14 files changed, 1198 insertions, 0 deletions

diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
new file mode 100644
index 0000000..ef004ca
--- /dev/null
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -0,0 +1,20 @@
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter5 =								\
+	../slides/7/symmetrien.tex					\
+	../slides/7/algebraisch.tex					\
+	../slides/7/parameter.tex					\
+	../slides/7/mannigfaltigkeit.tex				\
+	../slides/7/sl2.tex						\
+	../slides/7/drehung.tex						\
+	../slides/7/semi.tex						\
+	../slides/7/kurven.tex						\
+	../slides/7/einparameter.tex					\
+	../slides/7/ableitung.tex					\
+	../slides/7/liealgebra.tex					\
+	../slides/7/dg.tex						\
+	../slides/7/chapter.tex
+
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex
new file mode 100644
index 0000000..b061b9a
--- /dev/null
+++ b/vorlesungen/slides/7/ableitung.tex
@@ -0,0 +1,62 @@
+%
+% ableitung.tex -- Ableitung in der Lie-Gruppe
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ableitung in der Matrix-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{O}(n)$}
+$s \mapsto A(s)\in\operatorname{O}(n)$
+\begin{align*}
+I
+&=
+A(s)^tA(s)
+\\
+0
+=
+\frac{d}{ds} I
+&=
+\frac{d}{ds} (A(s)^t A(s))
+\\
+&=
+\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)
+\intertext{An der Stelle $s=0$, d.~h.~$A(0)=I$}
+0
+&=
+\dot{A}(0)^t
++
+\dot{A}(0)
+\\
+\Leftrightarrow
+\qquad
+\dot{A}(0)^t &= -\dot{A}(0)
+\end{align*}
+``Tangentialvektoren'' sind antisymmetrische Matrizen
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$}
+$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$
+\begin{align*}
+1 &= \det A(t)
+\\
+0
+=
+\frac{d}{dt}1
+&=
+\frac{d}{dt} \det A(t)
+\intertext{mit dem Entwicklungssatz kann man nachrechnen:}
+0&=\operatorname{Spur}\dot{A}(0)
+\end{align*}
+``Tangentialvektoren'' sind spurlose Matrizen
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex
new file mode 100644
index 0000000..5b33566
--- /dev/null
+++ b/vorlesungen/slides/7/algebraisch.tex
@@ -0,0 +1,105 @@
+%
+% algebraisch.tex -- algebraische Definition der Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Erhaltungsgrössen und Algebra}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Längen und Winkel}
+Längenmessung mit Skalarprodukt
+\begin{align*}
+\|\vec{v}\|^2
+&=
+\langle \vec{v},\vec{v}\rangle
+=
+\vec{v}\cdot \vec{v}
+=
+\vec{v}^t\vec{v}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Flächeninhalt/Volumen}
+$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$
+\\
+Volumen des Parallelepipeds: $\det V$
+\end{block}
+\end{column}
+\end{columns}
+%
+\vspace{-7pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Längenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\vec{x}^t\vec{y}
+&=
+(A\vec{x})
+\cdot
+(A\vec{y})
+=
+(A\vec{x})^t
+(A\vec{y})
+\\
+\vec{x}^tI\vec{y}
+&=
+\vec{x}^tA^tA\vec{y}
+\Rightarrow I=A^tA
+\end{align*}
+Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Volumenerhaltende Transformationen}
+$A\in\operatorname{GL}_n(\mathbb{R})$
+\begin{align*}
+\det(V)
+&=
+\det(AV)
+=
+\det(A)\det(V)
+\\
+1&=\det(A)
+\end{align*}
+(Produktsatz für Determinante)
+\end{block}
+\end{column}
+\end{columns}
+%
+\vspace{-3pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Orthogonale Matrizen}
+Längentreue Abbildungen = orthogonale Matrizen:
+\[
+O(n)
+=
+\{
+A \in \operatorname{GL}_n(\mathbb{R})
+\;|\;
+A^tA=I
+\}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{``Spezielle'' Matrizen}
+Volumen-/Orientierungserhaltende Transformationen:
+\[
+\operatorname{SL}_n(\mathbb R) 
+=
+\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\}
+\]
+\end{block}
+\end{column}
+\end{columns}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
new file mode 100644
index 0000000..44d46a6
--- /dev/null
+++ b/vorlesungen/slides/7/chapter.tex
@@ -0,0 +1,17 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{7/symmetrien.tex}
