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authorAndreas Müller <andreas.mueller@ost.ch>2021-06-14 07:26:10 +0200
committerGitHub <noreply@github.com>2021-06-14 07:26:10 +0200
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tree18e61c7d69883a1c9b69098b7d36856abaed5c1e /vorlesungen/slides/7
parentDelete buch.pdf (diff)
parentFix references.bib (diff)
downloadSeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.tar.gz
SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.zip
Merge branch 'master' into master
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-rw-r--r--vorlesungen/slides/7/Makefile.inc35
-rw-r--r--vorlesungen/slides/7/ableitung.tex68
-rw-r--r--vorlesungen/slides/7/algebraisch.tex115
-rw-r--r--vorlesungen/slides/7/bch.tex76
-rw-r--r--vorlesungen/slides/7/chapter.tex32
-rw-r--r--vorlesungen/slides/7/dg.tex92
-rw-r--r--vorlesungen/slides/7/drehanim.tex155
-rw-r--r--vorlesungen/slides/7/drehung.tex132
-rw-r--r--vorlesungen/slides/7/einparameter.tex93
-rw-r--r--vorlesungen/slides/7/exponentialreihe.tex24
-rw-r--r--vorlesungen/slides/7/haar.tex84
-rw-r--r--vorlesungen/slides/7/hopf.tex69
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-rw-r--r--vorlesungen/slides/7/integration.tex66
-rw-r--r--vorlesungen/slides/7/interpolation.tex112
-rw-r--r--vorlesungen/slides/7/kommutator.tex166
-rw-r--r--vorlesungen/slides/7/kurven.tex104
-rw-r--r--vorlesungen/slides/7/liealgbeispiel.tex78
-rw-r--r--vorlesungen/slides/7/liealgebra.tex85
-rw-r--r--vorlesungen/slides/7/logarithmus.tex82
-rw-r--r--vorlesungen/slides/7/mannigfaltigkeit.tex46
-rw-r--r--vorlesungen/slides/7/parameter.tex107
-rw-r--r--vorlesungen/slides/7/qdreh.tex110
-rw-r--r--vorlesungen/slides/7/quaternionen.tex74
-rw-r--r--vorlesungen/slides/7/semi.tex117
-rw-r--r--vorlesungen/slides/7/sl2.tex242
-rw-r--r--vorlesungen/slides/7/symmetrien.tex145
-rw-r--r--vorlesungen/slides/7/ueberlagerung.tex98
-rw-r--r--vorlesungen/slides/7/vektorlie.tex206
-rw-r--r--vorlesungen/slides/7/zusammenhang.tex99
102 files changed, 3528 insertions, 0 deletions
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
new file mode 100644
index 0000000..ffd5091
--- /dev/null
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -0,0 +1,35 @@
+#
+# 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/liealgbeispiel.tex \
+ ../slides/7/vektorlie.tex \
+ ../slides/7/kommutator.tex \
+ ../slides/7/bch.tex \
+ ../slides/7/dg.tex \
+ ../slides/7/interpolation.tex \
+ ../slides/7/exponentialreihe.tex \
+ ../slides/7/logarithmus.tex \
+ ../slides/7/zusammenhang.tex \
+ ../slides/7/quaternionen.tex \
+ ../slides/7/qdreh.tex \
+ ../slides/7/ueberlagerung.tex \
+ ../slides/7/hopf.tex \
+ ../slides/7/haar.tex \
+ ../slides/7/integration.tex \
+ ../slides/7/chapter.tex
+
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex
new file mode 100644
index 0000000..12f9084
--- /dev/null
+++ b/vorlesungen/slides/7/ableitung.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex
new file mode 100644
index 0000000..31d209a
--- /dev/null
+++ b/vorlesungen/slides/7/algebraisch.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/bch.tex b/vorlesungen/slides/7/bch.tex
new file mode 100644
index 0000000..0148dc4
--- /dev/null
+++ b/vorlesungen/slides/7/bch.tex
@@ -0,0 +1,76 @@
+%
+% bch.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Baker-Campbell-Hausdorff-Formel}
+$g(t),h(t)\in G
+\uncover<2->{\Rightarrow \exists A,B\in LG\text{ mit }
+g(t)=\exp At, h(t)=\exp Bt}$
+\uncover<3->{%
+\begin{align*}
+g(t)
+&=
+I + At + \frac{A^2t^2}{2!} + \frac{A^3t^3}{3!} + \dots,
+&
+h(t)
+&=
+I + Bt + \frac{B^2t^2}{2!} + \frac{B^3t^3}{3!} + \dots
+\end{align*}}
+\uncover<5->{%
+\begin{block}{Kommutator in G: $c(t) = g(t)h(t)g(t)^{-1}h(t)^{-1}$}
+\begin{align*}
+\uncover<6->{c(t)
+&=
+\biggl(
+ {\color<7,9-11,13-15,19-21>{red}I}
+ + {\color<8,16-19>{red}A}t
+ + \frac{{\color<12>{red}A^2}t^2}{2!}
+ + \dots
+\biggr)
+\biggl(
+ {\color<7,8,10-12,14-15,17-18,21>{red}I}
+ + {\color<9,16,19-20>{red}B}t
+ + \frac{{\color<13>{red}B^2}t^2}{2!}
+ + \dots
+\biggr)
+\exp(-{\color<10,14,17,19,21>{red}A}t)
+\exp(-{\color<11,15,18,20-21>{red}B}t)
+}
+\\
+&\uncover<7->{={\color<7>{red}I}}
+\uncover<8->{+t(
+ \uncover<8->{ {\color<8>{red}A}}
+ \uncover<9->{+ {\color<9>{red}B}}
+ \uncover<10->{- {\color<10>{red}A}}
+ \uncover<11->{- {\color<11>{red}B}}
+)}
+\uncover<12->{+\frac{t^2}{2!}(
+ \uncover<12->{ {\color<12>{red}A^2}}
+ \uncover<13->{+ {\color<13>{red}B^2}}
+ \uncover<14->{+ {\color<14>{red}A^2}}
+ \uncover<15->{+ {\color<15>{red}B^2}}
+)}
+\\
+&\phantom{\mathstrut=I}
+\uncover<12->{+t^2(
+ \uncover<16->{ {\color<16>{red}AB}}
+ \uncover<17->{- {\color<17>{red}A^2}}
+ \uncover<18->{- {\color<18>{red}AB}}
+ \uncover<19->{- {\color<19>{red}BA}}
+ \uncover<20->{- {\color<20>{red}B^2}}
+ \uncover<21->{+ {\color<21>{red}AB}}
+)}
+\uncover<22->{+t^3(\dots)+\dots}
+\\
+&\uncover<23->{=
+I + \frac{t^2}{2}[A,B] + o(t^3)
+}
+\end{align*}}
+\end{block}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
new file mode 100644
index 0000000..3736e0f
--- /dev/null
+++ b/vorlesungen/slides/7/chapter.tex
@@ -0,0 +1,32 @@
+%
+% 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/liealgbeispiel.tex}
+\folie{7/vektorlie.tex}
+\folie{7/kommutator.tex}
+\folie{7/bch.tex}
+\folie{7/dg.tex}
+\folie{7/interpolation.tex}
+\folie{7/exponentialreihe.tex}
+\folie{7/logarithmus.tex}
+\folie{7/zusammenhang.tex}
+\folie{7/quaternionen.tex}
+\folie{7/qdreh.tex}
+\folie{7/ueberlagerung.tex}
+\folie{7/hopf.tex}
+\folie{7/haar.tex}
+\folie{7/integration.tex}
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex
new file mode 100644
index 0000000..f9528a4
--- /dev/null
+++ b/vorlesungen/slides/7/dg.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex
new file mode 100644
index 0000000..ac136f1
--- /dev/null
+++ b/vorlesungen/slides/7/drehanim.tex
@@ -0,0 +1,155 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) }
+
+\makeatletter
+\hoffset=-2cm
+\advance\textwidth2cm
+\hsize\textwidth
+\columnwidth\textwidth
+\makeatother
+
+\begin{frame}[t,plain]
