\uncover<4->{0

+=

+\frac{d}{ds} I

+&=

+\frac{d}{ds} (A(s)^t A(s))}

+\\

+&\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)}

+\\

+\uncover<8->{\Leftrightarrow

+\qquad

+\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*}

+\uncover<3->{1 &= \det A(t)}

+\\

+\uncover<10->{0

+=

+\frac{d}{dt}1

+&=

+\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*}

+\uncover<13->{``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
index 31d209a..fba42cf 100644
--- a/vorlesungen/slides/7/algebraisch.tex
+++ b/vorlesungen/slides/7/algebraisch.tex
@@ -1,115 +1,115 @@
-%
-% 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}
-\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{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*}
-\vec{x}^t\vec{y}
-&=
-(A\vec{x})
-\cdot
-(A\vec{y})
-\uncover<5->{=
-(A\vec{x})^t
-(A\vec{y})}
-\\
-\uncover<6->{
-\vec{x}^tI\vec{y}
-&=
-\vec{x}^tA^tA\vec{y}}
-\uncover<7->{
-\Rightarrow I=A^tA}
-\end{align*}
-\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)
-\uncover<10->{=
-\det(A)\det(V)}
-\\
-\uncover<11->{
-1&=\det(A)}
-\end{align*}
-\uncover<10->{
-(Produktsatz für Determinante)
-}
-\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:
-\[
-O(n)
-=
-\{
-A \in \operatorname{GL}_n(\mathbb{R})
-\;|\;
-A^tA=I
-\}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<13->{%
-\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
+%

+% 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}

+\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{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*}

+\vec{x}^t\vec{y}

+&=

+(A\vec{x})

+\cdot

+(A\vec{y})

+\uncover<5->{=

+(A\vec{x})^t

+(A\vec{y})}

+\\

+\uncover<6->{

+\vec{x}^tI\vec{y}

+&=

+\vec{x}^tA^tA\vec{y}}

+\uncover<7->{

+\Rightarrow I=A^tA}

+\end{align*}

+\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)

+\uncover<10->{=

+\det(A)\det(V)}

+\\

+\uncover<11->{

+1&=\det(A)}

+\end{align*}

+\uncover<10->{

+(Produktsatz für Determinante)

+}

+\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:

+\[

+O(n)

+=

+\{

+A \in \operatorname{GL}_n(\mathbb{R})

+\;|\;

+A^tA=I

+\}

+\]

+\end{block}}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<13->{%

+\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
index 079cf16..0f14a9a 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -1,19 +1,19 @@
-%
-% 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/drehanim.tex}
-\folie{7/semi.tex}
-\folie{7/kurven.tex}
-\folie{7/einparameter.tex}
-\folie{7/ableitung.tex}
-\folie{7/liealgebra.tex}
-\folie{7/kommutator.tex}
-\folie{7/dg.tex}
+%

+% 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/drehanim.tex}

+\folie{7/semi.tex}

+\folie{7/kurven.tex}

+\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/dg.tex b/vorlesungen/slides/7/dg.tex
index 4447bac..446b2ab 100644
--- a/vorlesungen/slides/7/dg.tex
+++ b/vorlesungen/slides/7/dg.tex
@@ -1,92 +1,92 @@
-%
-% 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)
-&\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}
-\[
-\dot{\gamma}(t) = \gamma(t) A
-\quad
-\text{mit}
-\quad
-A=\dot{\gamma}(0)\in LG
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.50\textwidth}
-\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}}
-\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} \frac{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{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% 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)

+&\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}

+\[

+\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}

+\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}}

+\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} \frac{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{column}

+\end{columns}

+\end{frame}

+\egroup

diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex
index ac136f1..776617f 100644
--- a/vorlesungen/slides/7/drehanim.tex
+++ b/vorlesungen/slides/7/drehanim.tex
@@ -1,155 +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
+%