+\folie{7/algebraisch.tex}
+\folie{7/parameter.tex}
+\folie{7/mannigfaltigkeit.tex}
+\folie{7/sl2.tex}
+\folie{7/drehung.tex}
+\folie{7/semi.tex}
+\folie{7/kurven.tex}
+\folie{7/einparameter.tex}
+\folie{7/ableitung.tex}
+\folie{7/liealgebra.tex}
+\folie{7/dg.tex}
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex
new file mode 100644
index 0000000..36b1ade
--- /dev/null
+++ b/vorlesungen/slides/7/dg.tex
@@ -0,0 +1,80 @@
+%
+% dg.tex -- Differentialgleichung für die Exponentialabbildung
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Zurück zur Lie-Gruppe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Tangentialvektor im Punkt $\gamma(t)$}
+Ableitung von $\gamma(t)$ an der Stelle $t$:
+\begin{align*}
+\dot{\gamma}(t)
+&=
+\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t}
+\\
+&=
+\frac{d}{ds}
+\gamma(t+s)
+\bigg|_{s=0}
+\\
+&=
+\frac{d}{ds}
+\gamma(t)\gamma(s)
+\bigg|_{s=0}
+\\
+&=
+\gamma(t)
+\frac{d}{ds}
+\gamma(s)
+\bigg|_{s=0}
+=
+\gamma(t) \dot{\gamma}(0)
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\begin{block}{Differentialgleichung}
+\vspace{-10pt}
+\[
+\dot{\gamma}(t) = \gamma(t) A
+\quad
+\text{mit}
+\quad
+A=\dot{\gamma}(0)\in LG
+\]
+\end{block}
+\end{column}
+\begin{column}{0.50\textwidth}
+\begin{block}{Lösung}
+Exponentialfunktion
+\[
+\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k
+\]
+\end{block}
+\vspace{-5pt}
+\begin{block}{Kontrolle: Tangentialvektor berechnen}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}e^{At}
+&=
+\sum_{k=1}^\infty A^k \frac{d}{dt} t^{k}{k!}
+\\
+&=
+\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A
+\\
+&=
+\sum_{k=0} A^k\frac{t^k}{k!}
+A
+=
+e^{At} A
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
new file mode 100644
index 0000000..ae0dbe3
--- /dev/null
+++ b/vorlesungen/slides/7/drehung.tex
@@ -0,0 +1,113 @@
+%
+% drehung.tex -- Drehung aus streckungen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehung aus Streckungen und Scherungen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.38\textwidth}
+\begin{block}{Drehung}
+\[
+\operatorname{SO}(2)
+=
+\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2)
+\]
+\end{block}
+\begin{block}{Zusammensetzung}
+Eine Drehung muss als Zusammensetzung geschrieben werden können:
+\[
+D_{\alpha} 
+=
+\begin{pmatrix}
+\cos\alpha & -\sin\alpha\\
+\sin\alpha &\phantom{-}\cos\alpha
+\end{pmatrix}
+=
+DST
+\]
+\end{block}
+\begin{block}{Beispiel}
+\vspace{-12pt}
+\[
+D_{60^\circ}
+=
+{\tiny
+\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}
+\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix}
+\begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix}
+}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.58\textwidth}
+\begin{block}{Ansatz}
+\vspace{-12pt}
+\begin{align*}
+DST
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+1&-s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&0\\
+t&1
+\end{pmatrix}
+\\
+&=
+\begin{pmatrix}
+c^{-1}&0\\
+  0   &c
+\end{pmatrix}
+\begin{pmatrix}
+-st&-s\\
+  t& 1
+\end{pmatrix}
+\\
+&=
+\begin{pmatrix}
+-stc^{-1}&{\color{darkgreen}sc^{-1}}\\
+{\color{blue}ct}&{\color{red}c}
+\end{pmatrix}
+=
+\begin{pmatrix}
+\cos\alpha & {\color{darkgreen}- \sin\alpha} \\
+{\color{blue}\sin\alpha} & \phantom{-}  {\color{red}\cos\alpha}
+\end{pmatrix}
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\begin{block}{Koeffizientenvergleich}
+\vspace{-15pt}
+\begin{align*}
+{\color{red} c}
+&=
+{\color{red}\cos\alpha }
+&&
+&
+{\color{blue}
+t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\
+{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha}
+&
+&\Rightarrow&
+{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha
+\\
+{\color{orange} -stc^{-t}}
+&=
+\rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
new file mode 100644
index 0000000..52924bf
--- /dev/null
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -0,0 +1,87 @@
+%
