new images

70 files changed, 427 insertions, 5 deletions

diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc
index ef004ca..540b080 100644
--- a/vorlesungen/slides/7/Makefile.inc
+++ b/vorlesungen/slides/7/Makefile.inc
@@ -15,6 +15,7 @@ chapter5 =								\
 	../slides/7/einparameter.tex					\
 	../slides/7/ableitung.tex					\
 	../slides/7/liealgebra.tex					\
+	../slides/7/kommutator.tex					\
 	../slides/7/dg.tex						\
 	../slides/7/chapter.tex
 
diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex
index 44d46a6..5878164 100644
--- a/vorlesungen/slides/7/chapter.tex
+++ b/vorlesungen/slides/7/chapter.tex
@@ -14,4 +14,5 @@
 \folie{7/einparameter.tex}
 \folie{7/ableitung.tex}
 \folie{7/liealgebra.tex}
+\folie{7/kommutator.tex}
 \folie{7/dg.tex}
diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile
index 1b9590d..cc67c8a 100644
--- a/vorlesungen/slides/7/images/Makefile
+++ b/vorlesungen/slides/7/images/Makefile
@@ -3,10 +3,17 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 # 
-all:	rodriguez.jpg
+all:	rodriguez.jpg 
 
 rodriguez.png:	rodriguez.pov
 	povray +A0.1 -W1920 -H1080 -Orodriguez.png rodriguez.pov
 