+% 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 2d7b317..e7b4a92 100644
--- a/vorlesungen/slides/7/drehung.tex
+++ b/vorlesungen/slides/7/drehung.tex
@@ -1,132 +1,132 @@
-%
-% 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}
-{\color{blue}Längen}, {\color<2->{blue}Winkel},
-{\color<2->{darkgreen}Orientierung}
-erhalten
-\uncover<2->{
-\[
-\operatorname{SO}(2)
-=
-{\color{blue}\operatorname{O}(2)}
-\cap
-{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})}
-\]}
-\vspace{-20pt}
-\end{block}
-\uncover<3->{%
-\begin{block}{Zusammensetzung}
-Eine Drehung muss als Zusammensetzung geschrieben werden können:
-\[
-D_{\alpha} 
-=
-\begin{pmatrix}
-\cos\alpha & -\sin\alpha\\
-\sin\alpha &\phantom{-}\cos\alpha
-\end{pmatrix}
-=
-DST
-\]
-\end{block}}
-\vspace{-10pt}
-\uncover<12->{%
-\begin{block}{Beispiel}
-\vspace{-12pt}
-\[
-D_{60^\circ}
-=
-{\tiny
-\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
-\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
-\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix}
-}
-\]
-\end{block}}
-\end{column}
-\begin{column}{0.58\textwidth}
-\uncover<4->{%
-\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}
-\\
-&\uncover<5->{=
-\begin{pmatrix}
-c^{-1}&0\\
-  0   &c
-\end{pmatrix}
-\begin{pmatrix}
-1-st&-s\\
-   t& 1
-\end{pmatrix}
-}
-\\
-&\uncover<6->{=
-\begin{pmatrix}
-{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\
-{\color<9->{blue}ct}&{\color<8->{red}c}
-\end{pmatrix}}
-\uncover<7->{=
-\begin{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}}
-\vspace{-10pt}
-\uncover<7->{%
-\begin{block}{Koeffizientenvergleich}
-\vspace{-15pt}
-\begin{align*}
-\uncover<8->{
-{\color{red} c}
-&=
-{\color{red}\cos\alpha }}
-&&
-&
-\uncover<9->{
-{\color{blue}
-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} (1-st)c^{-t}}
-&=
-\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}
-}
-\end{align*}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% 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}

+{\color{blue}Längen}, {\color<2->{blue}Winkel},

+{\color<2->{darkgreen}Orientierung}

+erhalten

+\uncover<2->{

+\[

+\operatorname{SO}(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:

+\[

+D_{\alpha} 

+=

+\begin{pmatrix}

+\cos\alpha & -\sin\alpha\\

+\sin\alpha &\phantom{-}\cos\alpha

+\end{pmatrix}

+=

+DST

+\]

+\end{block}}

+\vspace{-10pt}

+\uncover<12->{%

+\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\\\sqrt{3}&1\end{pmatrix}

+}

+\]

+\end{block}}

+\end{column}

+\begin{column}{0.58\textwidth}

+\uncover<4->{%

+\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}

+\\

+&\uncover<5->{=

+\begin{pmatrix}

+c^{-1}&0\\

+  0   &c

+\end{pmatrix}

+\begin{pmatrix}

+1-st&-s\\

+   t& 1

+\end{pmatrix}

+}

+\\

+&\uncover<6->{=

+\begin{pmatrix}

+{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\

+{\color<9->{blue}ct}&{\color<8->{red}c}

+\end{pmatrix}}

+\uncover<7->{=

+\begin{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}}

+\vspace{-10pt}

+\uncover<7->{%

+\begin{block}{Koeffizientenvergleich}

+\vspace{-15pt}

+\begin{align*}

+\uncover<8->{

+{\color{red} c}

+&=

+{\color{red}\cos\alpha }}

+&&

+&

+\uncover<9->{

+{\color{blue}

+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} (1-st)c^{-t}}

+&=

+\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $}

+}

+\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
index 5171085..e9699a6 100644
--- a/vorlesungen/slides/7/einparameter.tex
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -1,93 +1,93 @@
-%
-% 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 {\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*}
-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}
-\uncover<5->{%
-\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}
-\uncover<6->{%
-\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}
-\uncover<7->{%
-\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
+%

+% 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 {\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*}

+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}

+\uncover<5->{%

+\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}

+\uncover<6->{%

+\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}

+\uncover<7->{%

+\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/images/Makefile b/vorlesungen/slides/7/images/Makefile
index cc67c8a..9de1c34 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -1,19 +1,19 @@
-#
-# 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
-
-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
-
-
+#