+% einparameter.tex -- Einparameter Untergruppen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Einparameter Untergruppen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$,
+die gleichzeitig eine Untergruppe von $G$ ist mit
+\[
+\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R}
+\]
+\end{block}
+\begin{block}{Drehungen}
+Drehmatrizen bilden Einparameter- Untergruppen
+\begin{align*}
+t \mapsto D_{x,t}
+&=
+\begin{pmatrix}
+1&0&0\\
+0&\cos t&-\sin t\\
+0&\sin t& \cos t
+\end{pmatrix}
+\\
+D_{x,t}D_{x,s}
+&=
+D_{x,t+s}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
+\vspace{-12pt}
+\[
+\begin{pmatrix}
+1&s\\
+0&1
+\end{pmatrix}
+\begin{pmatrix}
+1&t\\
+0&1
+\end{pmatrix}
+=
+\begin{pmatrix}
+1&s+t\\
+0&1
+\end{pmatrix}
+\]
+\end{block}
+\vspace{-12pt}
+\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
+\vspace{-12pt}
+\[
+\begin{pmatrix}
+e^s&0\\0&e^{-s}
+\end{pmatrix}
+\begin{pmatrix}
+e^t&0\\0&e^{-t}
+\end{pmatrix}
+=
+\begin{pmatrix}
+e^{t+s}&0\\0&e^{-(t+s)}
+\end{pmatrix}
+\]
+\end{block}
+\vspace{-12pt}
+\begin{block}{Gemischt}
+\vspace{-12pt}
+\begin{gather*}
+A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
+\\
+\text{dank}\quad
+\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2
+=I
+\end{gather*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex
new file mode 100644
index 0000000..196fa2a
--- /dev/null
+++ b/vorlesungen/slides/7/kurven.tex
@@ -0,0 +1,85 @@
+%
+% kurven.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kurven und Tangenten}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Kurven}
+Kurve in $\mathbb{R}^n$:
+\vspace{-12pt}
+\[
+\gamma
+\colon
+I=[a,b] \to \mathbb{R}^n
+:
+t\mapsto \gamma(t)
+=
+\begin{pmatrix}
+x_1(t)\\
+x_2(t)\\
+\vdots\\
+x_n(t)
+\end{pmatrix}
+\]
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A) at (1,0.5);
+\coordinate (B) at (4,0.5);
+\coordinate (C) at (2,2.2);
+\coordinate (D) at (5,2);
+\coordinate (E) at ($(C)+(80:2)$);
+
+\draw[color=red,line width=1.4pt]
+	(A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D);
+\fill[color=red] (C) circle[radius=0.06];
+\node[color=red] at (C) [left] {$\gamma(t)$};
+
+\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E);
+\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$};
+
+\draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}];
+\draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}];
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Tangenten}
+Ableitung
+\[
+\frac{d}{dt}\gamma(t)
+=
+\dot{\gamma}(t)
+=
+\begin{pmatrix}
+\dot{x}_1(t)\\
+\dot{x}_2(t)\\
+\vdots\\
+\dot{x}_n(t)
+\end{pmatrix}
+\]
+Lineare Approximation:
+\[
+\gamma(t+h)
+=
+\gamma(t)
++
+\dot{\gamma}(t) \cdot h
++
+o(h)
+\]
+Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$
+in $\mathbb{R}^n$ liegen
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex
new file mode 100644
index 0000000..16a7aa0
--- /dev/null
+++ b/vorlesungen/slides/7/liealgebra.tex
@@ -0,0 +1,69 @@
+%
+% liealgebra.tex -- Lie-Algebra
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Lie-Algebra}
+\vspace{-20pt}
+\begin{block}{Vektorraum}
+Tangentialvektoren im Punkt $I$:
+\begin{center}
+\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}}
+\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\
+\hline
+\operatorname{GL}_n(\mathbb{R})
+& beliebige Matrizen
+& M_n(\mathbb{R})
+\\
+\operatorname{O(n)}
+& antisymmetrische Matrizen
+& \operatorname{o}(n)
+\\
+\operatorname{SL}_n(\mathbb{R})
+& spurlose Matrizen
+& \operatorname{sl}_2(\mathbb{R})
+\\
+\operatorname{U(n)}
+& antihermitesche Matrizen
+& \operatorname{u}(n)
+\\
+\operatorname{SU(n)}
+& spurlose, antihermitesche Matrizen
+& \operatorname{su}(n)
+\end{tabular}
+\end{center}
+\end{block}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Lie-Klammer}
+Kommutator: $[A,B] = AB-BA$
+\end{block}
+\begin{block}{Nachprüfen}
+$[A,B]\in LG$
+für $A,B\in LG$