+\vspace{-5pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=white] (-4,-4) rectangle (9,4.5);
+
+\def\a{60}
+
+\pgfmathparse{tan(\a)}
+\xdef\T{\pgfmathresult}
+
+\pgfmathparse{-sin(\a)*cos(\a)}
+\xdef\S{\pgfmathresult}
+
+\pgfmathparse{1/cos(\a)}
+\xdef\E{\pgfmathresult}
+
+\def\N{20}
+\pgfmathparse{2*\N}
+\xdef\Nzwei{\pgfmathresult}
+\pgfmathparse{3*\N}
+\xdef\Ndrei{\pgfmathresult}
+
+\node at (4.2,4.2) [below right] {\begin{minipage}{7cm}
+\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$}
+\begin{itemize}
+\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized
+as shear mappings and axis rescalings
+\[
+A=
+\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix}
+\begin{pmatrix}1&s\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\t&1\end{pmatrix}
+\]
+\item Most rotations can be decomposed into a product of
+shear mappings and axis rescalings
+\end{itemize}
+\end{block}
+\end{minipage}};
+
+\foreach \d in {1,2,...,\Ndrei}{
+ % Scherung in Y-Richtung
+ \ifnum \d>\N
+ \pgfmathparse{\T}
+ \else
+ \pgfmathparse{\T*(\d-1)/(\N-1)}
+ \fi
+ \xdef\t{\pgfmathresult}
+
+ % Scherung in X-Richtung
+ \ifnum \d>\Nzwei
+ \xdef\s{\S}
+ \else
+ \ifnum \d<\N
+ \xdef\s{0}
+ \else
+ \ifnum \d=\N
+ \xdef\s{0}
+ \else
+ \pgfmathparse{\S*(\d-\N-1)/(\N-1)}
+ \xdef\s{\pgfmathresult}
+ \fi
+ \fi
+ \fi
+
+ % Reskalierung der Achsen
+ \ifnum \d>\Nzwei
+ \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))}
+ \else
+ \pgfmathparse{1}
+ \fi
+ \xdef\e{\pgfmathresult}
+
+ % Matrixelemente
+ \pgfmathparse{(\e)*((\s)*(\t)+1)}
+ \xdef\A{\pgfmathresult}
+
+ \pgfmathparse{(\e)*(\s)}
+ \xdef\B{\pgfmathresult}
+
+ \pgfmathparse{(\t)/(\e)}
+ \xdef\C{\pgfmathresult}
+
+ \pgfmathparse{1/(\e)}
+ \xdef\D{\pgfmathresult}
+
+ \only<\d>{
+ \node at (5.0,-0.9) [below right] {$
+ \begin{aligned}
+ t &= \t \\
+ s &= \s \\
+ d &= \e \\
+ D &= \begin{pmatrix}
+ \A&\B\\
+ \C&\D
+ \end{pmatrix}
+ \qquad
+ \only<60>{\checkmark}
+ \end{aligned}
+ $};
+ }
+
+ \begin{scope}
+ \clip (-4.05,-4.05) rectangle (4.05,4.05);
+ \only<\d>{
+ \foreach \x in {-6,...,6}{
+ \draw[color=blue,line width=0.5pt]
+ \punkt{\x}{-12} -- \punkt{\x}{12};
+ }
+ \foreach \y in {-12,...,12}{
+ \draw[color=darkgreen,line width=0.5pt]
+ \punkt{-6}{\y} -- \punkt{6}{\y};
+ }
+
+ \foreach \r in {1,2,3,4}{
+ \draw[color=red] plot[domain=0:359,samples=360]
+ ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))})
+ --
+ cycle;
+ }
+ }
+ \end{scope}
+
+% \uncover<\d>{
+% \node at (5,4) {\d};
+% }
+}
+
+\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}];
+\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}];
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
new file mode 100644
index 0000000..02201d4
--- /dev/null
+++ b/vorlesungen/slides/7/drehung.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
new file mode 100644
index 0000000..a32affd
--- /dev/null
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/exponentialreihe.tex b/vorlesungen/slides/7/exponentialreihe.tex
new file mode 100644
index 0000000..b1aeda6
--- /dev/null
+++ b/vorlesungen/slides/7/exponentialreihe.tex
@@ -0,0 +1,24 @@
+%
+% exponentialreihe.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Exponentialreihe}
+\begin{align*}
+h(s) &= \exp(tA_0 + sB) = \sum_{k=0}^\infty \frac{1}{k!} (tA_0 + sB)^k
+\\
+&=
+I + (tA_0 + sB) + \frac{1}{2!}(t^2A_0^2 + ts(A_0B + BA_0) + s^2B^2)
++ \frac{1}{3!}(t^3A_0^3 + t^2s(A_0^2B + A_0BA_0 + BA_0^2) + \dots)
++ \dots
+\\
+\frac{dg(s)}{ds}
+&=
+B + \frac1{2!}t(A_0B+BA_0) + \frac{1}{3!}t^2(A_0^2B+A_0BA_0+BA_0^2) + \dots
+\end{align*}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/haar.tex b/vorlesungen/slides/7/haar.tex
new file mode 100644
index 0000000..454dd69
--- /dev/null
+++ b/vorlesungen/slides/7/haar.tex
@@ -0,0 +1,84 @@
+%
+% haar.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Haar-Mass}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Invariantes Mass}
+Auf jeder lokalkompakten Gruppe $G$ gibt es ein \only<2->{invariantes }%
+Integral
+\begin{align*}
+\uncover<2->{\text{rechts:}}&&
+\int_G f(g)\,d\mu(g)
+&\uncover<2->{=
+\int_G f(gh)\,d\mu(g)}
+\\
+\uncover<3->{
+\text{links:}&&
+\int_G f(g)\,d\mu(g)
+&=
+\int_G f(hg)\,d\mu(g)}
+\end{align*}
+
+\end{block}
+\uncover<7->{%
+\begin{block}{Modulus-Funktion}
+$\mu$ linksinvariant, dann ist die Rechtsverschiebung ebenfalls
+linksinvariant
+\[
+\int_G f(gh) \, d\mu(g)
+\uncover<8->{
+=
+\int_G f(g) \Delta(h)\, d\mu(g)
+}
+\]
+\uncover<9->{$\Delta(h)$ heisst Modulus-Funktion}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Beispiel: $G=\mathbb{R}$}
+\[
+\int_Gf(g)\,d\mu(g)
+=
+\int_{-\infty}^{\infty} f(x)\,dx
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<5->{%
+\begin{block}{Beispiel: $\operatorname{SO}(2)$}
+\[
+\int_{\operatorname{SO}(2)}
+f(g)\,d\mu(g)
+=
+\frac{1}{2\pi}
+\int_{0}^{2\pi} f(D_{\alpha})\,d\alpha
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<6->{%
+\begin{block}{Beispiel: $G$ endlich}
+\[
+\int_G f(g)\,d\mu(g) = \frac{1}{|G|}\sum_{g\in G}f(g)
+\]
+\end{block}}
+\vspace{-10pt}
+\uncover<10->{%
+\begin{block}{Unimodular}
+$\Delta(h)=1$ heisst rechtsinvariant = linksinvariant
+\\
+\uncover<11->{%
+$G$ kompakt $\Rightarrow$ unimodular
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/hopf.tex b/vorlesungen/slides/7/hopf.tex
new file mode 100644
index 0000000..a90737f
--- /dev/null
+++ b/vorlesungen/slides/7/hopf.tex
@@ -0,0 +1,69 @@
+%
+% hopf.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Orbit-Räume}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aktion von $\operatorname{SO}(3)$ auf $S^2$}
+\begin{align*}
+S^2 &= \{x\in\mathbb{R}^3\;|\; |x|=1\}
+\\
+\operatorname{SO}(3) \times S^2 &\to S^2: (g, x) \mapsto gx
+\end{align*}
+\uncover<2->{%
+Allgemein: Aktion von $G$ auf $X$
+\begin{align*}
+\text{links:}&&
+G\times X \to X &: (g,x) \mapsto gx
+\\
+\text{rechts:}&&
+X\times G \to X &: (x,g) \mapsto xg
+\end{align*}}
+\end{block}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{block}{Stabilisator}
+Zu $x\in X$ gibt es eine Untergruppe
+\begin{align*}
+G_x = \{g\in G\;|\; gx=x\},
+\end{align*}
+der {\em Stabilisator} von $x$.