+# 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

+

+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/common.inc b/vorlesungen/slides/7/images/common.inc
index 0e27c9a..b028956 100644
--- a/vorlesungen/slides/7/images/common.inc
+++ b/vorlesungen/slides/7/images/common.inc
@@ -1,70 +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)
-
+//

+// 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
index 8c2211e..44a5ac5 100644
--- a/vorlesungen/slides/7/images/commutator.ini
+++ b/vorlesungen/slides/7/images/commutator.ini
@@ -1,8 +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
-
+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
index 5a448db..3f5ea17 100644
--- a/vorlesungen/slides/7/images/commutator.m
+++ b/vorlesungen/slides/7/images/commutator.m
@@ -1,111 +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
+#

+# 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
index 9ae11b9..8229a06 100644
--- a/vorlesungen/slides/7/images/commutator.pov
+++ b/vorlesungen/slides/7/images/commutator.pov
@@ -1,59 +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
-	}
-}
-
+//

+// 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/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov
index 07aec19..62306f8 100644
--- a/vorlesungen/slides/7/images/rodriguez.pov
+++ b/vorlesungen/slides/7/images/rodriguez.pov
@@ -1,118 +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)
+//

+// 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/kommutator.tex b/vorlesungen/slides/7/kommutator.tex
index 84bf034..9000160 100644
--- a/vorlesungen/slides/7/kommutator.tex
+++ b/vorlesungen/slides/7/kommutator.tex
@@ -1,166 +1,166 @@
-%
-% template.tex -- slide template
-%
-% (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{Kommutator in $\operatorname{SO}(3)$}
-\vspace{-20pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\t{14.0cm}
-\ifthenelse{\boolean{presentation}}{
-\only<1>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};}
-\only<2>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};}
-\only<3>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};}
-\only<4>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};}
-\only<5>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};}
-\only<6>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};}
-\only<7>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c07.jpg}};}
-\only<8>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};}
-\only<9>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c09.jpg}};}
-\only<10>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c10.jpg}};}
-\only<11>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c11.jpg}};}
-\only<12>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c12.jpg}};}
-\only<13>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c13.jpg}};}
-\only<14>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c14.jpg}};}
-\only<15>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c15.jpg}};}
-\only<16>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c16.jpg}};}
-\only<17>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c17.jpg}};}
-\only<18>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c18.jpg}};}
-\only<19>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c19.jpg}};}
-\only<20>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c20.jpg}};}
-\only<21>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c21.jpg}};}
-\only<22>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c22.jpg}};}
-\only<23>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c23.jpg}};}
-\only<24>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c24.jpg}};}
-\only<25>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c25.jpg}};}
-\only<26>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c26.jpg}};}
-\only<27>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c27.jpg}};}
-\only<28>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c28.jpg}};}
-\only<29>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c29.jpg}};}
-\only<30>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c30.jpg}};}
-\only<31>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c31.jpg}};}
-\only<32>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c32.jpg}};}
-\only<33>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c33.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c34.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c35.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c36.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c37.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c38.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c40.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c42.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c43.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c44.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c45.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c46.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c48.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c49.jpg}};}
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-\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};}
-\only<53>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};}
-\only<54>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};}
-\only<55>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};}
-\only<56>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};}
-\only<57>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};}
-\only<58>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};}
-\only<59>{\node at (0,0) {
-\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};}
-}{}
-\only<60>{\node at (0,0) {
-\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}
-\egroup
+%

+% template.tex -- slide template

+%

+% (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{Kommutator in $\operatorname{SO}(3)$}

