Merge remote-tracking branch 'upstream/master'

24 files changed, 1340 insertions, 13 deletions

diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index 7afeea1..4d291ed 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -16,7 +16,19 @@ chapter5 =								\
 	../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/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/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
index 079cf16..36e0bb1 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -15,5 +15,17 @@
 \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/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
index 4447bac..f9528a4 100644
--- a/vorlesungen/slides/7/dg.tex
+++ b/vorlesungen/slides/7/dg.tex
@@ -45,7 +45,7 @@ Ableitung von $\gamma(t)$ an der Stelle $t$:
 \vspace{-10pt}
 \uncover<7->{%
 \begin{block}{Differentialgleichung}
-\vspace{-10pt}
+%\vspace{-10pt}
 \[
 \dot{\gamma}(t) = \gamma(t) A
 \quad
@@ -66,7 +66,7 @@ Exponentialfunktion
 \vspace{-5pt}
 \uncover<9->{%
 \begin{block}{Kontrolle: Tangentialvektor berechnen}
-\vspace{-10pt}
+%\vspace{-10pt}
 \begin{align*}
 \frac{d}{dt}e^{At}
 &\uncover<10->{=
diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex
index 2d7b317..02201d4 100644
--- a/vorlesungen/slides/7/drehung.tex
+++ b/vorlesungen/slides/7/drehung.tex
@@ -58,7 +58,7 @@ D_{60^\circ}
 \begin{column}{0.58\textwidth}
 \uncover<4->{%
 \begin{block}{Ansatz}
-\vspace{-12pt}
+%\vspace{-12pt}
 \begin{align*}
 DST
 &=
@@ -101,7 +101,7 @@ c^{-1}&0\\
 \vspace{-10pt}
 \uncover<7->{%
 \begin{block}{Koeffizientenvergleich}
-\vspace{-15pt}
+%\vspace{-15pt}
 \begin{align*}
 \uncover<8->{
 {\color{red} c}
diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex
index 5171085..a32affd 100644
--- a/vorlesungen/slides/7/einparameter.tex
+++ b/vorlesungen/slides/7/einparameter.tex
@@ -41,7 +41,7 @@ D_{x,t+s}
 \begin{column}{0.48\textwidth}
 \uncover<5->{%
 \begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
+%\vspace{-12pt}
 \[
 \begin{pmatrix}
 1&s\\
@@ -61,7 +61,7 @@ D_{x,t+s}
 \vspace{-12pt}
 \uncover<6->{%
 \begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$}
-\vspace{-12pt}
+%\vspace{-12pt}
 \[
 \begin{pmatrix}
 e^s&0\\0&e^{-s}
@@ -78,7 +78,7 @@ e^{t+s}&0\\0&e^{-(t+s)}
 \vspace{-12pt}
 \uncover<7->{%
 \begin{block}{Gemischt}
-\vspace{-12pt}
+%\vspace{-12pt}
 \begin{gather*}
 A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t
 \\
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
index cc67c8a..6f99bc3 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -3,7 +3,7 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 # 
-all:	rodriguez.jpg 
+all:	rodriguez.jpg  test.png
 
 rodriguez.png:	rodriguez.pov
 	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
@@ -16,4 +16,14 @@ commutator:	commutator.ini commutator.pov common.inc
 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/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/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/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/parameter.tex b/vorlesungen/slides/7/parameter.tex
index 52c8e4a..f3579a3 100644
--- a/vorlesungen/slides/7/parameter.tex
+++ b/vorlesungen/slides/7/parameter.tex
@@ -14,7 +14,7 @@
 \begin{columns}[t,onlytextwidth]
 \begin{column}{0.4\textwidth}
 \begin{block}{Drehung um Achsen}
-\vspace{-12pt}
+%\vspace{-12pt}
 \begin{align*}
 \uncover<2->{
 D_{x,\alpha}
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
index 66b8d27..cd974c9 100644
--- a/vorlesungen/slides/7/semi.tex
+++ b/vorlesungen/slides/7/semi.tex
@@ -41,7 +41,7 @@ Wirkung auf $\mathbb{R}^2$:
 \begin{column}{0.48\textwidth}
 \uncover<3->{%
 \begin{block}{Verknüpfung}
-\vspace{-15pt}
+%\vspace{-15pt}
 \begin{align*}
 (e^{s_1},t_1)(e^{s_2},t_2)x
 &\uncover<4->{=
@@ -60,7 +60,7 @@ e^{s_1+s_2}x + e^{s_1}t_2+t_1}
 \begin{column}{0.48\textwidth}
 \uncover<7->{%
 \begin{block}{Verknüpfung}
-\vspace{-15pt}
+%\vspace{-15pt}
 \begin{align*}
 (\alpha_1,\vec{t}_1)
 (\alpha_2,\vec{t}_2)
@@ -85,7 +85,7 @@ e^{s_1+s_2}x + e^{s_1}t_2+t_1}
 \begin{column}{0.48\textwidth}
 \uncover<11->{%
 \begin{block}{Matrixschreibweise}
-\vspace{-12pt}
+%\vspace{-12pt}
 \[
 g=(e^s,t) =
 \begin{pmatrix}
@@ -100,7 +100,7 @@ e^s&t\\
 \begin{column}{0.48\textwidth}
 \uncover<12->{%
 \begin{block}{Matrixschreibweise}
-\vspace{-12pt}
+%\vspace{-12pt}
 \[
 g=(\alpha,\vec{t}) =
 \begin{pmatrix}
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