+
+\uncover<4->{%
+Der Stabilisator von $v\in S^2$ ist die Gruppe der Drehungen um
+die Achse $v$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Quotient}
+$G$ operiert von rechts auf $X$
+\[
+X/G = \{ xG \;|\; x\in X\}
+\]
+heisst Quotient
+\end{block}}
+\uncover<6->{
+\begin{block}{$\operatorname{SO}(3)/\operatorname{SO}(2)$}
+Wähle $\operatorname{SO}(2)$ als Drehungen um die $z$-Achse:
+\[
+\operatorname{SO}(3) \to S^2
+:
+g \mapsto \text{letzte Spalte von $g$}
+\]
+\uncover<7->{Daher
+\[
+S^2 \cong \operatorname{SO}(3) / \operatorname{SO}(2)
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
new file mode 100644
index 0000000..6f99bc3
--- /dev/null
+++ b/vorlesungen/slides/7/images/Makefile
@@ -0,0 +1,29 @@
+#
+# Makefile -- Illustrationen zu
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+# 
+all: rodriguez.jpg test.png
+
+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
+
+dreibein/timestamp: interpolation.m
+ octave interpolation.m
+ touch dreibein/timestamp
+
+test.png: test.pov drehung.inc dreibein/d025.inc dreibein/timestamp
+ povray +A0.1 -W1080 -H1080 -Otest.png test.pov
+
+dreibein/d025.inc: dreibein/timestamp
+
+animation:
+ povray +A0.1 -W1080 -H1080 -Ointerpolation/i.png interpolation.ini
diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg
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diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc
new file mode 100644
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+++ b/vorlesungen/slides/7/images/common.inc
@@ -0,0 +1,70 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.025;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+ location <18, 15, -50>
+ look_at <0.0, 0.5, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-40, 30, -50> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#declare l = 1.2;
+
+arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
+arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
+arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
+
diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini
new file mode 100644
index 0000000..8c2211e
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.ini
@@ -0,0 +1,8 @@
+Input_File_Name=commutator.pov
+Initial_Frame=1
+Final_Frame=60
+Initial_Clock=1
+Final_Clock=60
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m
new file mode 100644
index 0000000..5a448db
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.m
@@ -0,0 +1,111 @@
+#
+# commutator.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+X = [
+ 0, 0, 0;
+ 0, 0, -1;
+ 0, 1, 0
+];
+
+Y = [
+ 0, 0, 1;
+ 0, 0, 0;
+ -1, 0, 0
+];
+
+Z = [
+ 0, -1, 0;
+ 1, 0, 0;
+ 0, 0, 0
+];
+
+function retval = Dx(alpha)
+ retval = [
+ 1, 0, 0 ;
+ 0, cos(alpha), -sin(alpha);
+ 0, sin(alpha), cos(alpha)
+ ];
+end
+
+function retval = Dy(beta)
+ retval = [
+ cos(beta), 0, sin(beta);
+ 0, 1, 0 ;
+ -sin(beta), 0, cos(beta)
+ ];
+end
+
+t = 0.9;
+P = Dx(t) * Dy(t)
+Q = Dy(t) * Dx(t)
+P - Q
+(P - Q) * [0;0;1]
+
+function retval = kurven(filename, t)
+ retval = -1;
+ N = 20;
+ fn = fopen(filename, "w");
+ fprintf(fn, "//\n");
+ fprintf(fn, "// %s\n", filename);
+ fprintf(fn, "//\n");
+ fprintf(fn, "#macro XYkurve()\n");
+ for i = (0:N-1)
+ v1 = Dx(t * i / N) * [0;0;1];
+ v2 = Dx(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ for i = (0:N-1)
+ v1 = Dx(t) * Dy(t * i / N) * [0;0;1];
+ v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#end\n");
+ fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#macro YXkurve()\n");
+ for i = (0:N-1)
+ v1 = Dy(t * i / N) * [0;0;1];
+ v2 = Dy(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ for i = (0:N-1)
+ v1 = Dy(t) * Dx(t * i / N) * [0;0;1];
+ v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1];
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1));
+ fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n",
+ v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1));
+ end
+ fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n",
+ v2(1,1), v2(3,1), v2(2,1));
+ fprintf(fn, "#end\n");
+ fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n",
+ v2(1,1), v2(3,1), v2(2,1));
+
+ fclose(fn);
+ retval = 0;
+end
+
+function retval = kurve(i)
+ n = pi / 180;
+ filename = sprintf("f/%04d.inc", i);
+ kurven(filename, n * i);
+end
+
+for i = (1:60)
+ kurve(i);
+end
diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov
new file mode 100644
index 0000000..9ae11b9
--- /dev/null
+++ b/vorlesungen/slides/7/images/commutator.pov
@@ -0,0 +1,59 @@
+//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "common.inc"
+
+sphere { O, 0.99
+ pigment {
+ color rgbt<1,1,1,0.5>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+#declare filename = concat("f/", str(clock, -4, 0), ".inc");
+
+#include filename
+
+#declare n1 = vcross(<0,1,0>, finalXY);
+#declare n2 = vcross(<0,1,0>, finalYX);
+
+intersection {
+ sphere { O, 1 }
+ plane { -n1, 0 }
+ plane { n2, 0 }
+ pigment {
+ color rgb<0,0.4,0.1>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+union {
+ XYkurve()
+ pigment {
+ color Red
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+union {
+ YXkurve()
+ pigment {
+ color Blue
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
diff --git a/vorlesungen/slides/7/images/drehung.inc b/vorlesungen/slides/7/images/drehung.inc
new file mode 100644
index 0000000..c9b4bb7
--- /dev/null
+++ b/vorlesungen/slides/7/images/drehung.inc
@@ -0,0 +1,142 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.23;