+\vspace{-20pt}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\t{14.0cm}

+\ifthenelse{\boolean{presentation}}{

+\only<1>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};}

+\only<2>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};}

+\only<3>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};}

+\only<4>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};}

+\only<5>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};}

+\only<6>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};}

+\only<7>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c07.jpg}};}

+\only<8>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};}

+\only<9>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c09.jpg}};}

+\only<10>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c10.jpg}};}

+\only<11>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c11.jpg}};}

+\only<12>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c12.jpg}};}

+\only<13>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c13.jpg}};}

+\only<14>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c14.jpg}};}

+\only<15>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c15.jpg}};}

+\only<16>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c16.jpg}};}

+\only<17>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c17.jpg}};}

+\only<18>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c18.jpg}};}

+\only<19>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c19.jpg}};}

+\only<20>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c20.jpg}};}

+\only<21>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c21.jpg}};}

+\only<22>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c22.jpg}};}

+\only<23>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c23.jpg}};}

+\only<24>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c24.jpg}};}

+\only<25>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c25.jpg}};}

+\only<26>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c26.jpg}};}

+\only<27>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c27.jpg}};}

+\only<28>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c28.jpg}};}

+\only<29>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c29.jpg}};}

+\only<30>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c30.jpg}};}

+\only<31>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c31.jpg}};}

+\only<32>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c32.jpg}};}

+\only<33>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c33.jpg}};}

+\only<34>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c34.jpg}};}

+\only<35>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c35.jpg}};}

+\only<36>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c36.jpg}};}

+\only<37>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c37.jpg}};}

+\only<38>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c38.jpg}};}

+\only<39>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c39.jpg}};}

+\only<40>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c40.jpg}};}

+\only<41>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c41.jpg}};}

+\only<42>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c42.jpg}};}

+\only<43>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c43.jpg}};}

+\only<44>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c44.jpg}};}

+\only<45>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c45.jpg}};}

+\only<46>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c46.jpg}};}

+\only<47>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c47.jpg}};}

+\only<48>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c48.jpg}};}

+\only<49>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c49.jpg}};}

+\only<50>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c50.jpg}};}

+\only<51>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};}

+\only<52>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};}

+\only<53>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};}

+\only<54>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};}

+\only<55>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};}

+\only<56>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};}

+\only<57>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};}

+\only<58>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};}

+\only<59>{\node at (0,0) {

+\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};}

+}{}

+\only<60>{\node at (0,0) {

+\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}

+\egroup

diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex
index e0690eb..bca8417 100644
--- a/vorlesungen/slides/7/kurven.tex
+++ b/vorlesungen/slides/7/kurven.tex
@@ -1,104 +1,104 @@
-%
-% 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}
-\[
-\gamma
-\colon
-I=[a,b] \to \mathbb{R}^n
-:
-t\mapsto \gamma(t)
-\uncover<2->{
-=
-\begin{pmatrix}
-x_1(t)\\
-x_2(t)\\
-\vdots\\
-x_n(t)
-\end{pmatrix}
-}
-\]
-\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] {$\gamma(t)$};
-
-\uncover<4->{
-	\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
-	\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
-}
-
-\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
-\[
-\frac{d}{dt}\gamma(t)
-=
-\dot{\gamma}(t)
-=
-\begin{pmatrix}
-\dot{x}_1(t)\\
-\dot{x}_2(t)\\
-\vdots\\
-\dot{x}_n(t)
-\end{pmatrix}
-\]
-\uncover<5->{%
-Lineare Approximation:
-\[
-\gamma(t+h)
-=
-\gamma(t)
-+
-\dot{\gamma}(t) \cdot h
-+
-o(h)
-\]}%
-\vspace{-10pt}
-\begin{itemize}
-\item<6->
-Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
-in $\mathbb{R}^n$ liegen
-\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}
-\egroup
+%

+% 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)

+\uncover<2->{

+=

+\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] {$\gamma(t)$};

+

+\uncover<4->{

+	\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);

+	\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};

+}

+

+\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

+\[

+\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}

+\]

+\uncover<5->{%

+Lineare Approximation:

+\[

+\gamma(t+h)

+=

+\gamma(t)

++

+\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

+\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}

+\egroup

diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex
index 574467b..59c9121 100644
--- a/vorlesungen/slides/7/liealgebra.tex
+++ b/vorlesungen/slides/7/liealgebra.tex
@@ -1,85 +1,85 @@
-%
-% 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}
-\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}}
-\uncover<8->{%
-\begin{block}{Nachprüfen}
-$[A,B]\in LG$
-für $A,B\in LG$
-\end{block}}
-\end{column}
-\begin{column}{0.56\textwidth}
-\uncover<9->{%
-\begin{block}{Algebraische Eigenschaften}
-\begin{itemize}
-\item<10-> antisymmetrisch: $[A,B]=-[B,A]$
-\item<11-> Jacobi-Identität
-\[
-[A,[B,C]]+
-[B,[C,A]]+
-[C,[A,B]]
-= 0
-\]
-\end{itemize}
-\vspace{-13pt}
-\uncover<12->{%
-{\usebeamercolor[fg]{title}
-Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
-}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% 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}

+\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}}

+\uncover<8->{%

+\begin{block}{Nachprüfen}

+$[A,B]\in LG$

+für $A,B\in LG$

+\end{block}}

+\end{column}

+\begin{column}{0.56\textwidth}

+\uncover<9->{%

+\begin{block}{Algebraische Eigenschaften}

+\begin{itemize}

+\item<10-> antisymmetrisch: $[A,B]=-[B,A]$

+\item<11-> Jacobi-Identität

+\[

+[A,[B,C]]+

+[B,[C,A]]+

+[C,[A,B]]

+= 0

+\]

+\end{itemize}

+\vspace{-13pt}

+\uncover<12->{%

+{\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
index 077dc9d..f88042a 100644
--- a/vorlesungen/slides/7/mannigfaltigkeit.tex
+++ b/vorlesungen/slides/7/mannigfaltigkeit.tex
@@ -1,46 +1,46 @@
-%
-% 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<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
-\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
-\[
-\varphi_\beta\circ\varphi_\alpha^{-1}
-\colon
-\varphi_\alpha(U_\alpha\cap U_\beta)
-\to
-\varphi_\beta(U_\alpha\cap U_\beta)
-\]
-\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}}
-\vspace{-3pt}
-\uncover<6->{%
-\begin{block}{Lie-Gruppe}
-Gruppe und Mannigfaltigkeit
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% 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<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

+\varphi_\alpha(U_\alpha\cap U_\beta)

+\to

+\varphi_\beta(U_\alpha\cap U_\beta)

+\]

+\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}}

+\vspace{-3pt}

+\uncover<6->{%

+\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
index 52c8e4a..afc67c5 100644
--- a/vorlesungen/slides/7/parameter.tex
+++ b/vorlesungen/slides/7/parameter.tex
@@ -1,107 +1,107 @@
-%
-% parameter.tex -- Parametrisierung der Matrizen
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-\definecolor{darkyellow}{rgb}{1,0.8,0}
-\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}
-\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}
-\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
-}
-\[
-\uncover<9->{
-{\color{red}\vec{x}}
-\mapsto
-}
-\uncover<10->{
-({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})
-\cos\omega
-+
-}
-\uncover<11->{
-({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega
-+
-}
-\uncover<9->{
-{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) 
-}
-\]
-\vspace{-40pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\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) {$\vec{x}\times\vec{k}$};
-}
-\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{column}
-\end{columns}
-\vspace{-15pt}
-\uncover<5->{%
-{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
-dreidimensionale Gruppe}
-\end{frame}
-\egroup
+%

+% parameter.tex -- Parametrisierung der Matrizen

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\bgroup

+\definecolor{darkgreen}{rgb}{0,0.6,0}

+\definecolor{darkyellow}{rgb}{1,0.8,0}

+\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}

+\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}

+\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

+}

+\[

+\uncover<9->{

+{\color{red}\vec{x}}

+\mapsto

+}

+\uncover<10->{

+({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}})

+\cos\omega

++

+}

+\uncover<11->{

+({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega

++

+}

+\uncover<9->{

+{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) 

+}

+\]

+\vspace{-40pt}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\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) {$\vec{x}\times\vec{k}$};

+}

+\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{column}

+\end{columns}

+\vspace{-15pt}

+\uncover<5->{%

+{\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
index 66b8d27..d74b7d0 100644
--- a/vorlesungen/slides/7/semi.tex
+++ b/vorlesungen/slides/7/semi.tex
@@ -1,117 +1,117 @@
-%
-% 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}
-\uncover<2->{%
-\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}
-\uncover<3->{%
-\begin{block}{Verknüpfung}
-\vspace{-15pt}
-\begin{align*}
-(e^{s_1},t_1)(e^{s_2},t_2)x
-&\uncover<4->{=
-(e^{s_1},t_1)(e^{s_2}x+t_2)}
-\\
-&\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)}
-\end{align*}
-\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}
-&\uncover<8->{=
-(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}
-\\
-&\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{column}
-\end{columns}
-\vspace{-10pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\uncover<11->{%
-\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}
-\uncover<12->{%
-\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
+%