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\begin{block}{Algebraische Eigenschaften}
+\begin{itemize}
+\item antisymmetrisch: $[A,B]=-[B,A]$
+\item Jacobi-Identität
+\[
+[A,[B,C]]+
+[B,[C,A]]+
+[C,[A,B]]
+= 0
+\]
+\end{itemize}
+{\usebeamercolor[fg]{title}
+Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex
new file mode 100644
index 0000000..7809ea5
--- /dev/null
+++ b/vorlesungen/slides/7/mannigfaltigkeit.tex
@@ -0,0 +1,44 @@
+%
+% mannigfaltigkeit.tex -- Mannigfaltigkeit
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Mannigfaltigkeit}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+\begin{itemize}
+\item Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$
+\item differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet
+\[
+\varphi_\beta\circ\varphi_\alpha^{-1}
+\colon
+\varphi_\alpha(U_\alpha\cap U_\beta)
+\to
+\varphi_\beta(U_\alpha\cap U_\beta)
+\]
+\item Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken
+\end{itemize}
+\end{block}
+\vspace{-7pt}
+\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$}
+Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus
+\end{block}
+\vspace{-3pt}
+\begin{block}{Lie-Gruppe}
+Gruppe und Mannigfaltigkeit
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex
new file mode 100644
index 0000000..ec129bb
--- /dev/null
+++ b/vorlesungen/slides/7/parameter.tex
@@ -0,0 +1,58 @@
+%
+% parameter.tex -- Parametrisierung der Matrizen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehungen Parametrisieren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.4\textwidth}
+\begin{block}{Drehung um Achsen}
+\begin{align*}
+D_{x,\alpha}
+&=
+\begin{pmatrix}
+1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha
+\end{pmatrix}
+\\
+D_{y,\beta}
+&=
+\begin{pmatrix}
+\cos\beta&0&-\sin\beta\\0&1&0\\\sin\beta&0&\cos\beta
+\end{pmatrix}
+\\
+D_{z,\gamma}
+&=
+\begin{pmatrix}
+\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1
+\end{pmatrix}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\begin{block}{Drehung um $\vec{\omega}$}
+$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel
+\\
+$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse
+\[
+\vec{x}
+\mapsto
+\cos\omega
+\vec{x}
++
+(\vec{k}\times\vec{x})\sin\omega
++
+\vec{k}(\vec{k}\cdot\vec{x}) (1-\cos\omega)
+\]
+XXX TODO: Bild für Rodriguez Formel
+\end{block}
+\end{column}
+\end{columns}
+{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine
+dreidimensionale Gruppe
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex
new file mode 100644
index 0000000..46f6d03
--- /dev/null
+++ b/vorlesungen/slides/7/semi.tex
@@ -0,0 +1,109 @@
+%
+% semi.tex -- Beispiele: semidirekte Produkte
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Drehung/Skalierung und Verschiebung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Skalierung und Verschiebung}
+Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$
+\\
+Wirkung auf $\mathbb{R}$:
+\[
+x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Drehung und Verschiebung}
+Gruppe
+$G=
+\{ (\alpha,\vec{t})
+\;|\;
+\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2
+\}$
+Wirkung auf $\mathbb{R}^2$:
+\[
+\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(e^{s_1},t_1)(e^{s_2},t_2)x
+&=
+(e^{s_1},t_1)(e^{s_2}x+t_2)
+\\
+&=
+e^{s_1+s_2}x + e^{s_1}t_2+t_1
+\\
+(e^{s_1},t_1)(e^{s_2},t_2)
+&=
+(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Verknüpfung}
+\vspace{-15pt}
+\begin{align*}
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+\vec{x}
+&=
+(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)
+\\
+&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1
+\\
+(\alpha_1,\vec{t}_1)
+(\alpha_2,\vec{t}_2)
+&=
+(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1)
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Matrixschreibweise}
+\vspace{-12pt}
+\[
+g=(e^s,t) =
+\begin{pmatrix}
+e^s&t\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}x\\1\end{pmatrix}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Matrixschreibweise}