 rodriguez.jpg:	rodriguez.png 
 	convert -extract 1740x1070+135+10 rodriguez.png rodriguez.jpg
+
+commutator:	commutator.ini commutator.pov common.inc
+	povray +A0.1 -W1920 -H1080 -Oc/c.png commutator.ini
+jpg:
+	for f in c/c*.png; do convert $${f} c/`basename $${f} .png`.jpg; done
+
+
diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg
new file mode 100644
index 0000000..b2dbdb2
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c01.jpg
diff --git a/vorlesungen/slides/7/images/c/c02.jpg b/vorlesungen/slides/7/images/c/c02.jpg
new file mode 100644
index 0000000..9b45ba3
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c02.jpg
diff --git a/vorlesungen/slides/7/images/c/c03.jpg b/vorlesungen/slides/7/images/c/c03.jpg
new file mode 100644
index 0000000..882be40
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c03.jpg
diff --git a/vorlesungen/slides/7/images/c/c04.jpg b/vorlesungen/slides/7/images/c/c04.jpg
new file mode 100644
index 0000000..5d26572
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c04.jpg
diff --git a/vorlesungen/slides/7/images/c/c05.jpg b/vorlesungen/slides/7/images/c/c05.jpg
new file mode 100644
index 0000000..458f565
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c05.jpg
diff --git a/vorlesungen/slides/7/images/c/c06.jpg b/vorlesungen/slides/7/images/c/c06.jpg
new file mode 100644
index 0000000..cd40cda
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c06.jpg
diff --git a/vorlesungen/slides/7/images/c/c07.jpg b/vorlesungen/slides/7/images/c/c07.jpg
new file mode 100644
index 0000000..3349795
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c07.jpg
diff --git a/vorlesungen/slides/7/images/c/c08.jpg b/vorlesungen/slides/7/images/c/c08.jpg
new file mode 100644
index 0000000..65048cf
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c08.jpg
diff --git a/vorlesungen/slides/7/images/c/c09.jpg b/vorlesungen/slides/7/images/c/c09.jpg
new file mode 100644
index 0000000..000d502
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c09.jpg
diff --git a/vorlesungen/slides/7/images/c/c10.jpg b/vorlesungen/slides/7/images/c/c10.jpg
new file mode 100644
index 0000000..064c96b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c10.jpg
diff --git a/vorlesungen/slides/7/images/c/c11.jpg b/vorlesungen/slides/7/images/c/c11.jpg
new file mode 100644
index 0000000..c67bc5d
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c11.jpg
diff --git a/vorlesungen/slides/7/images/c/c12.jpg b/vorlesungen/slides/7/images/c/c12.jpg
new file mode 100644
index 0000000..f2174de
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c12.jpg
diff --git a/vorlesungen/slides/7/images/c/c13.jpg b/vorlesungen/slides/7/images/c/c13.jpg
new file mode 100644
index 0000000..3cb5ae6
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c13.jpg
diff --git a/vorlesungen/slides/7/images/c/c14.jpg b/vorlesungen/slides/7/images/c/c14.jpg
new file mode 100644
index 0000000..6985e4f
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c14.jpg
diff --git a/vorlesungen/slides/7/images/c/c15.jpg b/vorlesungen/slides/7/images/c/c15.jpg
new file mode 100644
index 0000000..5bb5aed
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c15.jpg
diff --git a/vorlesungen/slides/7/images/c/c16.jpg b/vorlesungen/slides/7/images/c/c16.jpg
new file mode 100644
index 0000000..cc77fa1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c16.jpg
diff --git a/vorlesungen/slides/7/images/c/c17.jpg b/vorlesungen/slides/7/images/c/c17.jpg
new file mode 100644
index 0000000..a60e2e9
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c17.jpg
diff --git a/vorlesungen/slides/7/images/c/c18.jpg b/vorlesungen/slides/7/images/c/c18.jpg
new file mode 100644
index 0000000..1d08c29
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c18.jpg
diff --git a/vorlesungen/slides/7/images/c/c19.jpg b/vorlesungen/slides/7/images/c/c19.jpg
new file mode 100644
index 0000000..9210c60
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c19.jpg
diff --git a/vorlesungen/slides/7/images/c/c20.jpg b/vorlesungen/slides/7/images/c/c20.jpg
new file mode 100644
index 0000000..8951883
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c20.jpg
diff --git a/vorlesungen/slides/7/images/c/c21.jpg b/vorlesungen/slides/7/images/c/c21.jpg
new file mode 100644
index 0000000..5169c7e
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c21.jpg
diff --git a/vorlesungen/slides/7/images/c/c22.jpg b/vorlesungen/slides/7/images/c/c22.jpg
new file mode 100644
index 0000000..bdeb90b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c22.jpg
diff --git a/vorlesungen/slides/7/images/c/c23.jpg b/vorlesungen/slides/7/images/c/c23.jpg
new file mode 100644
index 0000000..fa3eac7
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c23.jpg
diff --git a/vorlesungen/slides/7/images/c/c24.jpg b/vorlesungen/slides/7/images/c/c24.jpg
new file mode 100644
index 0000000..52adc13
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c24.jpg
diff --git a/vorlesungen/slides/7/images/c/c25.jpg b/vorlesungen/slides/7/images/c/c25.jpg
new file mode 100644
index 0000000..d557497
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c25.jpg
diff --git a/vorlesungen/slides/7/images/c/c26.jpg b/vorlesungen/slides/7/images/c/c26.jpg
new file mode 100644
index 0000000..f825f49
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c26.jpg
diff --git a/vorlesungen/slides/7/images/c/c27.jpg b/vorlesungen/slides/7/images/c/c27.jpg
new file mode 100644
index 0000000..0de6c3e
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c27.jpg
diff --git a/vorlesungen/slides/7/images/c/c28.jpg b/vorlesungen/slides/7/images/c/c28.jpg
new file mode 100644
index 0000000..d9f89a4
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c28.jpg
diff --git a/vorlesungen/slides/7/images/c/c29.jpg b/vorlesungen/slides/7/images/c/c29.jpg
new file mode 100644
index 0000000..937f692
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c29.jpg
diff --git a/vorlesungen/slides/7/images/c/c30.jpg b/vorlesungen/slides/7/images/c/c30.jpg
new file mode 100644
index 0000000..45901cc
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c30.jpg