+% 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}

+\uncover<2->{%

+\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}

+\uncover<3->{%

+\begin{block}{Verknüpfung}

+\vspace{-15pt}

+\begin{align*}

+(e^{s_1},t_1)(e^{s_2},t_2)x

+&\uncover<4->{=

+(e^{s_1},t_1)(e^{s_2}x+t_2)}

+\\

+&\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)}

+\end{align*}

+\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}

+&\uncover<8->{=

+(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)}

+\\

+&\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{column}

+\end{columns}

+\vspace{-10pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\uncover<11->{%

+\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}

+\uncover<12->{%

+\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
index a65b4f6..58e87a1 100644
--- a/vorlesungen/slides/7/sl2.tex
+++ b/vorlesungen/slides/7/sl2.tex
@@ -1,242 +1,242 @@
-%
-% 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);
-	}
-	\ifthenelse{\boolean{presentation}}{
-	\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}
-	$};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
-	\clip (-2.1,-2.1) rectangle (2.3,2.3);
-	\foreach \x in {-2,...,2}{
-		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
-	}
-	\foreach \y in {-2,...,2}{
-		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
-	}
-	\ifthenelse{\boolean{presentation}}{
-	\foreach \d in {11,...,17}{
-		\only<\d>{
-			\pgfmathparse{(\d-11)/10}
-			\xdef\t{\pgfmathresult}
-			\fill[color=red!40,opacity=0.5]
-				({-1+\t*(-1)},{-1})
-				--
-				({1+\t*(-1)},{-1})
-				--
-				({1+\t},{1})
-				--
-				({-1+\t},{1})
-				-- cycle;
-			\foreach \x in {-3,...,3}{
-				\draw[color=red,line width=0.3pt]
-					({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
-			}
-			\foreach \y in {-3,...,3}{
-				\draw[color=red,line width=0.3pt]
-					({-3+\t*\y},\y) -- ({3+\t*\y},\y);
-			}
-		}
-	}
-	}{}
-	\uncover<18->{
-		\xdef\t{0.6}
-		\fill[color=red!40,opacity=0.5]
-			({-1+\t*(-1)},{-1})
-			--
-			({1+\t*(-1)},{-1})
-			--
-			({1+\t},{1})
-			--
-			({-1+\t},{1})
-			-- cycle;
-		\foreach \x in {-3,...,3}{
-			\draw[color=red,line width=0.3pt]
-				({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
-		}
-		\foreach \y in {-3,...,3}{
-			\draw[color=red,line width=0.3pt]
-				({-3+\t*\y},\y) -- ({3+\t*\y},\y);
-		}
-	}
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$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}
-	$};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\begin{column}{0.33\textwidth}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick,scale=\s]
-\fill[color=blue!20] (-1,-1) rectangle (1,1);
-\begin{scope}
-	\clip (-2.1,-2.1) rectangle (2.3,2.3);
-	\foreach \x in {-2,...,2}{
-		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
-	}
-	\foreach \y in {-2,...,2}{
-		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
-	}
-	\ifthenelse{\boolean{presentation}}{
-	\foreach \d in {18,...,24}{
-		\only<\d>{
-			\pgfmathparse{(\d-18)/10}
-			\xdef\t{\pgfmathresult}
-			\fill[color=red!40,opacity=0.5]
-				(-1,{\t*(-1)-1})
-				--
-				(1,{\t*1-1})
-				--
-				(1,{\t*1+1})
-				--
-				(-1,{\t*(-1)+1})
-				-- cycle;
-			\foreach \x in {-3,...,3}{
-				\draw[color=red,line width=0.3pt]
-					(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
-			}
-			\foreach \y in {-3,...,3}{
-				\draw[color=red,line width=0.3pt]
-					(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
-			}
-		}
-	}
-	}{}
-	\uncover<25->{
-		\xdef\t{0.6}
-		\fill[color=red!40,opacity=0.5]
-			(-1,{\t*(-1)-1})
-			--
-			(1,{\t*1-1})
-			--
-			(1,{\t*1+1})
-			--
-			(-1,{\t*(-1)+1})
-			-- cycle;
-		\foreach \x in {-3,...,3}{
-			\draw[color=red,line width=0.3pt]
-				(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
-		}
-		\foreach \y in {-3,...,3}{
-			\draw[color=red,line width=0.3pt]
-				(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
-		}
-	}
-\end{scope}
-\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
-\uncover<18->{%
-\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
-	\node at (0,-2.1) {$
-	T
-	=
-	\begin{pmatrix} 1&0\\t&1\end{pmatrix}
-	$};
-}
-\end{tikzpicture}
-\end{center}
-\end{column}
-\end{columns}}
-\end{frame}
-\egroup
+%