+\vspace{-12pt}
+\[
+g=(\alpha,\vec{t}) =
+\begin{pmatrix}
+D_{\alpha}&\vec{t}\\
+0&1
+\end{pmatrix}
+\quad\text{auf}\quad
+\begin{pmatrix}\vec{x}\\1\end{pmatrix}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
new file mode 100644
index 0000000..3480460
--- /dev/null
+++ b/vorlesungen/slides/7/sl2.tex
@@ -0,0 +1,216 @@
+%
+% sl2.tex -- Beispiel: Parametrisierung von SL_2(R)
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t,fragile]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Determinante}
+\[
+A=\begin{pmatrix}
+a&b\\
+c&d
+\end{pmatrix}
+\;\Rightarrow\;
+\det A = ad-bc
+\]
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\begin{block}{Dimension}
+\[
+4\; \text{Variablen}
+-
+1\; \text{Bedingung}
+=
+3\; \text{Dimensionen}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<3->{%
+\begin{columns}[t,onlytextwidth]
+\def\s{0.94}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.3,2.3);
+	\fill[color=blue!20] (-1,-1) rectangle (1,1);
+	\foreach \x in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+	}
+	\foreach \y in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+	}
+	\foreach \d in {4,...,10}{
+		\only<\d>{
+			\pgfmathparse{1+(\d-4)/10}
+			\xdef\t{\pgfmathresult}
+			\fill[color=red!40,opacity=0.5]
+				({-\t},{-1/\t}) rectangle (\t,{1/\t});
+			\foreach \x in {-2,...,2}{
+				\draw[color=red,line width=0.3pt]
+					({\x*\t},-3) -- ({\x*\t},3);
+			}
+			\foreach \y in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+				(-3,{\y/\t}) -- (3,{\y/\t});
+			}
+		}
+	}
+	\uncover<11->{
+		\xdef\t{1.6}
+		\fill[color=red!40,opacity=0.5]
+			({-\t},{-1/\t}) rectangle (\t,{1/\t});
+		\foreach \x in {-2,...,2}{
+			\draw[color=red,line width=0.3pt]
+				({\x*\t},-3) -- ({\x*\t},3);
+		}
+		\foreach \y in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+			(-3,{\y/\t}) -- (3,{\y/\t});
+		}
+	}
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<3->{%
+	\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+	\node at (0,-2.1) {$
+	D
+	=
+	\begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix}
+	$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.3,2.3);
+	\foreach \x in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+	}
+	\foreach \y in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+	}
+	\foreach \d in {11,...,17}{
+		\only<\d>{
+			\pgfmathparse{(\d-11)/10}
+			\xdef\t{\pgfmathresult}
+			\fill[color=red!40,opacity=0.5]
+				({-1+\t*(-1)},{-1})
+				--
+				({1+\t*(-1)},{-1})
+				--
+				({1+\t},{1})
+				--
+				({-1+\t},{1})
+				-- cycle;
+			\foreach \x in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+			}
+			\foreach \y in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+			}
+		}
+	}
+	\uncover<18->{
+		\xdef\t{0.6}
+		\fill[color=red!40,opacity=0.5]
+			({-1+\t*(-1)},{-1})
+			--
+			({1+\t*(-1)},{-1})
+			--
+			({1+\t},{1})
+			--
+			({-1+\t},{1})
+			-- cycle;
+		\foreach \x in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+				({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3);
+		}
+		\foreach \y in {-3,...,3}{
+			\draw[color=red,line width=0.3pt]
+				({-3+\t*\y},\y) -- ({3+\t*\y},\y);
+		}
+	}
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<11->{
+	\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+	\node at (0,-2.1) {$
+	S
+	=
+	\begin{pmatrix} 1&s\\ 0&1\end{pmatrix}
+	$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.33\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=\s]
+\fill[color=blue!20] (-1,-1) rectangle (1,1);
+\begin{scope}
+	\clip (-2.1,-2.1) rectangle (2.3,2.3);
+	\foreach \x in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3);
+	}
+	\foreach \y in {-2,...,2}{
+		\draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y);
+	}
+	\foreach \d in {18,...,24}{
+		\only<\d>{
+			\pgfmathparse{(\d-18)/10}
+			\xdef\t{\pgfmathresult}