diff --git a/vorlesungen/slides/7/images/c/c31.jpg b/vorlesungen/slides/7/images/c/c31.jpg
new file mode 100644
index 0000000..eb52bec
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c31.jpg
diff --git a/vorlesungen/slides/7/images/c/c32.jpg b/vorlesungen/slides/7/images/c/c32.jpg
new file mode 100644
index 0000000..a011640
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c32.jpg
diff --git a/vorlesungen/slides/7/images/c/c33.jpg b/vorlesungen/slides/7/images/c/c33.jpg
new file mode 100644
index 0000000..ecbb8bd
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c33.jpg
diff --git a/vorlesungen/slides/7/images/c/c34.jpg b/vorlesungen/slides/7/images/c/c34.jpg
new file mode 100644
index 0000000..8a624d1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c34.jpg
diff --git a/vorlesungen/slides/7/images/c/c35.jpg b/vorlesungen/slides/7/images/c/c35.jpg
new file mode 100644
index 0000000..33765a1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c35.jpg
diff --git a/vorlesungen/slides/7/images/c/c36.jpg b/vorlesungen/slides/7/images/c/c36.jpg
new file mode 100644
index 0000000..e38a448
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c36.jpg
diff --git a/vorlesungen/slides/7/images/c/c37.jpg b/vorlesungen/slides/7/images/c/c37.jpg
new file mode 100644
index 0000000..9f823f4
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c37.jpg
diff --git a/vorlesungen/slides/7/images/c/c38.jpg b/vorlesungen/slides/7/images/c/c38.jpg
new file mode 100644
index 0000000..19c96fc
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c38.jpg
diff --git a/vorlesungen/slides/7/images/c/c39.jpg b/vorlesungen/slides/7/images/c/c39.jpg
new file mode 100644
index 0000000..c00482b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c39.jpg
diff --git a/vorlesungen/slides/7/images/c/c40.jpg b/vorlesungen/slides/7/images/c/c40.jpg
new file mode 100644
index 0000000..de9fca4
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c40.jpg
diff --git a/vorlesungen/slides/7/images/c/c41.jpg b/vorlesungen/slides/7/images/c/c41.jpg
new file mode 100644
index 0000000..1b8a3d1
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c41.jpg
diff --git a/vorlesungen/slides/7/images/c/c42.jpg b/vorlesungen/slides/7/images/c/c42.jpg
new file mode 100644
index 0000000..6d4e5bc
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c42.jpg
diff --git a/vorlesungen/slides/7/images/c/c43.jpg b/vorlesungen/slides/7/images/c/c43.jpg
new file mode 100644
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diff --git a/vorlesungen/slides/7/images/c/c44.jpg b/vorlesungen/slides/7/images/c/c44.jpg
new file mode 100644
index 0000000..0fe56f6
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c44.jpg
diff --git a/vorlesungen/slides/7/images/c/c45.jpg b/vorlesungen/slides/7/images/c/c45.jpg
new file mode 100644
index 0000000..9196ad3
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+++ b/vorlesungen/slides/7/images/c/c45.jpg
diff --git a/vorlesungen/slides/7/images/c/c46.jpg b/vorlesungen/slides/7/images/c/c46.jpg
new file mode 100644
index 0000000..a4f5823
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+++ b/vorlesungen/slides/7/images/c/c46.jpg
diff --git a/vorlesungen/slides/7/images/c/c47.jpg b/vorlesungen/slides/7/images/c/c47.jpg
new file mode 100644
index 0000000..18474dd
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+++ b/vorlesungen/slides/7/images/c/c47.jpg
diff --git a/vorlesungen/slides/7/images/c/c48.jpg b/vorlesungen/slides/7/images/c/c48.jpg
new file mode 100644
index 0000000..d839014
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+++ b/vorlesungen/slides/7/images/c/c48.jpg
diff --git a/vorlesungen/slides/7/images/c/c49.jpg b/vorlesungen/slides/7/images/c/c49.jpg
new file mode 100644
index 0000000..3cb5cb0
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diff --git a/vorlesungen/slides/7/images/c/c50.jpg b/vorlesungen/slides/7/images/c/c50.jpg
new file mode 100644
index 0000000..de32f8b
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c50.jpg
diff --git a/vorlesungen/slides/7/images/c/c51.jpg b/vorlesungen/slides/7/images/c/c51.jpg
new file mode 100644
index 0000000..4bbc224
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c51.jpg
diff --git a/vorlesungen/slides/7/images/c/c52.jpg b/vorlesungen/slides/7/images/c/c52.jpg
new file mode 100644
index 0000000..ebf1db6
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c52.jpg
diff --git a/vorlesungen/slides/7/images/c/c53.jpg b/vorlesungen/slides/7/images/c/c53.jpg
new file mode 100644
index 0000000..2ece537
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c53.jpg
diff --git a/vorlesungen/slides/7/images/c/c54.jpg b/vorlesungen/slides/7/images/c/c54.jpg
new file mode 100644
index 0000000..0ffbac2
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c54.jpg
diff --git a/vorlesungen/slides/7/images/c/c55.jpg b/vorlesungen/slides/7/images/c/c55.jpg
new file mode 100644
index 0000000..5f75419
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c55.jpg
diff --git a/vorlesungen/slides/7/images/c/c56.jpg b/vorlesungen/slides/7/images/c/c56.jpg
new file mode 100644
index 0000000..5c0f9ae
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c56.jpg
diff --git a/vorlesungen/slides/7/images/c/c57.jpg b/vorlesungen/slides/7/images/c/c57.jpg
new file mode 100644
index 0000000..9b61179
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c57.jpg
diff --git a/vorlesungen/slides/7/images/c/c58.jpg b/vorlesungen/slides/7/images/c/c58.jpg
new file mode 100644
index 0000000..81d2173
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c58.jpg
diff --git a/vorlesungen/slides/7/images/c/c59.jpg b/vorlesungen/slides/7/images/c/c59.jpg
new file mode 100644
index 0000000..0ae9bac
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c59.jpg
diff --git a/vorlesungen/slides/7/images/c/c60.jpg b/vorlesungen/slides/7/images/c/c60.jpg
new file mode 100644
index 0000000..2bc77e7
--- /dev/null
+++ b/vorlesungen/slides/7/images/c/c60.jpg
diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc
new file mode 100644
index 0000000..0e27c9a
--- /dev/null
+++ 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/kommutator.tex b/vorlesungen/slides/7/kommutator.tex
new file mode 100644
index 0000000..f01debe
--- /dev/null
+++ b/vorlesungen/slides/7/kommutator.tex
@@ -0,0 +1,139 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kommutator in $\operatorname{SO}(3)$}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\ifthenelse{\boolean{presentation}}{
+\only<1>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c01.jpg}};}
+\only<2>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c02.jpg}};}
+\only<3>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c03.jpg}};}
+\only<4>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c04.jpg}};}
+\only<5>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c05.jpg}};}
+\only<6>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c06.jpg}};}
+\only<7>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c07.jpg}};}
+\only<8>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c08.jpg}};}
+\only<9>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c09.jpg}};}
+\only<10>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c10.jpg}};}
+\only<11>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c11.jpg}};}
+\only<12>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c12.jpg}};}
+\only<13>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c13.jpg}};}
+\only<14>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c14.jpg}};}
+\only<15>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c15.jpg}};}
+\only<16>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c16.jpg}};}
+\only<17>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c17.jpg}};}
+\only<18>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c18.jpg}};}
+\only<19>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c19.jpg}};}
+\only<20>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c20.jpg}};}
+\only<21>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c21.jpg}};}
+\only<22>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c22.jpg}};}
+\only<23>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c23.jpg}};}
+\only<24>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c24.jpg}};}
+\only<25>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c25.jpg}};}
+\only<26>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c26.jpg}};}
+\only<27>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c27.jpg}};}
+\only<28>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c28.jpg}};}
+\only<29>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c29.jpg}};}
+\only<30>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c30.jpg}};}
+\only<31>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c31.jpg}};}
+\only<32>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c32.jpg}};}
+\only<33>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c33.jpg}};}
+\only<34>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c34.jpg}};}
+\only<35>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c35.jpg}};}
+\only<36>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c36.jpg}};}
+\only<37>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c37.jpg}};}
+\only<38>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c38.jpg}};}
+\only<39>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c39.jpg}};}
+\only<40>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c40.jpg}};}
+\only<41>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c41.jpg}};}
+\only<42>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c42.jpg}};}
+\only<43>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c43.jpg}};}
+\only<44>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c44.jpg}};}
+\only<45>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c45.jpg}};}
+\only<46>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c46.jpg}};}
+\only<47>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c47.jpg}};}
+\only<48>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c48.jpg}};}
+\only<49>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c49.jpg}};}
+\only<50>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c50.jpg}};}
+\only<51>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c51.jpg}};}
+\only<52>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c52.jpg}};}
+\only<53>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c53.jpg}};}
+\only<54>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c54.jpg}};}
+\only<55>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c55.jpg}};}
+\only<56>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c56.jpg}};}
+\only<57>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c57.jpg}};}
+\only<58>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c58.jpg}};}
+\only<59>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c59.jpg}};}
+\only<60>{\node at (0,0) {
+\includegraphics[width=\textwidth]{../slides/7/images/c/c60.jpg}};}
+}{}
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex
index 3480460..a65b4f6 100644
--- a/vorlesungen/slides/7/sl2.tex
+++ b/vorlesungen/slides/7/sl2.tex
@@ -50,6 +50,7 @@ c&d
 	\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}
@@ -66,6 +67,7 @@ c&d
 			}
 		}
 	}
+	}{}
 	\uncover<11->{
 		\xdef\t{1.6}
 		\fill[color=red!40,opacity=0.5]
@@ -105,6 +107,7 @@ c&d
 	\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}
@@ -128,6 +131,7 @@ c&d
 			}
 		}
 	}
+	}{}
 	\uncover<18->{
 		\xdef\t{0.6}
 		\fill[color=red!40,opacity=0.5]
@@ -174,6 +178,7 @@ c&d
 	\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}
@@ -197,6 +202,27 @@ c&d
 			}
 		}
 	}
+	}{}
+	\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:$y$}];
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index 7062509..ee6829e 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -15,8 +15,8 @@
 \folie{7/mannigfaltigkeit.tex}
 % Weitere Beispiele
 % SL_2(R)
-%\folie{7/sl2.tex}
-%\folie{7/drehung.tex}
+\folie{7/sl2.tex}
+\folie{7/drehung.tex}
 % Semidirekte Produkte SO(2) x R^2, R^+ x R
 \folie{7/semi.tex}
 
@@ -29,10 +29,10 @@
 \folie{7/ableitung.tex}
 % Lie-Algebra
 \folie{7/liealgebra.tex}
+% Kommutator
+\folie{7/kommutator.tex}
 
 \section{Exponentialabbildung}
 % Differentialgleichung für die Exponentialabbildung
 \folie{7/dg.tex}
-% Exponentialabbilung
-%\folie{7/exponential.tex}