+% 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);

+	}

+	\ifthenelse{\boolean{presentation}}{

+	\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);

+	}

+	\ifthenelse{\boolean{presentation}}{

+	\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);

+	}

+	\ifthenelse{\boolean{presentation}}{

+	\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});

+			}

+		}

+	}

+	}{}

+	\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:$y$}];

+\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
index 35d62d8..8931a24 100644
--- a/vorlesungen/slides/7/symmetrien.tex
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -1,145 +1,145 @@
-%
-% 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<2->
-Ebenen-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
-{\tiny
-\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\uncover<4->{\!,\;
-\vec{x}
-\mapsto
-\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
-}
-\]
-\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)});
-\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=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,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<5->
-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}
-\uncover<6->{%
-\begin{block}{Kontinuierliche Symmetrien}
-\begin{itemize}
-\item<7-> Translation:
-\(
-\vec{x} \mapsto \vec{x} + \vec{t}
-\)
-\item<8-> Drehung:
-\vspace{-3pt}
-\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,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}
-\[
-\uncover<9->{%
-\begin{pmatrix}x\\y\end{pmatrix}
-\mapsto
-\begin{pmatrix}
-{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
-{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
-\end{pmatrix}
-\begin{pmatrix}x\\y\end{pmatrix}
-}
-\]
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<10->{%
-\begin{block}{Definition}
-Längen/Winkel bleiben erhalten
-\\
-\uncover<11->{%
-$\Rightarrow$ $\exists$ Erhaltungsgrösse}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% 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<2->

+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*}

+}

+\uncover<4->{\!,\;

+\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)});

+\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=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,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<5->

+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}

+\uncover<6->{%

+\begin{block}{Kontinuierliche Symmetrien}

+\begin{itemize}

+\item<7-> Translation:

+\(

+\vec{x} \mapsto \vec{x} + \vec{t}

+\)

+\item<8-> Drehung:

+\vspace{-3pt}

+\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,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}

+{\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}

+\uncover<10->{%

+\begin{block}{Definition}

+Längen/Winkel bleiben erhalten

+\\

+\uncover<11->{%

+$\Rightarrow$ $\exists$ Erhaltungsgrösse}

+\end{block}}

+\end{column}

+\end{columns}

+\end{frame}

+\egroup

diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc
index e2271b8..6454463 100644
--- a/vorlesungen/slides/Makefile.inc
+++ b/vorlesungen/slides/Makefile.inc
@@ -1,18 +1,18 @@
-#
-# Makefile.inc -- additional depencencies
-#
-# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-#
-include ../slides/0/Makefile.inc
-include ../slides/1/Makefile.inc
-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) $(chapter7) $(chapter8) $(chapter9)
+#

+# Makefile.inc -- additional depencencies

+#

+# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil

+#

+include ../slides/0/Makefile.inc

+include ../slides/1/Makefile.inc

+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) $(chapter7) $(chapter8) $(chapter9)

diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 4673f76..6c102f2 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -1,39 +1,39 @@
-%
-% test.tex collection of all slides
-%
-% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-
-\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}
-%\folie{7/drehanim.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}
-% Kommutator
-%\folie{7/kommutator.tex}
-
-\section{Exponentialabbildung}
-% Differentialgleichung für die Exponentialabbildung
-%\folie{7/dg.tex}
-
+%

+% test.tex collection of all slides

+%

+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil

+%

+

+\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}

+%\folie{7/drehanim.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}

+% Kommutator

+%\folie{7/kommutator.tex}

+

+\section{Exponentialabbildung}

+% Differentialgleichung für die Exponentialabbildung

+%\folie{7/dg.tex}

+