+#declare O = <0, 0, 0>;
+#declare at = 0.02;
+
+camera {
+ location <8.5, 2, 6.5>
+ look_at <0, 0, 0>
+ right x * imagescale
+ up y * imagescale
+}
+
+//light_source {
+// <-14, 20, -50> color White
+// area_light <1,0,0> <0,0,1>, 10, 10
+// adaptive 1
+// jitter
+//}
+
+light_source {
+ <41, 20, 10> 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, 1.0 * 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 r = 1.0;
+
+arrow(< -r-0.2, 0.0, 0 >, < r+0.2, 0.0, 0.0 >, at, Gray)
+arrow(< 0.0, 0.0, -r-0.2>, < 0.0, 0.0, r+0.2 >, at, Gray)
+arrow(< 0.0, -r-0.2, 0 >, < 0.0, r+0.2, 0.0 >, at, Gray)
+
+#declare farbeX = rgb<1.0,0.2,0.6>;
+#declare farbeY = rgb<0.0,0.8,0.4>;
+#declare farbeZ = rgb<0.4,0.6,1.0>;
+
+#declare farbex = rgb<1.0,0.0,0.0>;
+#declare farbey = rgb<0.0,0.6,0.0>;
+#declare farbez = rgb<0.0,0.0,1.0>;
+
+#macro quadrant(X, Y, Z)
+ intersection {
+ sphere { O, 0.5 }
+ plane { -X, 0 }
+ plane { -Y, 0 }
+ plane { -Z, 0 }
+ pigment {
+ color rgb<1.0,0.6,0.2>
+ }
+ finish {
+ specular 0.95
+ metallic
+ }
+ }
+ arrow(O, X, 1.1*at, farbex)
+ arrow(O, Y, 1.1*at, farbey)
+ arrow(O, Z, 1.1*at, farbez)
+#end
+
+#macro drehung(X, Y, Z)
+// intersection {
+// sphere { O, 0.5 }
+// plane { -X, 0 }
+// plane { -Y, 0 }
+// plane { -Z, 0 }
+// pigment {
+// color Gray
+// }
+// finish {
+// specular 0.95
+// metallic
+// }
+// }
+ arrow(O, 1.1*X, 0.9*at, farbeX)
+ arrow(O, 1.1*Y, 0.9*at, farbeY)
+ arrow(O, 1.1*Z, 0.9*at, farbeZ)
+#end
+
+#macro achse(H)
+ cylinder { H, -H, at
+ pigment {
+ color rgb<0.6,0.4,0.2>
+ }
+ finish {
+ specular 0.95
+ metallic
+ }
+ }
+ cylinder { 0.003 * H, -0.003 * H, 1
+ pigment {
+ color rgbt<0.6,0.4,0.2,0.5>
+ }
+ finish {
+ specular 0.95
+ metallic
+ }
+ }
+#end
diff --git a/vorlesungen/slides/7/images/interpolation.ini b/vorlesungen/slides/7/images/interpolation.ini
new file mode 100644
index 0000000..f07c079
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.ini
@@ -0,0 +1,8 @@
+Input_File_Name=interpolation.pov
+Initial_Frame=0
+Final_Frame=50
+Initial_Clock=0
+Final_Clock=50
+Cyclic_Animation=off
+Pause_when_Done=off
+
diff --git a/vorlesungen/slides/7/images/interpolation.m b/vorlesungen/slides/7/images/interpolation.m
new file mode 100644
index 0000000..31554e8
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.m
@@ -0,0 +1,54 @@
+#
+# interpolation.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global N;
+N = 50;
+global A;
+global B;
+
+A = (pi / 2) * [
+ 0, 0, 0;
+ 0, 0, -1;
+ 0, 1, 0
+];
+g0 = expm(A)
+
+B = (pi / 2) * [
+ 0, 0, 1;
+ 0, 0, 0;
+ -1, 0, 0
+];
+g1 = expm(B)
+
+function retval = g(t)
+ global A;
+ global B;
+ retval = expm((1-t)*A+t*B);
+endfunction
+
+function dreibein(fn, M, funktion)
+ fprintf(fn, "%s(<%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>)\n",
+ funktion,
+ M(1,1), M(3,1), M(2,1),
+ M(1,2), M(3,2), M(2,2),
+ M(1,3), M(3,3), M(2,3));
+endfunction
+
+G = g1 * inverse(g0);
+[V, lambda] = eig(G);
+H = real(V(:,3));
+
+D = logm(g1*inverse(g0));
+
+for i = (0:N)
+ filename = sprintf("dreibein/d%03d.inc", i);
+ fn = fopen(filename, "w");
+ t = i/N;
+ dreibein(fn, g(t), "quadrant");
+ dreibein(fn, expm(t*D)*g0, "drehung");
+ fprintf(fn, "achse(<%.4f,%.4f,%.4f>)\n", H(1,1), H(3,1), H(2,1));
+ fclose(fn);
+endfor
+
diff --git a/vorlesungen/slides/7/images/interpolation.pov b/vorlesungen/slides/7/images/interpolation.pov
new file mode 100644
index 0000000..71e0257
--- /dev/null
+++ b/vorlesungen/slides/7/images/interpolation.pov
@@ -0,0 +1,10 @@
+//
+// commutator.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "drehung.inc"
+
+#declare filename = concat("dreibein/d", str(clock, -3, 0), ".inc");
+#include filename
+
diff --git a/vorlesungen/slides/7/images/rodriguez.jpg b/vorlesungen/slides/7/images/rodriguez.jpg
new file mode 100644
index 0000000..5c49700
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.jpg
Binary files differ
diff --git a/vorlesungen/slides/7/images/rodriguez.png b/vorlesungen/slides/7/images/rodriguez.png
new file mode 100644
index 0000000..6d9e9e4
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.png
Binary files differ
diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov
new file mode 100644
index 0000000..07aec19
--- /dev/null
+++ b/vorlesungen/slides/7/images/rodriguez.pov
@@ -0,0 +1,118 @@
+//
+// rodriguez.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+ assumed_gamma 1
+}
+
+#declare imagescale = 0.020;
+#declare O = <0, 0, 0>;
+#declare at = 0.015;
+
+camera {
+ location <8, 15, -50>
+ look_at <0.1, 0.475, 0>
+ right 16/9 * x * imagescale
+ up y * imagescale
+}
+
+light_source {
+ <-4, 20, -50> color White
+ area_light <1,0,0> <0,0,1>, 10, 10
+ adaptive 1
+ jitter
+}
+
+sky_sphere {
+ pigment {
+ color rgb<1,1,1>
+ }
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union {
+ sphere {
+ from, 1.1 * arrowthickness
+ }
+ cylinder {
+ from,
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ arrowthickness
+ }
+ cone {
+ from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+ 2 * arrowthickness,
+ to,
+ 0
+ }
+ pigment {
+ color c
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+#end
+
+#declare K = vnormalize(<0.2,1,0.1>);
+#declare X = vnormalize(<1.1,1,-1.2>);
+#declare O = <0,0,0>;
+
+#declare r = vlength(vcross(K, X)) / vlength(K);
+
+#declare l = 1.0;
+
+arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White)
+arrow(< 0, 0, -l >, < 0, 0, l >, at, White)
+arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White)