+			\fill[color=red!40,opacity=0.5]
+				(-1,{\t*(-1)-1})
+				--
+				(1,{\t*1-1})
+				--
+				(1,{\t*1+1})
+				--
+				(-1,{\t*(-1)+1})
+				-- cycle;
+			\foreach \x in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					(\x,{\x*\t-3}) -- (\x,{\x*\t+3});
+			}
+			\foreach \y in {-3,...,3}{
+				\draw[color=red,line width=0.3pt]
+					(-3,{-3*\t+\y}) -- (3,{3*\t+\y});
+			}
+		}
+	}
+\end{scope}
+\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}];
+\uncover<18->{%
+\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3);
+	\node at (0,-2.1) {$
+	T
+	=
+	\begin{pmatrix} 1&0\\t&1\end{pmatrix}
+	$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex
new file mode 100644
index 0000000..79f9ef7
--- /dev/null
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -0,0 +1,133 @@
+%
+% symmetrien.tex -- Symmetrien
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Symmetrien}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diskrete Symmetrien}
+\begin{itemize}
+\item
+Ebenen-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+{\tiny
+\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*},
+}
+\;
+\vec{x}
+\mapsto
+\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
+\]
+\vspace{-10pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{10}
+\def\b{50}
+\def\r{2}
+\coordinate (O) at (0,0);
+\coordinate (A) at (\b:\r);
+\coordinate (B) at ({180+2*\a-\b}:\r);
+\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
+\coordinate (N) at (\a:2);
+\coordinate (D) at (\a:{\r*cos(\b-\a)});
+\clip (-2.5,-0.45) rectangle (2.5,1.95);
+
+\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) -- cycle;
+\draw[->,color=darkgreen] (O) -- (N);
+\node[color=darkgreen] at (N) [above] {$\vec{n}$};
+
+
+\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) -- cycle;
+\fill[color=red] (O) circle[radius=0.06];
+\draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
+\fill[color=blue] (C) circle[radius=0.06];
+\draw[color=blue,line width=0.1pt] (A) -- (D);
+\node[color=darkgreen] at (D) [below,rotate=\a] {$(\vec{n}\cdot\vec{x})\vec{n}$};
+\draw[color=blue,line width=0.5pt] (A)--(B);
+
+\node[color=blue] at (A) [above right] {$\vec{x}$};
+\node[color=blue] at (B) [above left] {$\vec{x}'$};
+
+\node[color=red] at (O) [below left] {$O$};
+
+\draw[->,color=blue,shorten <= 0.06cm] (O) -- (A);
+\draw[->,color=blue,shorten <= 0.06cm] (O) -- (B);
+
+\end{tikzpicture}
+\end{center}
+\vspace{-5pt}
+$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
+\item
+Punkt-Spiegelung:
+\[
+{\tiny
+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\mapsto
+-
+{\tiny
+\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
+}
+\]
+\end{itemize}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Kontinuierliche Symmetrien}
+\begin{itemize}
+\item Translation:
+\(
+\vec{x} \mapsto \vec{x} + \vec{t}
+\)
+\item Drehung:
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\a{25}
+\def\r{1.3}
+\coordinate (O) at (0,0);
+\begin{scope}
+\clip (-1.1,-0.1) rectangle (2.3,2.3);
+\draw[color=red] (O) circle[radius=2];
+\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
+\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
+\node at ({0.5*\a}:1) {$\alpha$};
+\node at ({90+0.5*\a}:1) {$\alpha$};
+\draw[->,color=blue] (O) -- (\a:2);
+\draw[->,color=darkgreen] (O) -- ({90+\a}:2);
+\end{scope}
+\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
+\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
+\end{tikzpicture}
+\end{center}
+\[
+\begin{pmatrix}x\\y\end{pmatrix}
+\mapsto
+\begin{pmatrix}
+{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
+{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
+\end{pmatrix}
+\begin{pmatrix}x\\y\end{pmatrix}
+\]
+\end{itemize}
+\end{block}
+\vspace{-10pt}
+\begin{block}{Definition}
+Längen/Winkel bleiben erhalten
+\\
+$\Rightarrow$ $\exists$ Erhaltungsgrösse
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup