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-rw-r--r--vorlesungen/slides/7/Makefile.inc7
-rw-r--r--vorlesungen/slides/7/bch.tex76
-rw-r--r--vorlesungen/slides/7/chapter.tex7
-rw-r--r--vorlesungen/slides/7/dg.tex4
-rw-r--r--vorlesungen/slides/7/einparameter.tex6
-rw-r--r--vorlesungen/slides/7/exponentialreihe.tex24
-rw-r--r--vorlesungen/slides/7/images/Makefile12
-rw-r--r--vorlesungen/slides/7/images/drehung.inc142
-rw-r--r--vorlesungen/slides/7/images/interpolation.ini8
-rw-r--r--vorlesungen/slides/7/images/interpolation.m54
-rw-r--r--vorlesungen/slides/7/images/interpolation.pov10
-rw-r--r--vorlesungen/slides/7/images/test.pov7
-rw-r--r--vorlesungen/slides/7/integration.tex66
-rw-r--r--vorlesungen/slides/7/interpolation.tex112
-rw-r--r--vorlesungen/slides/7/liealgbeispiel.tex78
-rw-r--r--vorlesungen/slides/7/logarithmus.tex82
-rw-r--r--vorlesungen/slides/7/vektorlie.tex206
17 files changed, 895 insertions, 6 deletions
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index 7512612..ffd5091 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -16,13 +16,20 @@ 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/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/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 1c78ccc..3736e0f 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -15,11 +15,18 @@
\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
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/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/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/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/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/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