+
+arrow(O, X, at, Red)
+arrow(O, K, at, Blue)
+
+#macro punkt(H,phi)
+ ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K)
+#end
+
+cone { vdot(K, X) * K, r, O, 0
+ pigment {
+ color rgbt<0.6,0.6,0.6,0.5>
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+
+union {
+ #declare phistep = pi / 100;
+ #declare phi = 0;
+ #while (phi < 2 * pi - phistep/2)
+ sphere { punkt(K, phi), at/2 }
+ cylinder {
+ punkt(X, phi),
+ punkt(X, phi + phistep),
+ at/2
+ }
+ #declare phi = phi + phistep;
+ #end
+ pigment {
+ color Orange
+ }
+ finish {
+ specular 0.9
+ metallic
+ }
+}
+
+arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow)
+#declare Darkgreen = rgb<0,0.5,0>;
+arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen)
diff --git a/vorlesungen/slides/7/images/test.pov b/vorlesungen/slides/7/images/test.pov
new file mode 100644
index 0000000..5707be1
--- /dev/null
+++ b/vorlesungen/slides/7/images/test.pov
@@ -0,0 +1,7 @@
+//
+// test.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#include "drehung.inc"
+#include "dreibein/d025.inc"
diff --git a/vorlesungen/slides/7/integration.tex b/vorlesungen/slides/7/integration.tex
new file mode 100644
index 0000000..525e6de
--- /dev/null
+++ b/vorlesungen/slides/7/integration.tex
@@ -0,0 +1,66 @@
+%
+% integration.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Invariante Integration}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Koordinatenwechsel}
+Die Koordinatentransformation
+$f\colon\mathbb{R}^n\to\mathbb{R}^n:x\to y$
+hat die Ableitungsmatrix
+\[
+t_{ij}
+=
+\frac{\partial y_i}{\partial x_j}
+\]
+\uncover<2->{%
+$n$-faches Integral
+\begin{gather*}
+\int\dots\int
+h(f(x))
+\det
+\biggl(
+\frac{\partial y_i}{\partial x_j}
+\biggr)
+\,dx_1\,\dots dx_n
+\\
+=
+\int\dots\int
+h(y)
+\,dy_1\,\dots dy_n
+\end{gather*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{auf einer Lie-Gruppe}
+Koordinatenwechsel sind Multiplikationen mit einer
+Matrix $g\in G$
+\end{block}}
+\uncover<4->{%
+\begin{block}{Volumenelement in $I$}
+Man muss nur das Volumenelement in $I$ in einem beliebigen
+Koordinatensystem definieren:
+\[
+dV = dy_1\,\dots\,dy_n
+\]
+\end{block}}
+\uncover<5->{%
+\begin{block}{Volumenelement in $g$}
+\[
+\text{``\strut}g\cdot dV\text{\strut''}
+=
+\det(g) \, dy_1\,\dots\,dy_n
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/interpolation.tex b/vorlesungen/slides/7/interpolation.tex
new file mode 100644
index 0000000..249ee26
--- /dev/null
+++ b/vorlesungen/slides/7/interpolation.tex
@@ -0,0 +1,112 @@
+%
+% interpolation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\bild#1#2{\only<#1|handout:0>{\includegraphics[width=\textwidth]{../slides/7/images/interpolation/#2.png}}}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Interpolation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Aufgabe}
+Finde einen Weg $g(t)\in \operatorname{SO}(3)$ zwischen
+$g_0\in\operatorname{SO}(3)$
+und
+$g_1\in\operatorname{SO}(3)$:
+\[
+g_0=g(0)
+\quad\wedge\quad
+g_1=g(1)
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Lösung}
+$g_i=\exp(A_i) \uncover<3->{\Rightarrow A_i^t=-A_i}$
+\begin{align*}
+\uncover<4->{A(t) &= (1-t)A_0 + tA_1}\uncover<8->{ \in \operatorname{so}(3)}
+\\
+\uncover<5->{A(t)^t
+&=(1-t)A_0^t + tA_1^t}
+\\
+&\uncover<6->{=
+-(1-t)A_0 - t A_1}
+\uncover<7->{=
+-A(t)}
+\\
+\uncover<9->{\Rightarrow
+g(t) &= \exp A(t) \in \operatorname{SO}(3)}
+\\
+&\uncover<10->{\ne
+\exp (\log(g_1g_0^{-1})t) g_0}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Animation}
+\centering
+\ifthenelse{\boolean{presentation}}{
+\bild{12}{i00}
+\bild{13}{i01}
+\bild{14}{i02}
+\bild{15}{i03}
+\bild{16}{i04}
+\bild{17}{i05}
+\bild{18}{i06}
+\bild{19}{i07}
+\bild{20}{i08}
+\bild{21}{i09}
+\bild{22}{i10}
+\bild{23}{i11}
+\bild{24}{i12}
+\bild{25}{i13}
+\bild{26}{i14}
+\bild{27}{i15}
+\bild{28}{i16}
+\bild{29}{i17}
+\bild{30}{i18}
+\bild{31}{i19}
+\bild{32}{i20}
+\bild{33}{i21}
+\bild{34}{i22}
+\bild{35}{i23}
+\bild{36}{i24}
+\bild{37}{i25}
+\bild{38}{i26}
+\bild{39}{i27}
+\bild{40}{i28}
+\bild{41}{i29}
+\bild{42}{i30}
+\bild{43}{i31}
+\bild{44}{i32}
+\bild{45}{i33}
+\bild{46}{i34}
+\bild{47}{i35}
+\bild{48}{i36}
+\bild{49}{i37}
+\bild{50}{i38}
+\bild{51}{i39}
+\bild{52}{i40}
+\bild{53}{i41}
+\bild{54}{i42}
+\bild{55}{i43}
+\bild{56}{i44}
+\bild{57}{i45}
+\bild{58}{i46}
+\bild{59}{i47}
+\bild{60}{i48}
+\bild{61}{i49}
+\bild{62}{i50}
+}{
+\includegraphics[width=\textwidth]{../slides/7/images/interpolation/i25.png}
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex
new file mode 100644
index 0000000..84bf034
--- /dev/null
+++ b/vorlesungen/slides/7/kommutator.tex
@@ -0,0 +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}};}
+\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
new file mode 100644
index 0000000..e0690eb
--- /dev/null
+++ b/vorlesungen/slides/7/kurven.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/liealgbeispiel.tex b/vorlesungen/slides/7/liealgbeispiel.tex
new file mode 100644
index 0000000..a17de40
--- /dev/null
+++ b/vorlesungen/slides/7/liealgbeispiel.tex
@@ -0,0 +1,78 @@
+%
+% liealgbeispiel.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lie-Algebra Beispiele}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{sl}_2(\mathbb{R})$}
+Spurlose Matrizen:
+\[
+\operatorname{sl}_2(\mathbb{R})
+=
+\{A\in M_n(\mathbb{R})\;|\; \operatorname{Spur}A=0\}
+\]
+\end{block}
+\begin{block}{Lie-Algebra?}
+Nachrechnen: $[A,B]\in \operatorname{sl}_2(\mathbb{R})$:
+\begin{align*}
+\operatorname{Spur}([A,B])
+&=
+\operatorname{Spur}(AB-BA)
+\\
+&=
+\operatorname{Spur}(AB)-\operatorname{Spur}(BA)
+\\
+&=
+\operatorname{Spur}(AB)-\operatorname{Spur}(AB)
+\\
+&=0
+\end{align*}
+$\Rightarrow$ $\operatorname{sl}_2(\mathbb{R})$ ist eine Lie-Algebra
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{so}(n)$}
+Antisymmetrische Matrizen:
+\[
+\operatorname{so}(n)
+=
+\{A\in M_n(\mathbb{R})
+\;|\;
+A=-A^t
+\}
+\]
+\end{block}
+\begin{block}{Lie-Algebra?}
+Nachrechnen: $A,B\in \operatorname{so}(n)$
+\begin{align*}
+[A,B]^t
+&=
+(AB-BA)^t
+\\
+&=
+B^tA^t - A^tB^t
+\\
+&=
+(-B)(-A)-(-A)(-B)
+\\
+&=
+BA-AB
+=
+-(AB-BA)
+\\
+&=
+-[A,B]
+\end{align*}
+$\Rightarrow$ $\operatorname{so}(n)$ ist eine Lie-Algebra
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex
new file mode 100644
index 0000000..574467b
--- /dev/null
+++ b/vorlesungen/slides/7/liealgebra.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex
new file mode 100644
index 0000000..58065d7
--- /dev/null
+++ b/vorlesungen/slides/7/logarithmus.tex
@@ -0,0 +1,82 @@
+%
+% logarithmus.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Logarithmus}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Taylor-Reihe}
+\begin{align*}
+\frac{d}{dx}\log(1+x)
+&= \frac{1}{1+x}
+\\
+\uncover<2->{
+\Rightarrow\quad
+\log (1+x)
+&=
+\int_0^x \frac{1}{1+t}\,dt}
+\end{align*}
+\begin{align*}
+\uncover<3->{\frac{1}{1+t}
+&=
+1-t+t^2-t^3+\dots}
+\\
+\uncover<4->{\log(1+x)
+&=\int_0^x
+1-t+t^2-t^3+\dots
+\,dt}
+\\
+&\only<5>{=
+x-\frac{x^2}{2}  + \frac{x^3}{3} - \frac{x^4}4 + \dots}
+\uncover<6->{=
+\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}}
+\\
+\uncover<7->{\log (I+A)
+&=
+\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Konvergenzradius}
+Polstelle bei $x=-1$
+\(
+\varrho =1
+\)
+\end{block}}
+\vspace{-5pt}
+\begin{block}{\uncover<9->{Alternative: Spektraltheorie}}
+\uncover<9->{
+Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$
+definiert:}
+\vspace{-15pt}
+\uncover<8->{
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<9->{
+ \fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1);
+}
+\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}];
+\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}];
+\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1];
+\draw[color=blue] (1,0) circle[radius=1];
+\uncover<9->{
+ \draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0);
+}
+\fill (1,0) circle[radius=0.08];
+\node at (2.3,1.9) {$\mathbb{C}$};
+\node at (1,0) [below] {$1$};
+\end{tikzpicture}
+\end{center}}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex
new file mode 100644
index 0000000..077dc9d
--- /dev/null
+++ b/vorlesungen/slides/7/mannigfaltigkeit.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex
new file mode 100644
index 0000000..f3579a3
--- /dev/null
+++ b/vorlesungen/slides/7/parameter.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/qdreh.tex b/vorlesungen/slides/7/qdreh.tex
new file mode 100644
index 0000000..8ed512a
--- /dev/null
+++ b/vorlesungen/slides/7/qdreh.tex
@@ -0,0 +1,110 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehungen mit Quaternionen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Drehung?}
+Abbildung von $\vec{x}$ mit $\operatorname{Re}\vec{x}=0$:
+\[
+\varrho_{q}
+\colon
+\vec{x}\mapsto q\vec{x}q^{-1} = q\vec{x}\overline{q}
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Achse}
+\begin{align*}
+\varrho_q(q)
+&=
+qq\overline{q}
+\uncover<3->{=
+q(qq^{-1})}
+\uncover<4->{=
+q}
+\end{align*}
+\end{block}}
+\uncover<4->{%
+\begin{block}{Norm}
+\begin{align*}
+|\varrho_q(\vec{x})|^2
+&=
+q\vec{x}\overline{q}\overline{(q\vec{x}\overline{q})}
+\uncover<5->{=
+q\vec{x}\overline{q}\overline{\overline{q}}\overline{\vec{x}}\overline{q}
+}
+\\
+&\uncover<6->{=
+q\vec{x}(\overline{q}q)\overline{\vec{x}}\overline{q}}
+\uncover<7->{=
+q(\vec{x}\overline{\vec{x}})\overline{q}}
+\uncover<8->{=
+q\overline{q}|\vec{x}|^2}
+\\
+&\uncover<9->{=
+|\vec{x}|^2}
+\end{align*}
+\uncover<10->{%
+$\Rightarrow$ $\varrho_q\in\operatorname{O}(3)$}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<11->{%
+\begin{block}{Drehung!}
+$\vec{a},\vec{b},\vec{n}$ bilden ein on.~Rechtssystem
+\begin{align*}
+\uncover<12->{
+qa
+&=
+c\vec{a}+s\vec{n}\times \vec{a}}
+\uncover<13->{=
+c\vec{a} + s\vec{b}}
+\\
+\uncover<14->{
+q\vec{a}\overline{q}
+&=
+(c\vec{a}+s\vec{b}) c
+-(c\vec{a}+s\vec{b})\times s\vec{n}}
+\\
+&\uncover<15->{=
+c^2 \vec{a}+ sc\vec{b}
++sc\vec{b} - s^2 \vec{a}}
+\\
+&\uncover<16->{=
+\vec{a} \cos\alpha +\vec{b} \sin\alpha }
+\end{align*}
+\vspace{-5pt}
+\uncover<17->{wegen
+%\vspace{-5pt}
+\[
+\begin{aligned}
+\cos\alpha &= \cos^2\frac{\alpha}2 - \sin^2\frac{\alpha}2 &&=c^2-s^2
+\\
+\sin\alpha &= 2\cos\frac{\alpha}2\sin\frac{\alpha}2&&=2cs
+\end{aligned}\]}
+\end{block}}
+\vspace{-18pt}
+\uncover<18->{%
+\begin{block}{Matrix}
+\[
+D
+=
+\tiny
+\begin{pmatrix}
+1-2(q_2^2+q_3^2)&-2q_0q_3+2q_1q_2&-2q_0q_2+2q_1q_3\\
+ 2q_0q_3+2q_1q_2&1-2(q_1^2+q_3^2)&-2q_0q_1+2q_2q_3\\
+-2q_0q_2+2q_1q_3& 2q_0q_1+2q_2q_3&1-2(q_1^2+q_2^2)
+\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/quaternionen.tex b/vorlesungen/slides/7/quaternionen.tex
new file mode 100644
index 0000000..f526366
--- /dev/null
+++ b/vorlesungen/slides/7/quaternionen.tex
@@ -0,0 +1,74 @@
+%
+% quaternionen.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Quaternionen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Quaternionen}
+$4$-dimensionaler $\mathbb{R}$-Vektorraum
+\[
+\mathbb{H}
+=
+\langle 1,i,j,k\rangle_{\mathbb{R}}
+\]
+mit Rechenregeln
+\[
+i^2=j^2=k^2=ijk=-1
+\]
+$x=x_0+x_1i+x_2j+x_3k\in\mathbb{H}$
+\begin{itemize}
+\item<2-> Realteil: $\operatorname{Re}x=x_0$
+\item<3-> Vektorteil: $\operatorname{Im}x=x_1i+x_2j+x_3k$
+\item<4-> Konjugation: $\overline{x}=\operatorname{Re}x-\operatorname{Im}x$
+\item<5-> Norm: $|x|^2 = x\overline{x} = x_0^2+x_1^2+x_2^2+x_3^2$
+\item<6-> Inverse: $x^{1}= \overline{x}/x\overline{x}$
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.50\textwidth}
+\uncover<7->{%
+\begin{block}{Skalarprodukt und Vektorprodukt}
+\begin{align*}
+pq
+&=
+\operatorname{Re}p \operatorname{Re}q
+-
+\operatorname{Im}p\cdot \operatorname{Im}q
+\\
+&\phantom{=}
++
+\operatorname{Re}p\operatorname{Im}q
++
+\operatorname{Im}p\operatorname{Re}q
++
+\operatorname{Im}p\times\operatorname{Im}q
+\end{align*}
+\end{block}}
+\uncover<8->{%
+\begin{block}{Einheitsquaternionen}
+$q\in \mathbb{H}$, $|q|=1, q^{-1}=\overline{q}$
+\end{block}}
+\uncover<9->{%
+\begin{block}{Polardarstellung}
+\[
+q = \cos\frac{\alpha}2 + \vec{n} \sin\frac{\alpha}2
+\]
+\vspace{-8pt}
+\begin{itemize}
+\item<10->
+Drehmatrix: 9 Parameter, 6 Bedingungen
+\item<11->
+Quaternionen: 4 Parameter, 1 Bedingung
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex
new file mode 100644
index 0000000..cd974c9
--- /dev/null
+++ b/vorlesungen/slides/7/semi.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
new file mode 100644
index 0000000..a65b4f6
--- /dev/null
+++ b/vorlesungen/slides/7/sl2.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex
new file mode 100644
index 0000000..35d62d8
--- /dev/null
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -0,0 +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
diff --git a/vorlesungen/slides/7/ueberlagerung.tex b/vorlesungen/slides/7/ueberlagerung.tex
new file mode 100644
index 0000000..426641a
--- /dev/null
+++ b/vorlesungen/slides/7/ueberlagerung.tex
@@ -0,0 +1,98 @@
+%
+% ueberlagerung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$S^3$, $\operatorname{SU}(2)$ und $\operatorname{SO}(3)$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\uncover<6->{%
+\begin{block}{Überlagerung}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (0,0);
+\coordinate (B) at (2,0);
+\coordinate (C) at (2,-2);
+\coordinate (D) at (0,-2);
+
+\uncover<7->{
+\node at (A) {$\{\pm 1\}\mathstrut$};
+}
+\uncover<6->{
+\node at (B) {$S^3\mathstrut$};
+\node at ($(B)+(0.1,0)$) [right] {$=\operatorname{SU}(2)\mathstrut$};
+}
+\uncover<7->{
+\node at (C) {$\operatorname{SO}(3)\mathstrut$};
+\node at (D) {$\{I\}\mathstrut$};
+}
+
+\uncover<7->{
+\draw[->,shorten >= 0.3cm,shorten <= 0.5cm] (A) -- (B);
+\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (D);
+\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C);
+\draw[->,shorten >= 0.6cm,shorten <= 0.3cm] (D) -- (C);
+}
+
+\end{tikzpicture}
+\end{center}
+\begin{itemize}
+\item<7->
+$\pm q\in S^3$ $\Rightarrow$ $\varrho_{q}=\varrho_{-q}$
+\item<8->
+In der Nähe von $I$ sehen die Gruppen
+$\operatorname{SO}(3)$
+und
+$\operatorname{SU}(2)$
+``gleich'' aus
+\item<9->
+$\operatorname{SU}(2)$ ist geometrisch ``einfacher''
+\end{itemize}
+\end{block}}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{block}{Pauli-Matrizen}
+Quaternionen als $2\times 2$-Matrizen schreiben
+\begin{align*}
+1&=\begin{pmatrix}1&0\\0&1\end{pmatrix}=\sigma_0,
+&
+i&=\begin{pmatrix}0&i\\i&0\end{pmatrix}=-i\sigma_1
+\\
+j&=\begin{pmatrix}0&-1\\1&0\end{pmatrix}=-i\sigma_2,
+&
+k&=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=-i\sigma_3
+\end{align*}
+\uncover<2->{%
+erfüllen $i^2=j^2=k^2=ijk=-1$.}
+\end{block}
+\uncover<3->{%
+\begin{block}{$S^3 = \operatorname{SU}(2)$}
+\[
+a+bi+cj+dk
+=
+\begin{pmatrix}
+a+id&-c+bi\\
+c+ib&a-id
+\end{pmatrix}
+=
+A
+\]
+\begin{align*}
+\uncover<4->{
+\det A &= a^2 + b^2 + c^2 + d^2 = 1
+}
+\\
+\uncover<5->{
+A^* &= a - ib - jc - kd
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/vektorlie.tex b/vorlesungen/slides/7/vektorlie.tex
new file mode 100644
index 0000000..621a832
--- /dev/null
+++ b/vorlesungen/slides/7/vektorlie.tex
@@ -0,0 +1,206 @@
+%
+% viktorlie.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{Vektorprodukt als Lie-Algebra}
+%\vspace{-10pt}
+\centering
+\begin{tikzpicture}[>=latex,thick]
+\arraycolsep=2.4pt
+\def\Ax{0}
+\def\Ux{4.1}
+\def\Kx{7.2}
+\def\Rx{13.1}
+
+\def\Lx{2.2}
+\def\Ly{0}
+\def\Lz{-2.2}
+
+\fill[color=red!20] (\Ax,{\Lx-1.55}) rectangle ({\Ux-0.1},{\Lx+0.55});
+\fill[color=red!20] (\Ux,{\Lx-1.55}) rectangle ({\Kx-0.1},{\Lx+0.55});
+\fill[color=red!20] (\Kx,{\Lx-1.55}) rectangle ({\Rx},{\Lx+0.55});
+
+\fill[color=darkgreen!20] (\Ax,{\Ly-1.55}) rectangle ({\Ux-0.1},{\Ly+0.55});
+\fill[color=darkgreen!20] (\Ux,{\Ly-1.55}) rectangle ({\Kx-0.1},{\Ly+0.55});
+\fill[color=darkgreen!20] (\Kx,{\Ly-1.55}) rectangle ({\Rx},{\Ly+0.55});
+
+\fill[color=blue!20] (\Ax,{\Lz-1.55}) rectangle ({\Ux-0.1},{\Lz+0.55});
+\fill[color=blue!20] (\Ux,{\Lz-1.55}) rectangle ({\Kx-0.1},{\Lz+0.55});
+\fill[color=blue!20] (\Kx,{\Lz-1.55}) rectangle ({\Rx},{\Lz+0.55});
+
+\coordinate (A) at (\Ax,3.2);
+\coordinate (Ax) at (\Ax,\Lx);
+\coordinate (Ay) at (\Ax,\Ly);
+\coordinate (Az) at (\Ax,\Lz);
+
+\node at (A) [right]
+ {\usebeamercolor[fg]{title}Drehmatrix, $\operatorname{SO}(n)$\strut};
+
+\node at (Ax) [right] {$\displaystyle\tiny
+D_{x,\alpha}=\begin{pmatrix}
+1&0&0\\
+0&\cos\alpha&-\sin\alpha\\
+0&\sin\alpha&\cos\alpha
+\end{pmatrix}$};
+
+\node at (Ay) [right] {$\displaystyle\tiny
+D_{y,\alpha}=\begin{pmatrix}
+\cos\alpha&0&\sin\alpha\\
+0&1&0\\
+-\sin\alpha&0&\cos\alpha
+\end{pmatrix}$};
+
+\node at (Az) [right] {$\displaystyle\tiny
+D_{z,\alpha}=\begin{pmatrix}
+\cos\alpha&-\sin\alpha&0\\
+\sin\alpha&\cos\alpha&0\\
+0&0&1
+\end{pmatrix}$};
+
+\coordinate (U) at (\Ux,3.2);
+\coordinate (Ux) at (\Ux,\Lx);
+\coordinate (Uy) at (\Ux,\Ly);
+\coordinate (Uz) at (\Ux,\Lz);
+\coordinate (Ex) at (\Ux,{\Lx-1});
+\coordinate (Ey) at (\Ux,{\Ly-1});
+\coordinate (Ez) at (\Ux,{\Lz-1});
+
+\uncover<2->{
+\node at (U) [right]
+ {\usebeamercolor[fg]{title}Ableitung, $\operatorname{so}(n)$\strut};
+
+\node at (Ux) [right] {$\displaystyle\tiny
+U_x=\begin{pmatrix*}[r]
+0&0&0\\
+0&0&-1\\
+0&1&0
+\end{pmatrix*}
+$};
+
+\node at (Uy) [right] {$\displaystyle\tiny
+U_y=\begin{pmatrix*}[r]
+0&0&1\\
+0&0&0\\
+-1&0&0
+\end{pmatrix*}
+$};
+
+\node at (Uz) [right] {$\displaystyle\tiny
+U_z=\begin{pmatrix*}[r]
+0&-1&0\\
+1&0&0\\
+0&0&0
+\end{pmatrix*}
+$};
+}
+
+\uncover<9->{
+\node at (Ex) [right] {$\displaystyle
+\, e_x = \tiny\begin{pmatrix}1\\0\\0\end{pmatrix}
+$};
+
+\node at (Ey) [right] {$\displaystyle
+\, e_y = \tiny\begin{pmatrix}0\\1\\0\end{pmatrix}
+$};
+
+\node at (Ez) [right] {$\displaystyle
+\, e_z = \tiny\begin{pmatrix}0\\0\\1\end{pmatrix}
+$};
+}
+
+\coordinate (K) at (\Kx,3.2);
+\coordinate (Kx) at (\Kx,\Lx);
+\coordinate (Ky) at (\Kx,\Ly);
+\coordinate (Kz) at (\Kx,\Lz);
+\coordinate (Vx) at (\Kx,{\Lx-1});
+\coordinate (Vy) at (\Kx,{\Ly-1});
+\coordinate (Vz) at (\Kx,{\Lz-1});
+
+\uncover<3->{
+\node at (K) [right]
+ {\usebeamercolor[fg]{title}Kommutator\strut};
+
+\node at (Kx) [right] {$\displaystyle
+\begin{aligned}
+[U_y,U_z] &\uncover<4->{=
+{\tiny
+\begin{pmatrix}
+0&0&0\\
+0&0&0\\
+0&1&0
+\end{pmatrix}}
+\uncover<5->{\mathstrut-
+\tiny
+\begin{pmatrix}
+0&0&0\\
+0&0&1\\
+0&0&0
+\end{pmatrix}}}
+\uncover<6->{=U_x}
+\end{aligned}
+$};
+}
+
+\uncover<7->{
+\node at (Ky) [right] {$\displaystyle
+\begin{aligned}
+[U_z,U_x] &=
+{\tiny
+\begin{pmatrix}
+0&0&1\\
+0&0&0\\
+0&0&0
+\end{pmatrix}
+-
+\begin{pmatrix}
+0&0&0\\
+0&0&0\\
+1&0&0
+\end{pmatrix}}
+=U_y
+\end{aligned}
+$};
+}
+
+\uncover<8->{
+\node at (Kz) [right] {$\displaystyle
+\begin{aligned}
+[U_x,U_y] &=
+{\tiny
+\begin{pmatrix}
+0&0&0\\
+1&0&0\\
+0&0&0
+\end{pmatrix}
+-
+\begin{pmatrix}
+0&1&0\\
+0&0&0\\
+0&0&0
+\end{pmatrix}}
+=U_z
+\end{aligned}
+$};
+}
+
+\uncover<10->{
+\node at (Vx) [right] {$\displaystyle \phantom{]}e_y\times e_z = e_x$};
+}
+
+\uncover<11->{
+\node at (Vy) [right] {$\displaystyle \phantom{]}e_z\times e_x = e_y$};
+}
+
+\uncover<12->{
+\node at (Vz) [right] {$\displaystyle \phantom{]}e_x\times e_y = e_z$};
+}
+
+\end{tikzpicture}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/zusammenhang.tex b/vorlesungen/slides/7/zusammenhang.tex
new file mode 100644
index 0000000..6a43cd8
--- /dev/null
+++ b/vorlesungen/slides/7/zusammenhang.tex
@@ -0,0 +1,99 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zusammenhang}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zusammenhängend --- oder nicht}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\ds{2.4}
+\coordinate (A) at (0,0);
+\coordinate (B) at (\ds,0);
+\coordinate (C) at ({2*\ds},0);
+
+\node at (A) {$\operatorname{SO}(n)$};
+\node at (B) {$\operatorname{O}(n)$};
+\node at (C) {$\{\pm 1\}$};
+
+\draw[->,shorten <= 0.6cm,shorten >= 0.5cm] (A) -- (B);
+\draw[->,shorten <= 0.5cm,shorten >= 0.5cm] (B) -- (C);
+\node at ($0.5*(B)+0.5*(C)$) [above] {$\det$};
+
+\coordinate (A2) at (0,-1.0);
+\coordinate (B2) at (\ds,-1.0);
+\coordinate (C2) at ({2*\ds},-1.0);
+
+\draw[color=blue] (A2) ellipse (1cm and 0.3cm);
+\draw[color=blue] (B2) ellipse (1cm and 0.3cm);
+\node[color=blue] at (C2) {$+1$};
+
+\coordinate (A3) at (0,-1.7);
+\coordinate (B3) at (\ds,-1.7);
+\coordinate (C3) at ({2*\ds},-1.7);
+
+\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B2) -- (C2);
+\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B3) -- (C3);
+
+\draw[color=red] (B3) ellipse (1cm and 0.3cm);
+\node[color=red] at (C3) {$-1$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\begin{block}{Zusammenhangskomponente von $e$}
+$G_e\subset G$ grösste zusammenhängende Menge, die $e$ enthält:
+\begin{align*}
+\operatorname{SO}(n)&\subset \operatorname{O}(n)
+\\
+\{A\in\operatorname{GL}_n(\mathbb{R})\,|\, \det A > 0\}
+ &\subset \operatorname{GL}_n(\mathbb{R})
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenschaften}
+\begin{itemize}
+\item
+{\bf Untergruppe}: $\gamma_i(t)$ Weg von $e$ nach $g_i$,
+dann ist
+\begin{itemize}
+\item
+$\gamma_1(t)\gamma_2(t)$ ein Weg von $e$ nach $g_1g_2$
+\item
+$\gamma_1(t)^{-1}$ Weg von $e$ nach $g_1^{-1}$
+\end{itemize}
+\item
+{\bf Normalteiler}: $\gamma(t)$ ein Weg von $e$ nach $g$, dann
+ist $h\gamma(t)h^{-1}$ ein Weg von $h$ nach $hgh^{-1}$
+$\Rightarrow hG_eh^{-1}\subset G_e$
+\end{itemize}
+\end{block}
+\begin{block}{Quotient}
+$G/G_e$ ist eine diskrete Gruppe
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (0,0);
+\coordinate (B) at (2,0);
+\coordinate (C) at (4,0);
+\node at (A) {$G_e$};
+\node at (B) {$G$};
+\node at (C) {$G/G_e$};
+\draw [->,shorten <= 0.3cm,shorten >= 0.3cm] (A) -- (B);
+\draw [->,shorten <= 0.3cm,shorten >= 0.5cm] (B) -- (C);
+\end{tikzpicture}
+\end{center}
+\vspace{-7pt}
+$\Rightarrow$ $G_e$ und $G/G_e$ separat studieren
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup