From 2db90bfe4b174570424c408f04000902411d8755 Mon Sep 17 00:00:00 2001 From: Joshua Baer Date: Mon, 12 Apr 2021 21:51:55 +0200 Subject: update to current state of book --- vorlesungen/06_msegalois/Makefile | 66 +-- vorlesungen/06_msegalois/MathSemMSE-06-galois.tex | 28 +- vorlesungen/06_msegalois/common.tex | 32 +- vorlesungen/06_msegalois/galois-handout.tex | 22 +- vorlesungen/06_msegalois/slides.tex | 46 +- vorlesungen/07_lie/Makefile | 66 +-- vorlesungen/07_lie/MathSem-07-lie.tex | 36 +- vorlesungen/07_lie/common.tex | 32 +- vorlesungen/07_lie/lie-handout.tex | 22 +- vorlesungen/07_lie/slides.tex | 52 +-- vorlesungen/common/README | 56 +-- vorlesungen/common/presentation-template.tex | 98 ++--- vorlesungen/common/slide-template.tex | 38 +- vorlesungen/slides/4/Makefile.inc | 72 ++-- vorlesungen/slides/4/chapter.tex | 62 +-- vorlesungen/slides/4/galois/aufloesbarkeit.tex | 240 +++++------ vorlesungen/slides/4/galois/automorphismus.tex | 236 +++++------ vorlesungen/slides/4/galois/erweiterung.tex | 130 +++--- vorlesungen/slides/4/galois/images/Makefile | 24 +- vorlesungen/slides/4/galois/images/common.inc | 178 ++++---- vorlesungen/slides/4/galois/images/wuerfel.pov | 18 +- vorlesungen/slides/4/galois/images/wuerfel2.pov | 18 +- vorlesungen/slides/4/galois/konstruktion.tex | 294 ++++++------- vorlesungen/slides/4/galois/quadratur.tex | 132 +++--- vorlesungen/slides/4/galois/radikale.tex | 138 +++--- vorlesungen/slides/4/galois/sn.tex | 174 ++++---- vorlesungen/slides/4/galois/winkeldreiteilung.tex | 188 ++++----- vorlesungen/slides/4/galois/wuerfel.tex | 128 +++--- vorlesungen/slides/7/Makefile.inc | 44 +- vorlesungen/slides/7/ableitung.tex | 136 +++--- vorlesungen/slides/7/algebraisch.tex | 230 +++++----- vorlesungen/slides/7/chapter.tex | 38 +- vorlesungen/slides/7/dg.tex | 184 ++++---- vorlesungen/slides/7/drehanim.tex | 310 +++++++------- vorlesungen/slides/7/drehung.tex | 264 ++++++------ vorlesungen/slides/7/einparameter.tex | 186 ++++----- vorlesungen/slides/7/images/Makefile | 38 +- vorlesungen/slides/7/images/common.inc | 140 +++---- vorlesungen/slides/7/images/commutator.ini | 16 +- vorlesungen/slides/7/images/commutator.m | 222 +++++----- vorlesungen/slides/7/images/commutator.pov | 118 +++--- vorlesungen/slides/7/images/rodriguez.pov | 236 +++++------ vorlesungen/slides/7/kommutator.tex | 332 +++++++-------- vorlesungen/slides/7/kurven.tex | 208 +++++----- vorlesungen/slides/7/liealgebra.tex | 170 ++++---- vorlesungen/slides/7/mannigfaltigkeit.tex | 92 ++-- vorlesungen/slides/7/parameter.tex | 214 +++++----- vorlesungen/slides/7/semi.tex | 234 +++++------ vorlesungen/slides/7/sl2.tex | 484 +++++++++++----------- vorlesungen/slides/7/symmetrien.tex | 290 ++++++------- vorlesungen/slides/Makefile.inc | 36 +- vorlesungen/slides/test.tex | 78 ++-- vorlesungen/stream/countdown.html | 80 ++-- vorlesungen/stream/ende.html | 60 +-- 54 files changed, 3533 insertions(+), 3533 deletions(-) (limited to 'vorlesungen') diff --git a/vorlesungen/06_msegalois/Makefile b/vorlesungen/06_msegalois/Makefile index a6e829d..4cdc3d1 100644 --- a/vorlesungen/06_msegalois/Makefile +++ b/vorlesungen/06_msegalois/Makefile @@ -1,33 +1,33 @@ -# -# Makefile -- galois -# -# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -# -all: galois-handout.pdf MathSemMSE-06-galois.pdf - -include ../slides/Makefile.inc - -SOURCES = common.tex slides.tex $(slides) - -MathSemMSE-06-galois.pdf: MathSemMSE-06-galois.tex $(SOURCES) - pdflatex MathSemMSE-06-galois.tex - -galois-handout.pdf: galois-handout.tex $(SOURCES) - pdflatex galois-handout.tex - -thumbnail: thumbnail.jpg # fix1.jpg - -thumbnail.pdf: MathSemMSE-06-galois.pdf - pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ - MathSemMSE-06-galois.pdf 1 -thumbnail.jpg: thumbnail.pdf - convert -density 300 thumbnail.pdf \ - -resize 1920x1080 -units PixelsPerInch thumbnail.jpg - -fix1.pdf: MathSemMSE-06-galois.pdf - pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ - MathSemMSE-06-galois.pdf 1 -fix1.jpg: fix1.pdf - convert -density 300 fix1.pdf \ - -resize 1920x1080 -units PixelsPerInch fix1.jpg - +# +# Makefile -- galois +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: galois-handout.pdf MathSemMSE-06-galois.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-06-galois.pdf: MathSemMSE-06-galois.tex $(SOURCES) + pdflatex MathSemMSE-06-galois.tex + +galois-handout.pdf: galois-handout.tex $(SOURCES) + pdflatex galois-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-06-galois.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-06-galois.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-06-galois.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-06-galois.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex b/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex index 2b8da6d..1f6b354 100644 --- a/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex +++ b/vorlesungen/06_msegalois/MathSemMSE-06-galois.tex @@ -1,14 +1,14 @@ -% -% MathSem-06-msegalois.tex -- Präsentation -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\documentclass[aspectratio=169]{beamer} -\input{common.tex} -\setboolean{presentation}{true} -\begin{document} -\begin{frame} -\titlepage -\end{frame} -\input{slides.tex} -\end{document} +% +% MathSem-06-msegalois.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/06_msegalois/common.tex b/vorlesungen/06_msegalois/common.tex index 50adc4f..f88a87b 100644 --- a/vorlesungen/06_msegalois/common.tex +++ b/vorlesungen/06_msegalois/common.tex @@ -1,16 +1,16 @@ -% -% common.tex -- gemeinsame definition -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\input{../common/packages.tex} -\input{../common/common.tex} -\mode{% -\usetheme[hideothersubsections,hidetitle]{Hannover} -} -\beamertemplatenavigationsymbolsempty -\title[Galois]{Galois-Theorie} -\author[A.~Müller]{Prof. Dr. Andreas Müller} -\date[]{} -\newboolean{presentation} - +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Galois]{Galois-Theorie} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/06_msegalois/galois-handout.tex b/vorlesungen/06_msegalois/galois-handout.tex index e3e80f8..54238f6 100644 --- a/vorlesungen/06_msegalois/galois-handout.tex +++ b/vorlesungen/06_msegalois/galois-handout.tex @@ -1,11 +1,11 @@ -% -% msegalois-handout.tex -- Handout XXX -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\documentclass[handout,aspectratio=169]{beamer} -\input{common.tex} -\setboolean{presentation}{false} -\begin{document} -\input{slides.tex} -\end{document} +% +% msegalois-handout.tex -- Handout XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/06_msegalois/slides.tex b/vorlesungen/06_msegalois/slides.tex index 95695c4..386d19f 100644 --- a/vorlesungen/06_msegalois/slides.tex +++ b/vorlesungen/06_msegalois/slides.tex @@ -1,23 +1,23 @@ -% -% slides.tex -- Slides für die kleine Einführung in die Galois-Theorie -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% - -\section{Körpererweiterungen} -\folie{4/galois/erweiterung.tex} - -\section{Geometrische Anwendungen} -\folie{4/galois/konstruktion.tex} -\folie{4/galois/wuerfel.tex} -\folie{4/galois/winkeldreiteilung.tex} -\folie{4/galois/quadratur.tex} - -\section{Galois-Gruppe} -\folie{4/galois/automorphismus.tex} - -\section{Lösbarkeit durch Radikale} -\folie{4/galois/radikale.tex} -\folie{4/galois/aufloesbarkeit.tex} -\folie{4/galois/sn.tex} - +% +% slides.tex -- Slides für die kleine Einführung in die Galois-Theorie +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Körpererweiterungen} +\folie{4/galois/erweiterung.tex} + +\section{Geometrische Anwendungen} +\folie{4/galois/konstruktion.tex} +\folie{4/galois/wuerfel.tex} +\folie{4/galois/winkeldreiteilung.tex} +\folie{4/galois/quadratur.tex} + +\section{Galois-Gruppe} +\folie{4/galois/automorphismus.tex} + +\section{Lösbarkeit durch Radikale} +\folie{4/galois/radikale.tex} +\folie{4/galois/aufloesbarkeit.tex} +\folie{4/galois/sn.tex} + diff --git a/vorlesungen/07_lie/Makefile b/vorlesungen/07_lie/Makefile index 1788301..7e925d8 100644 --- a/vorlesungen/07_lie/Makefile +++ b/vorlesungen/07_lie/Makefile @@ -1,33 +1,33 @@ -# -# Makefile -- lie -# -# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -# -all: lie-handout.pdf MathSem-07-lie.pdf - -include ../slides/Makefile.inc - -SOURCES = common.tex slides.tex $(slides) - -MathSem-07-lie.pdf: MathSem-07-lie.tex $(SOURCES) - pdflatex MathSem-07-lie.tex - -lie-handout.pdf: lie-handout.tex $(SOURCES) - pdflatex lie-handout.tex - -thumbnail: thumbnail.jpg fix1.jpg - -thumbnail.pdf: MathSem-07-lie.pdf - pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ - MathSem-07-lie.pdf 1 -thumbnail.jpg: thumbnail.pdf - convert -density 300 thumbnail.pdf \ - -resize 1920x1080 -units PixelsPerInch thumbnail.jpg - -fix1.pdf: MathSem-07-lie.pdf - pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ - MathSem-07-lie.pdf 205 -fix1.jpg: fix1.pdf - convert -density 300 fix1.pdf \ - -resize 1920x1080 -units PixelsPerInch fix1.jpg - +# +# Makefile -- lie +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: lie-handout.pdf MathSem-07-lie.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSem-07-lie.pdf: MathSem-07-lie.tex $(SOURCES) + pdflatex MathSem-07-lie.tex + +lie-handout.pdf: lie-handout.tex $(SOURCES) + pdflatex lie-handout.tex + +thumbnail: thumbnail.jpg fix1.jpg + +thumbnail.pdf: MathSem-07-lie.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSem-07-lie.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSem-07-lie.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSem-07-lie.pdf 205 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/07_lie/MathSem-07-lie.tex b/vorlesungen/07_lie/MathSem-07-lie.tex index 8a5557d..6cf5bd3 100644 --- a/vorlesungen/07_lie/MathSem-07-lie.tex +++ b/vorlesungen/07_lie/MathSem-07-lie.tex @@ -1,18 +1,18 @@ -% -% MathSem-07-lie.tex -- Präsentation -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\documentclass[aspectratio=169]{beamer} -\input{common.tex} -\setboolean{presentation}{true} -\begin{document} -\begin{frame} -\titlepage -\vspace{-1.5cm} -\begin{center} -\includegraphics[width=10cm]{../slides/7/images/rodriguez.jpg} -\end{center} -\end{frame} -\input{slides.tex} -\end{document} +% +% MathSem-07-lie.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\vspace{-1.5cm} +\begin{center} +\includegraphics[width=10cm]{../slides/7/images/rodriguez.jpg} +\end{center} +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/07_lie/common.tex b/vorlesungen/07_lie/common.tex index 8472b93..12f0700 100644 --- a/vorlesungen/07_lie/common.tex +++ b/vorlesungen/07_lie/common.tex @@ -1,16 +1,16 @@ -% -% common.tex -- gemeinsame definition -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\input{../common/packages.tex} -\input{../common/common.tex} -\mode{% -\usetheme[hideothersubsections,hidetitle]{Hannover} -} -\beamertemplatenavigationsymbolsempty -\title[Lie]{Lie-Gruppen und Lie-Algebren} -\author[A.~Müller]{Prof. Dr. Andreas Müller} -\date[]{} -\newboolean{presentation} - +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Lie]{Lie-Gruppen und Lie-Algebren} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/07_lie/lie-handout.tex b/vorlesungen/07_lie/lie-handout.tex index dbdb386..43053b8 100644 --- a/vorlesungen/07_lie/lie-handout.tex +++ b/vorlesungen/07_lie/lie-handout.tex @@ -1,11 +1,11 @@ -% -% lie-handout.tex -- Handout XXX -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\documentclass[handout,aspectratio=169]{beamer} -\input{common.tex} -\setboolean{presentation}{false} -\begin{document} -\input{slides.tex} -\end{document} +% +% lie-handout.tex -- Handout XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/07_lie/slides.tex b/vorlesungen/07_lie/slides.tex index 19131d8..7efc554 100644 --- a/vorlesungen/07_lie/slides.tex +++ b/vorlesungen/07_lie/slides.tex @@ -1,26 +1,26 @@ -% -% slides.tex -- Vorlesung über Lie-Theorie -% -% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Matrizen-Gruppen} -\folie{7/symmetrien.tex} -\folie{7/algebraisch.tex} -\folie{7/parameter.tex} -\folie{7/mannigfaltigkeit.tex} -\folie{7/sl2.tex} -\folie{7/drehung.tex} -\ifthenelse{\boolean{presentation}}{ -\folie{7/drehanim.tex} -}{} -\folie{7/semi.tex} - -\section{Ableitungen} -\folie{7/kurven.tex} -\folie{7/einparameter.tex} -\folie{7/ableitung.tex} -\folie{7/liealgebra.tex} -\folie{7/kommutator.tex} - -\section{Exponentialabbildung} -\folie{7/dg.tex} +% +% slides.tex -- Vorlesung über Lie-Theorie +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Matrizen-Gruppen} +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\ifthenelse{\boolean{presentation}}{ +\folie{7/drehanim.tex} +}{} +\folie{7/semi.tex} + +\section{Ableitungen} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/kommutator.tex} + +\section{Exponentialabbildung} +\folie{7/dg.tex} diff --git a/vorlesungen/common/README b/vorlesungen/common/README index 1ed40aa..3edcf14 100644 --- a/vorlesungen/common/README +++ b/vorlesungen/common/README @@ -1,28 +1,28 @@ -Die beiden Files - - presentation-template.tex - slide-template.tex - -können als Basis für die eigene Präsentation verwendet werden. -Dazu geht man wie folgt vor: - -1. In einem Arbeitsverzeichnis eine Kopie von presentation-template.tex -anlegen und im file Author und Titel anpassen. Im Folgenden wird diese -Kopie als beispiel-praesentation.tex bezeichnet. - -2. Für jede Folie der Präsentation im Arbeitsverzeichnis eine Kopie von -slide-template.tex anlegen und den Inhalt anpassen. - -3. Die Slides mit Hilfe von Input-Befehlen, die in presentation-template.tex -eingetragen werden, in die Präsentation importieren. - -4. Die Präsentation mit dem Befehl - - pdflatex beispiel-praesentation.tex - -erzeugen, es entsteht das File beispile-praesentation.pdf - -Diese Vorgehen erlaubt, die Reihenfolge der Folien während der Vorbereitung -zu ändern oder zwecks Beschleunigung des pdflatex-Laufs während der -Entwicklung auszukommentieren. - +Die beiden Files + + presentation-template.tex + slide-template.tex + +können als Basis für die eigene Präsentation verwendet werden. +Dazu geht man wie folgt vor: + +1. In einem Arbeitsverzeichnis eine Kopie von presentation-template.tex +anlegen und im file Author und Titel anpassen. Im Folgenden wird diese +Kopie als beispiel-praesentation.tex bezeichnet. + +2. Für jede Folie der Präsentation im Arbeitsverzeichnis eine Kopie von +slide-template.tex anlegen und den Inhalt anpassen. + +3. Die Slides mit Hilfe von Input-Befehlen, die in presentation-template.tex +eingetragen werden, in die Präsentation importieren. + +4. Die Präsentation mit dem Befehl + + pdflatex beispiel-praesentation.tex + +erzeugen, es entsteht das File beispile-praesentation.pdf + +Diese Vorgehen erlaubt, die Reihenfolge der Folien während der Vorbereitung +zu ändern oder zwecks Beschleunigung des pdflatex-Laufs während der +Entwicklung auszukommentieren. + diff --git a/vorlesungen/common/presentation-template.tex b/vorlesungen/common/presentation-template.tex index 9f92489..c872c58 100644 --- a/vorlesungen/common/presentation-template.tex +++ b/vorlesungen/common/presentation-template.tex @@ -1,49 +1,49 @@ -% -% presentation-template.tex -- Präsentation -% -% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\documentclass[aspectratio=169]{beamer} -\usepackage[utf8]{inputenc} -\usepackage[T1]{fontenc} -\usepackage{epic} -\usepackage{color} -\usepackage{array} -\usepackage{ifthen} -\usepackage{lmodern} -\usepackage{amsmath} -\usepackage{amssymb} -\usepackage{mathtools} -\usepackage{adjustbox} -\usepackage{multimedia} -\usepackage{verbatim} -\usepackage{wasysym} -\usepackage{stmaryrd} -\usepackage{tikz} -\usetikzlibrary{shapes.geometric} -\usetikzlibrary{decorations.pathreplacing} -\usetikzlibrary{calc} -\usetikzlibrary{arrows} -\usetikzlibrary{3d} -\usetikzlibrary{arrows,shapes,math,decorations.text,automata} -\usepackage{pifont} -\usepackage[all]{xy} -\usepackage[many]{tcolorbox} -\mode{% -\usetheme[hideothersubsections,hidetitle]{Hannover} -} -\beamertemplatenavigationsymbolsempty -\title[Titel]{Titel} -\author[A. Uthor]{A. Uthor} -\date[]{} -\newboolean{presentation} -\setboolean{presentation}{true} -\begin{document} - -\begin{frame} -\titlepage -\end{frame} - -%\input{slide.tex} - -\end{document} +% +% presentation-template.tex -- Präsentation +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{ifthen} +\usepackage{lmodern} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{adjustbox} +\usepackage{multimedia} +\usepackage{verbatim} +\usepackage{wasysym} +\usepackage{stmaryrd} +\usepackage{tikz} +\usetikzlibrary{shapes.geometric} +\usetikzlibrary{decorations.pathreplacing} +\usetikzlibrary{calc} +\usetikzlibrary{arrows} +\usetikzlibrary{3d} +\usetikzlibrary{arrows,shapes,math,decorations.text,automata} +\usepackage{pifont} +\usepackage[all]{xy} +\usepackage[many]{tcolorbox} +\mode{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Titel]{Titel} +\author[A. Uthor]{A. Uthor} +\date[]{} +\newboolean{presentation} +\setboolean{presentation}{true} +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +%\input{slide.tex} + +\end{document} diff --git a/vorlesungen/common/slide-template.tex b/vorlesungen/common/slide-template.tex index a1343f8..2dd4db1 100644 --- a/vorlesungen/common/slide-template.tex +++ b/vorlesungen/common/slide-template.tex @@ -1,19 +1,19 @@ -% -% template.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Template} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\end{column} -\begin{column}{0.48\textwidth} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Template} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\end{column} +\begin{column}{0.48\textwidth} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 5aac429..1ab27fa 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -1,36 +1,36 @@ - -# -# Makefile.inc -- additional depencencies -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# -chapter4 = \ - ../slides/4/ggt.tex \ - ../slides/4/euklidmatrix.tex \ - ../slides/4/euklidbeispiel.tex \ - ../slides/4/euklidtabelle.tex \ - ../slides/4/fp.tex \ - ../slides/4/division.tex \ - ../slides/4/gauss.tex \ - ../slides/4/dh.tex \ - ../slides/4/divisionpoly.tex \ - ../slides/4/euklidpoly.tex \ - ../slides/4/polynomefp.tex \ - ../slides/4/schieberegister.tex \ - ../slides/4/charakteristik.tex \ - ../slides/4/char2.tex \ - ../slides/4/frobenius.tex \ - ../slides/4/qundr.tex \ - ../slides/4/alpha.tex \ - ../slides/4/galois/erweiterung.tex \ - ../slides/4/galois/automorphismus.tex \ - ../slides/4/galois/konstruktion.tex \ - ../slides/4/galois/wuerfel.tex \ - ../slides/4/galois/winkeldreiteilung.tex \ - ../slides/4/galois/quadratur.tex \ - ../slides/4/galois/radikale.tex \ - ../slides/4/galois/aufloesbarkeit.tex \ - ../slides/4/galois/sn.tex \ - ../slides/4/chapter.tex - - + +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter4 = \ + ../slides/4/ggt.tex \ + ../slides/4/euklidmatrix.tex \ + ../slides/4/euklidbeispiel.tex \ + ../slides/4/euklidtabelle.tex \ + ../slides/4/fp.tex \ + ../slides/4/division.tex \ + ../slides/4/gauss.tex \ + ../slides/4/dh.tex \ + ../slides/4/divisionpoly.tex \ + ../slides/4/euklidpoly.tex \ + ../slides/4/polynomefp.tex \ + ../slides/4/schieberegister.tex \ + ../slides/4/charakteristik.tex \ + ../slides/4/char2.tex \ + ../slides/4/frobenius.tex \ + ../slides/4/qundr.tex \ + ../slides/4/alpha.tex \ + ../slides/4/galois/erweiterung.tex \ + ../slides/4/galois/automorphismus.tex \ + ../slides/4/galois/konstruktion.tex \ + ../slides/4/galois/wuerfel.tex \ + ../slides/4/galois/winkeldreiteilung.tex \ + ../slides/4/galois/quadratur.tex \ + ../slides/4/galois/radikale.tex \ + ../slides/4/galois/aufloesbarkeit.tex \ + ../slides/4/galois/sn.tex \ + ../slides/4/chapter.tex + + diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 0691e39..3015e7c 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -1,31 +1,31 @@ -% -% chapter.tex -% -% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi -% -\folie{4/ggt.tex} -\folie{4/euklidmatrix.tex} -\folie{4/euklidbeispiel.tex} -\folie{4/euklidtabelle.tex} -\folie{4/fp.tex} -\folie{4/division.tex} -\folie{4/gauss.tex} -\folie{4/dh.tex} -\folie{4/divisionpoly.tex} -\folie{4/euklidpoly.tex} -\folie{4/polynomefp.tex} -\folie{4/alpha.tex} -\folie{4/schieberegister.tex} -\folie{4/charakteristik.tex} -\folie{4/char2.tex} -\folie{4/frobenius.tex} -\folie{4/qundr.tex} -\folie{4/galois/erweiterung.tex} -\folie{4/galois/automorphismus.tex} -\folie{4/galois/konstruktion.tex} -\folie{4/galois/wuerfel.tex} -\folie{4/galois/winkeldreiteilung.tex} -\folie{4/galois/quadratur.tex} -\folie{4/galois/radikale.tex} -\folie{4/galois/aufloesbarkeit.tex} -\folie{4/galois/sn.tex} +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{4/ggt.tex} +\folie{4/euklidmatrix.tex} +\folie{4/euklidbeispiel.tex} +\folie{4/euklidtabelle.tex} +\folie{4/fp.tex} +\folie{4/division.tex} +\folie{4/gauss.tex} +\folie{4/dh.tex} +\folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} +\folie{4/polynomefp.tex} +\folie{4/alpha.tex} +\folie{4/schieberegister.tex} +\folie{4/charakteristik.tex} +\folie{4/char2.tex} +\folie{4/frobenius.tex} +\folie{4/qundr.tex} +\folie{4/galois/erweiterung.tex} +\folie{4/galois/automorphismus.tex} +\folie{4/galois/konstruktion.tex} +\folie{4/galois/wuerfel.tex} +\folie{4/galois/winkeldreiteilung.tex} +\folie{4/galois/quadratur.tex} +\folie{4/galois/radikale.tex} +\folie{4/galois/aufloesbarkeit.tex} +\folie{4/galois/sn.tex} diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex index ef5902b..3d52b00 100644 --- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex +++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex @@ -1,120 +1,120 @@ -% -% aufloesbarkeit.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Auflösbarkeit} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<2->{% -\begin{block}{Radikalerweiterung} -Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$ -einer Radikalerweiterung -\[ -\Bbbk \subset \Bbbk(\alpha) -\] -sind festgelegt durch Wahl von $f(\alpha)$. - -\begin{itemize} -\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt -\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein -\end{itemize} - -\end{block}} -\uncover<8->{% -\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$} -$\mathbb{Q}\subset \Bbbk$, -$n$ verschiedene Nullstellen $\mathbb{C}$: -\[ -\uncover<9->{ -\operatorname{Gal}(\Bbbk/\mathbb{Q}) -\cong -S_n} -\uncover<10->{ -\quad -\text{auflösbar?}} -\] -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\begin{block}{\uncover<5->{Galois-Gruppen}} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\s{1.2} - -\uncover<2->{ -\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3}); -\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen}; -} - -\node at (0,0) {$\mathbb{Q}$}; -\node at (0,{1*\s}) {$E_1$}; -\node at (0,{2*\s}) {$E_2$}; -\node at (0,{3*\s}) {$E_3$}; -\node at (0,{4*\s}) {$\vdots\mathstrut$}; -\node at (0,{5*\s}) {$\Bbbk$}; -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s}); -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s}); -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s}); -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s}); -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s}); - -\begin{scope}[xshift=0.5cm] -\uncover<7->{ -\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3}); -\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe}; -} -\uncover<5->{ -\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$}; -\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$}; -\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$}; -\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$}; -\node at (1,{4*\s}) {$\vdots\mathstrut$}; -\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$}; -\node at (1,{0.5*\s}) {$\cap\mathstrut$}; -\node at (1,{1.5*\s}) {$\cap\mathstrut$}; -\node at (1,{2.5*\s}) {$\cap\mathstrut$}; -\node at (1,{3.5*\s}) {$\cap\mathstrut$}; -\node at (1,{4.5*\s}) {$\cap\mathstrut$}; -} - -\uncover<6->{ -\begin{scope}[xshift=2.5cm] -\node at (0,{0*\s}) {$G_n$}; -\node at (0,{1*\s}) {$G_{n-1}$}; -\node at (0,{2*\s}) {$G_{n-2}$}; -\node at (0,{3*\s}) {$G_{n-3}$}; -\node at (0,{5*\s}) {$G_0=\{e\}$}; -\node at (0,{0.5*\s}) {$\cap\mathstrut$}; -\node at (0,{1.5*\s}) {$\cap\mathstrut$}; -\node at (0,{2.5*\s}) {$\cap\mathstrut$}; -\node at (0,{3.5*\s}) {$\cap\mathstrut$}; -\node at (0,{4.5*\s}) {$\cap\mathstrut$}; -} - -\uncover<7->{ -\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$}; -\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch}; - -\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$}; -\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch}; - -\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$}; -\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch}; -} - -\end{scope} -\end{scope} - - - -\end{tikzpicture} -\end{center} -\end{block} -\end{column} -\end{columns} -\end{frame} +% +% aufloesbarkeit.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Auflösbarkeit} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Radikalerweiterung} +Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$ +einer Radikalerweiterung +\[ +\Bbbk \subset \Bbbk(\alpha) +\] +sind festgelegt durch Wahl von $f(\alpha)$. + +\begin{itemize} +\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt +\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein +\end{itemize} + +\end{block}} +\uncover<8->{% +\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$} +$\mathbb{Q}\subset \Bbbk$, +$n$ verschiedene Nullstellen $\mathbb{C}$: +\[ +\uncover<9->{ +\operatorname{Gal}(\Bbbk/\mathbb{Q}) +\cong +S_n} +\uncover<10->{ +\quad +\text{auflösbar?}} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{\uncover<5->{Galois-Gruppen}} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{1.2} + +\uncover<2->{ +\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3}); +\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen}; +} + +\node at (0,0) {$\mathbb{Q}$}; +\node at (0,{1*\s}) {$E_1$}; +\node at (0,{2*\s}) {$E_2$}; +\node at (0,{3*\s}) {$E_3$}; +\node at (0,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) {$\Bbbk$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s}); + +\begin{scope}[xshift=0.5cm] +\uncover<7->{ +\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3}); +\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe}; +} +\uncover<5->{ +\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$}; +\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$}; +\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$}; +\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$}; +\node at (1,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$}; +\node at (1,{0.5*\s}) {$\cap\mathstrut$}; +\node at (1,{1.5*\s}) {$\cap\mathstrut$}; +\node at (1,{2.5*\s}) {$\cap\mathstrut$}; +\node at (1,{3.5*\s}) {$\cap\mathstrut$}; +\node at (1,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<6->{ +\begin{scope}[xshift=2.5cm] +\node at (0,{0*\s}) {$G_n$}; +\node at (0,{1*\s}) {$G_{n-1}$}; +\node at (0,{2*\s}) {$G_{n-2}$}; +\node at (0,{3*\s}) {$G_{n-3}$}; +\node at (0,{5*\s}) {$G_0=\{e\}$}; +\node at (0,{0.5*\s}) {$\cap\mathstrut$}; +\node at (0,{1.5*\s}) {$\cap\mathstrut$}; +\node at (0,{2.5*\s}) {$\cap\mathstrut$}; +\node at (0,{3.5*\s}) {$\cap\mathstrut$}; +\node at (0,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<7->{ +\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$}; +\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$}; +\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$}; +\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch}; +} + +\end{scope} +\end{scope} + + + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/automorphismus.tex b/vorlesungen/slides/4/galois/automorphismus.tex index 6051813..e59f9b9 100644 --- a/vorlesungen/slides/4/galois/automorphismus.tex +++ b/vorlesungen/slides/4/galois/automorphismus.tex @@ -1,118 +1,118 @@ -% -% automorphismus.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{4pt} -\setlength{\belowdisplayskip}{4pt} -\frametitle{Galois-Gruppe} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.40\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\s{3.0} -\begin{scope}[xshift=-1.5cm] -\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut}; -\node at (0,{\s}) {$G$}; -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0); -\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0); -\node at (0,{-0.5*\s}) [left] {$[F:E]$}; -\node at (0,{0.5*\s}) [left] {$[G:F]$}; -\node at (0,0) {$F$}; -\node at (0,{-\s}) {$E$}; -\end{scope} -\uncover<3->{ -\begin{scope}[xshift=1.8cm] -\node at (0,{\s+0.1}) [above] {Gruppe\strut}; -\fill (0,{-\s}) circle[radius=0.06]; -\fill (0,0) circle[radius=0.06]; -\fill (0,{\s}) circle[radius=0.06]; -\draw[shorten >= 0.1cm,shorten <= 0.1cm] - (0,{-\s}) to[out=100,in=-100] (0,{\s}); -\draw[shorten >= 0.1cm,shorten <= 0.1cm] - (0,{-\s}) to[out=80,in=-80] (0,0); -\draw[shorten >= 0.1cm,shorten <= 0.1cm] - (0,0) to[out=80,in=-80] (0,{\s}); -\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$}; -\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$}; -\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$}; -\end{scope} -\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s}); -\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$}; -} -\uncover<4->{ -\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s}); -\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$}; -} -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.56\textwidth} -\uncover<2->{% -\begin{block}{Automorphismus} -\vspace{-10pt} -\[ -\operatorname{Aut}(F) -= -\left\{ -f\colon F\to F -\left| -\begin{aligned} -f(x+y)&=f(x)+f(y)\\ -f(xy)&=f(x)f(y) -\end{aligned} -\right. -\right\} -\] -\end{block}} -\vspace{-10pt} -\uncover<3->{% -\begin{block}{Galois-Gruppe} -Automorphismen, die $E$ festlassen -\[ -{\color{red} -\operatorname{Gal}(F/E) -} -= -\left\{ -\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E -\right\} -\] -\end{block}} -\vspace{-10pt} -\uncover<4->{% -\begin{block}{Fixkörper} -$H\subset \operatorname{Aut}(F)$: -\begin{align*} -{\color{blue}F^H} -&= -\{x\in F\;|\; hx = x\forall h\in H\} -=\operatorname{Fix}(H) -\end{align*} -\end{block}} -\vspace{-13pt} -\uncover<5->{% -\begin{block}{Beispiel} -\begin{itemize} -\item<6-> -\( -\operatorname{Gal}(\mathbb{C}/\mathbb{R}) -= -\{ -\operatorname{id}_{\mathbb{C}}, -\operatorname{conj}\colon z\mapsto\overline{z} -\} -\) -\item<7-> -\( -\mathbb{C}^{\operatorname{conj}} -= -\mathbb{R} -\) -\end{itemize} -\end{block}} -\end{column} -\end{columns} -\end{frame} +% +% automorphismus.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{4pt} +\setlength{\belowdisplayskip}{4pt} +\frametitle{Galois-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{3.0} +\begin{scope}[xshift=-1.5cm] +\node at (0,{\s+0.1}) [above] {Körpererweiterung\strut}; +\node at (0,{\s}) {$G$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{-\s}) -- (0,0); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{\s}) -- (0,0); +\node at (0,{-0.5*\s}) [left] {$[F:E]$}; +\node at (0,{0.5*\s}) [left] {$[G:F]$}; +\node at (0,0) {$F$}; +\node at (0,{-\s}) {$E$}; +\end{scope} +\uncover<3->{ +\begin{scope}[xshift=1.8cm] +\node at (0,{\s+0.1}) [above] {Gruppe\strut}; +\fill (0,{-\s}) circle[radius=0.06]; +\fill (0,0) circle[radius=0.06]; +\fill (0,{\s}) circle[radius=0.06]; +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=100,in=-100] (0,{\s}); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,{-\s}) to[out=80,in=-80] (0,0); +\draw[shorten >= 0.1cm,shorten <= 0.1cm] + (0,0) to[out=80,in=-80] (0,{\s}); +\node at (-0.6,0) [rotate=90] {$\operatorname{Gal}(G/E)$}; +\node at (0.45,{0.5*\s}) [rotate=90] {$\operatorname{Gal}(G/F)$}; +\node at (0.45,{-0.5*\s}) [rotate=90] {$\operatorname{Gal}(F/E)$}; +\end{scope} +\draw[->,color=red!20,line width=14pt] (-1.4,{0.6*\s}) -- (1.4,{0.6*\s}); +\node[color=red] at (0,{0.6*\s}) {$\operatorname{Gal}$}; +} +\uncover<4->{ +\draw[<-,color=blue!20,line width=14pt] (-1.4,{-0.6*\s}) -- (1.4,{-0.6*\s}); +\node[color=blue] at (0,{-0.6*\s}) {$\operatorname{Fix}, F^H$}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<2->{% +\begin{block}{Automorphismus} +\vspace{-10pt} +\[ +\operatorname{Aut}(F) += +\left\{ +f\colon F\to F +\left| +\begin{aligned} +f(x+y)&=f(x)+f(y)\\ +f(xy)&=f(x)f(y) +\end{aligned} +\right. +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Galois-Gruppe} +Automorphismen, die $E$ festlassen +\[ +{\color{red} +\operatorname{Gal}(F/E) +} += +\left\{ +\varphi\in\operatorname{Aut}(F)\;|\; \varphi(x)=x\forall x\in E +\right\} +\] +\end{block}} +\vspace{-10pt} +\uncover<4->{% +\begin{block}{Fixkörper} +$H\subset \operatorname{Aut}(F)$: +\begin{align*} +{\color{blue}F^H} +&= +\{x\in F\;|\; hx = x\forall h\in H\} +=\operatorname{Fix}(H) +\end{align*} +\end{block}} +\vspace{-13pt} +\uncover<5->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<6-> +\( +\operatorname{Gal}(\mathbb{C}/\mathbb{R}) += +\{ +\operatorname{id}_{\mathbb{C}}, +\operatorname{conj}\colon z\mapsto\overline{z} +\} +\) +\item<7-> +\( +\mathbb{C}^{\operatorname{conj}} += +\mathbb{R} +\) +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/erweiterung.tex b/vorlesungen/slides/4/galois/erweiterung.tex index 6909849..20b278e 100644 --- a/vorlesungen/slides/4/galois/erweiterung.tex +++ b/vorlesungen/slides/4/galois/erweiterung.tex @@ -1,65 +1,65 @@ -% -% erweiterung.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Körpererweiterungen} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Körpererweiterung} -$E,F$ Körper: $E\subset F$ -\end{block} -\uncover<6->{% -\begin{block}{Vektorraum} -$F$ ist ein Vektorraum über $E$ -\end{block}} -\uncover<7->{% -\begin{block}{Endliche Körpererweiterung} -$\dim_E F < \infty$ -\end{block}} -\uncover<8->{% -\begin{block}{Adjunktion eines $\alpha$} -$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und -$\alpha$ enthält. -\end{block}} -\uncover<9->{% -\begin{block}{Algebraische Erweiterung} -$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von -$m(X)\in\Bbbk[X]$ -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<2->{% -\begin{block}{Beispiele} -\begin{enumerate} -\item<3-> -$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$ -\item<4-> -$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$ -\item<5-> -$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$ -\end{enumerate} -\end{block}} -\uncover<7->{% -\begin{block}{Grad} -$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls -\[ -\dim_E F = n =: [F:E] -\] -\uncover<8->{% -Gleichbedeutend: $\deg m(X) = n$} -\uncover<10->{% -\[ -E\subset F\subset G -\Rightarrow -[G:E] = [G:F]\cdot [F:E] -\] -(in unseren Fällen)} -\end{block}} -\end{column} -\end{columns} -\end{frame} +% +% erweiterung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Körpererweiterungen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Körpererweiterung} +$E,F$ Körper: $E\subset F$ +\end{block} +\uncover<6->{% +\begin{block}{Vektorraum} +$F$ ist ein Vektorraum über $E$ +\end{block}} +\uncover<7->{% +\begin{block}{Endliche Körpererweiterung} +$\dim_E F < \infty$ +\end{block}} +\uncover<8->{% +\begin{block}{Adjunktion eines $\alpha$} +$\Bbbk(\alpha)$ kleinster Körper, der $\Bbbk$ und +$\alpha$ enthält. +\end{block}} +\uncover<9->{% +\begin{block}{Algebraische Erweiterung} +$\alpha$ algebraisch über $\Bbbk$, i.~e.~Nullstelle von +$m(X)\in\Bbbk[X]$ +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Beispiele} +\begin{enumerate} +\item<3-> +$\mathbb{R} \subset \mathbb{R}(i) = \mathbb{C}$ +\item<4-> +$\mathbb{Q}\subset \mathbb{Q}(\sqrt{2})$ +\item<5-> +$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2}) \subset \mathbb{Q}(\sqrt[4]{2})$ +\end{enumerate} +\end{block}} +\uncover<7->{% +\begin{block}{Grad} +$E\subset F$ heisst Körpererweiterung vom Grad $n$, falls +\[ +\dim_E F = n =: [F:E] +\] +\uncover<8->{% +Gleichbedeutend: $\deg m(X) = n$} +\uncover<10->{% +\[ +E\subset F\subset G +\Rightarrow +[G:E] = [G:F]\cdot [F:E] +\] +(in unseren Fällen)} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/images/Makefile b/vorlesungen/slides/4/galois/images/Makefile index 444944e..fd197ce 100644 --- a/vorlesungen/slides/4/galois/images/Makefile +++ b/vorlesungen/slides/4/galois/images/Makefile @@ -1,12 +1,12 @@ -# -# Makefile -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# -all: wuerfel2.png wuerfel.png - -wuerfel.png: wuerfel.pov common.inc - povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov - -wuerfel2.png: wuerfel2.pov common.inc - povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov +# +# Makefile +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +all: wuerfel2.png wuerfel.png + +wuerfel.png: wuerfel.pov common.inc + povray +A0.1 -W1080 -H1080 -Owuerfel.png wuerfel.pov + +wuerfel2.png: wuerfel2.pov common.inc + povray +A0.1 -W1080 -H1080 -Owuerfel2.png wuerfel2.pov diff --git a/vorlesungen/slides/4/galois/images/common.inc b/vorlesungen/slides/4/galois/images/common.inc index 6cfcabe..44ee4c8 100644 --- a/vorlesungen/slides/4/galois/images/common.inc +++ b/vorlesungen/slides/4/galois/images/common.inc @@ -1,89 +1,89 @@ -// -// common.inc -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#version 3.7; -#include "colors.inc" -#include "textures.inc" -#include "stones.inc" - -global_settings { - assumed_gamma 1 -} - -#declare imagescale = 0.133; -#declare O = <0, 0, 0>; -#declare E = <1, 1, 1>; -#declare a = pow(2, 1/3); -#declare at = 0.02; - -camera { - location <3, 2, 12> - look_at E * (a / 2) * 0.93 - right x * imagescale - up y * imagescale -} - -light_source { - <11, 20, 16> color White - area_light <1,0,0> <0,0,1>, 10, 10 - adaptive 1 - jitter -} - -sky_sphere { - pigment { - color rgb<1,1,1> - } -} - -#macro wuerfelgitter(A, AT) - cylinder { O, , AT } - cylinder { O, <0, A, 0>, AT } - cylinder { O, <0, 0, A>, AT } - cylinder { , , AT } - cylinder { , , AT } - cylinder { <0, A, 0>, , AT } - cylinder { <0, A, 0>, <0, A, A>, AT } - cylinder { <0, 0, A>, , AT } - cylinder { <0, 0, A>, <0, A, A>, AT } - cylinder { , , AT } - cylinder { , , AT } - cylinder { <0, A, A>, , AT } - sphere { <0, 0, 0>, AT } - sphere { , AT } - sphere { <0, A, 0>, AT } - sphere { <0, 0, A>, AT } - sphere { , AT } - sphere { , AT } - sphere { <0, A, A>, AT } - sphere { , AT } -#end - -#macro wuerfel() - union { - box { O, E } - wuerfelgitter(1, 0.5*at) - texture { - T_Grnt24 - } - finish { - specular 0.9 - metallic - } - } -#end - -#macro wuerfel2() - union { - wuerfelgitter(a, at) - pigment { - color rgb<0.8,0.4,0.4> - } - finish { - specular 0.9 - metallic - } - } -#end +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" +#include "textures.inc" +#include "stones.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.133; +#declare O = <0, 0, 0>; +#declare E = <1, 1, 1>; +#declare a = pow(2, 1/3); +#declare at = 0.02; + +camera { + location <3, 2, 12> + look_at E * (a / 2) * 0.93 + right x * imagescale + up y * imagescale +} + +light_source { + <11, 20, 16> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro wuerfelgitter(A, AT) + cylinder { O, , AT } + cylinder { O, <0, A, 0>, AT } + cylinder { O, <0, 0, A>, AT } + cylinder { , , AT } + cylinder { , , AT } + cylinder { <0, A, 0>, , AT } + cylinder { <0, A, 0>, <0, A, A>, AT } + cylinder { <0, 0, A>, , AT } + cylinder { <0, 0, A>, <0, A, A>, AT } + cylinder { , , AT } + cylinder { , , AT } + cylinder { <0, A, A>, , AT } + sphere { <0, 0, 0>, AT } + sphere { , AT } + sphere { <0, A, 0>, AT } + sphere { <0, 0, A>, AT } + sphere { , AT } + sphere { , AT } + sphere { <0, A, A>, AT } + sphere { , AT } +#end + +#macro wuerfel() + union { + box { O, E } + wuerfelgitter(1, 0.5*at) + texture { + T_Grnt24 + } + finish { + specular 0.9 + metallic + } + } +#end + +#macro wuerfel2() + union { + wuerfelgitter(a, at) + pigment { + color rgb<0.8,0.4,0.4> + } + finish { + specular 0.9 + metallic + } + } +#end diff --git a/vorlesungen/slides/4/galois/images/wuerfel.pov b/vorlesungen/slides/4/galois/images/wuerfel.pov index a5db465..a0466f3 100644 --- a/vorlesungen/slides/4/galois/images/wuerfel.pov +++ b/vorlesungen/slides/4/galois/images/wuerfel.pov @@ -1,9 +1,9 @@ -// -// wuerfel.pov -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#include "common.inc" - -wuerfel() - +// +// wuerfel.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +wuerfel() + diff --git a/vorlesungen/slides/4/galois/images/wuerfel2.pov b/vorlesungen/slides/4/galois/images/wuerfel2.pov index ac32b2f..a11bab0 100644 --- a/vorlesungen/slides/4/galois/images/wuerfel2.pov +++ b/vorlesungen/slides/4/galois/images/wuerfel2.pov @@ -1,9 +1,9 @@ -// -// wuerfel.pov -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#include "common.inc" - -wuerfel() -wuerfel2() +// +// wuerfel.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +wuerfel() +wuerfel2() diff --git a/vorlesungen/slides/4/galois/konstruktion.tex b/vorlesungen/slides/4/galois/konstruktion.tex index 094b570..b461d44 100644 --- a/vorlesungen/slides/4/galois/konstruktion.tex +++ b/vorlesungen/slides/4/galois/konstruktion.tex @@ -1,147 +1,147 @@ -% -% konstruktion.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\frametitle{Konstruktion mit Zirkel und Lineal} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Strahlensatz} -\uncover<6->{% -Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$ -kann mit Zirkel und Lineal konstruiert werden.} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<7->{% -\begin{block}{Kreis--Gerade} -Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$ -\uncover<13->{% -$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden} -\end{block}} -\end{column} -\end{columns} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\s{0.5} -\def\t{0.45} - -\coordinate (A) at (0,0); -\coordinate (B) at ({10*\t},0); - -\uncover<2->{ - \draw (0,0) -- (30:{10.5*\s}); -} - -\uncover<3->{ - \foreach \x in {0,...,10}{ - \fill (30:{\x*\s}) circle[radius=0.03]; - } - \foreach \x in {0,1,2,3,4,7,8,9}{ - \node at (30:{\x*\s}) [above] {\tiny $\x$}; - } - \node at (30:{10*\s}) [above right] {$q=10$}; -} - -\uncover<4->{ - \foreach \x in {1,...,10}{ - \fill (0:{\x*\t}) circle[radius=0.03]; - \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t}); - } -} - -\draw (A) -- (0:{10.5*\t}); -\node at (A) [below left] {$A$}; -\node at (B) [below right] {$B$}; -\fill (A) circle[radius=0.05]; -\fill (B) circle[radius=0.05]; - -\uncover<5->{ - \node at (30:{6*\s}) [above left] {$p=6$}; - \draw[line width=0.2pt] (0,0) -- (0,-0.4); - \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4); - \draw[<->] (0,-0.3) -- ({6*\t},-0.3); - \node at ({3*\t},-0.4) [below] - {$\displaystyle\frac{p}{q}\cdot\overline{AB}$}; -} - -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<8->{% -\begin{center} -\begin{tikzpicture}[>=latex,thick] - -%\foreach \x in {8,...,14}{ -% \only<\x>{\node at (4,4) {$\x$};} -%} - -\def\r{4} -\def\a{50} - -\coordinate (A) at ({\r*cos(\a)},0); - -\uncover<10->{ - \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle; - \fill[color=gray] - (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle; -} - -\draw[->] (0,0) -- (95:\r); -\node at (95:{0.5*\r}) [left] {$c$}; - -\begin{scope} - \clip (-1,-0.3) rectangle (4.5,4.1); - \uncover<10->{ - \draw (-1,0) -- (5,0); - \draw[->] (0,0) -- (\r,0); - \draw (0,0) circle[radius=\r]; - \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5); - } -\end{scope} - -\uncover<11->{ - \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle; -} - -\uncover<9->{ - \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle; - \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07]; - \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1); -} - -\uncover<11->{ - \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r); - \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$}; -} - -\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0); -\fill[color=blue] (0,0) circle[radius=0.04]; -\fill[color=blue] (A) circle[radius=0.04]; -\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$}; - -\uncover<12->{ - \fill[color=white,opacity=0.8] - ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25}) - rectangle - ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25}); - - \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right] - {$b=\sqrt{c^2-a^2}$}; - \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r); - \fill[color=red] (\a:\r) circle[radius=0.05]; - \fill[color=red] (A) circle[radius=0.05]; -} - -\end{tikzpicture} -\end{center}} -\end{column} -\end{columns} -\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:} -Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$} -\end{frame} +% +% konstruktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Konstruktion mit Zirkel und Lineal} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Strahlensatz} +\uncover<6->{% +Jedes beliebige rationale Streckenverhältnis $\frac{p}{q}$ +kann mit Zirkel und Lineal konstruiert werden.} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Kreis--Gerade} +Aus $c$ und $a$ konstruiere $b=\sqrt{c^2-a^2}$ +\uncover<13->{% +$\Rightarrow$ jede beliebige Quadratwurzel kann konstruiert werden} +\end{block}} +\end{column} +\end{columns} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{0.5} +\def\t{0.45} + +\coordinate (A) at (0,0); +\coordinate (B) at ({10*\t},0); + +\uncover<2->{ + \draw (0,0) -- (30:{10.5*\s}); +} + +\uncover<3->{ + \foreach \x in {0,...,10}{ + \fill (30:{\x*\s}) circle[radius=0.03]; + } + \foreach \x in {0,1,2,3,4,7,8,9}{ + \node at (30:{\x*\s}) [above] {\tiny $\x$}; + } + \node at (30:{10*\s}) [above right] {$q=10$}; +} + +\uncover<4->{ + \foreach \x in {1,...,10}{ + \fill (0:{\x*\t}) circle[radius=0.03]; + \draw[->,line width=0.2pt] (30:{\x*\s}) -- (0:{\x*\t}); + } +} + +\draw (A) -- (0:{10.5*\t}); +\node at (A) [below left] {$A$}; +\node at (B) [below right] {$B$}; +\fill (A) circle[radius=0.05]; +\fill (B) circle[radius=0.05]; + +\uncover<5->{ + \node at (30:{6*\s}) [above left] {$p=6$}; + \draw[line width=0.2pt] (0,0) -- (0,-0.4); + \draw[line width=0.2pt] ({6*\t},0) -- ({6*\t},-0.4); + \draw[<->] (0,-0.3) -- ({6*\t},-0.3); + \node at ({3*\t},-0.4) [below] + {$\displaystyle\frac{p}{q}\cdot\overline{AB}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +%\foreach \x in {8,...,14}{ +% \only<\x>{\node at (4,4) {$\x$};} +%} + +\def\r{4} +\def\a{50} + +\coordinate (A) at ({\r*cos(\a)},0); + +\uncover<10->{ + \fill[color=gray] (\r,0) -- (\r,0.3) arc (90:180:0.3) -- cycle; + \fill[color=gray] + (95:\r) -- ($(95:\r)+(185:0.3)$) arc (185:275:0.3) -- cycle; +} + +\draw[->] (0,0) -- (95:\r); +\node at (95:{0.5*\r}) [left] {$c$}; + +\begin{scope} + \clip (-1,-0.3) rectangle (4.5,4.1); + \uncover<10->{ + \draw (-1,0) -- (5,0); + \draw[->] (0,0) -- (\r,0); + \draw (0,0) circle[radius=\r]; + \draw ({\r*cos(\a)},-1) -- ({\r*cos(\a)},5); + } +\end{scope} + +\uncover<11->{ + \fill[color=blue!20] (0,0) -- (A) -- (\a:\r) -- cycle; +} + +\uncover<9->{ + \fill[color=gray!80] (A) -- ($(A)+(0,0.5)$) arc (90:180:0.5) -- cycle; + \fill[color=gray!120] ($(A)+(-0.2,0.2)$) circle[radius=0.07]; + \draw ({\r*cos(\a)},-0.3) -- ({\r*cos(\a)},4.1); +} + +\uncover<11->{ + \draw[color=blue,line width=1.4pt] (0,0) -- (\a:\r); + \node[color=blue] at (\a:{0.5*\r}) [above left] {$c$}; +} + +\draw[color=blue,line width=1.4pt] (0,0) -- ({\r*cos(\a)},0); +\fill[color=blue] (0,0) circle[radius=0.04]; +\fill[color=blue] (A) circle[radius=0.04]; +\node[color=blue] at ({0.5*\r*cos(\a)},0) [below] {$a$}; + +\uncover<12->{ + \fill[color=white,opacity=0.8] + ({\r*cos(\a)+0.1},{0.5*\r*sin(\a)-0.25}) + rectangle + ({\r*cos(\a)+2},{0.5*\r*sin(\a)+0.25}); + + \node[color=red] at ({\r*cos(\a)},{0.5*\r*sin(\a)}) [right] + {$b=\sqrt{c^2-a^2}$}; + \draw[color=red,line width=1.4pt] ({\r*cos(\a)},0) -- (\a:\r); + \fill[color=red] (\a:\r) circle[radius=0.05]; + \fill[color=red] (A) circle[radius=0.05]; +} + +\end{tikzpicture} +\end{center}} +\end{column} +\end{columns} +\uncover<14->{{\usebeamercolor[fg]{title}Folgerung:} +Konstruierbar sind Körpererweiterungen $[F:E] = 2^l$} +\end{frame} diff --git a/vorlesungen/slides/4/galois/quadratur.tex b/vorlesungen/slides/4/galois/quadratur.tex index f5763b9..f9510ba 100644 --- a/vorlesungen/slides/4/galois/quadratur.tex +++ b/vorlesungen/slides/4/galois/quadratur.tex @@ -1,66 +1,66 @@ -% -% quadratur.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\frametitle{Quadratur des Kreises} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.44\textwidth} -\begin{center} -\uncover<2->{% -\begin{tikzpicture}[>=latex,thick] - -\def\r{2.8} -\pgfmathparse{sqrt(3.14159)*\r/2} -\xdef\s{\pgfmathresult} - -\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s); -\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r]; - -\uncover<3->{ - \draw[->,color=red] (0,0) -- (50:\r); - \fill[color=red] (0,0) circle[radius=0.04]; - \node[color=red] at (50:{0.5*\r}) [below right] {$r$}; -} - -\uncover<4->{ - \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7}); - \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7}); - \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6}); - \node[color=blue] at (0,{-\s-0.6}) [below] {$l$}; -} - -\uncover<5->{ - \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2} - \;\Rightarrow\; - {\color{blue}l}={\color{red}\sqrt{\pi}r}$}; -} - -\end{tikzpicture}} -\end{center} -\end{column} -\begin{column}{0.52\textwidth} -\begin{block}{Aufgabe} -Konstruiere ein zu einem Kreis flächengleiches Quadrat -\end{block} -\uncover<6->{% -\begin{block}{Modifizierte Aufgabe} -Konstruiere eine Strecke, deren Länge Lösung der Gleichung -$x^2-\pi=0$ ist. -\end{block}} -\uncover<7->{% -\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch] -\begin{itemize} -\item<8-> Lösung in einem Erweiterungskörper -\item<9-> Lösung ist Nullstelle eines Polynoms -\item<10-> Lösung ist algebraisch -\item<11-> $\pi$ ist {\bf nicht} algebraisch -\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})} -\qedhere -\end{itemize} -\end{proof}} -\end{column} -\end{columns} -\end{frame} +% +% quadratur.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Quadratur des Kreises} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{center} +\uncover<2->{% +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.8} +\pgfmathparse{sqrt(3.14159)*\r/2} +\xdef\s{\pgfmathresult} + +\fill[color=blue!20] (-\s,-\s) rectangle (\s,\s); +\fill[color=red!40,opacity=0.5] (0,0) circle[radius=\r]; + +\uncover<3->{ + \draw[->,color=red] (0,0) -- (50:\r); + \fill[color=red] (0,0) circle[radius=0.04]; + \node[color=red] at (50:{0.5*\r}) [below right] {$r$}; +} + +\uncover<4->{ + \draw[line width=0.3pt] (-\s,-\s) -- (-\s,{-\s-0.7}); + \draw[line width=0.3pt] (\s,-\s) -- (\s,{-\s-0.7}); + \draw[<->,color=blue] (-\s,{-\s-0.6}) -- (\s,{-\s-0.6}); + \node[color=blue] at (0,{-\s-0.6}) [below] {$l$}; +} + +\uncover<5->{ + \node at (0,{-\s/2}) {${\color{red}\pi r^2}={\color{blue}l^2} + \;\Rightarrow\; + {\color{blue}l}={\color{red}\sqrt{\pi}r}$}; +} + +\end{tikzpicture}} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Aufgabe} +Konstruiere ein zu einem Kreis flächengleiches Quadrat +\end{block} +\uncover<6->{% +\begin{block}{Modifizierte Aufgabe} +Konstruiere eine Strecke, deren Länge Lösung der Gleichung +$x^2-\pi=0$ ist. +\end{block}} +\uncover<7->{% +\begin{proof}[Unmöglichkeitsbeweis mit Widerspruch] +\begin{itemize} +\item<8-> Lösung in einem Erweiterungskörper +\item<9-> Lösung ist Nullstelle eines Polynoms +\item<10-> Lösung ist algebraisch +\item<11-> $\pi$ ist {\bf nicht} algebraisch +\uncover<12->{(Lindemann 1882\only<13>{, Weierstrass 1885})} +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/radikale.tex b/vorlesungen/slides/4/galois/radikale.tex index e9e4ce8..cb08dca 100644 --- a/vorlesungen/slides/4/galois/radikale.tex +++ b/vorlesungen/slides/4/galois/radikale.tex @@ -1,69 +1,69 @@ -% -% radikale.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Lösung durch Radikale} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Problemstellung} -Finde Nullstellen eines Polynomes -\[ -p(X) -= -a_nX^n + a_{n-1}X^{n-1} -+\dots+ -a_1X+a_0 -\] -$p\in\mathbb{Q}[X]$ -\end{block} -\uncover<2->{% -\begin{block}{Radikale} -Geschachtelte Wurzelausdrücke -\[ -\sqrt[3]{ --\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} -} -+ -\sqrt[3]{ --\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} -} -\] -\uncover<3->{(Lösung von $x^3+px+q=0$)} -\end{block}} -\uncover<4->{% -\begin{block}{Lösbar durch Radikale} -Nullstelle von $p(X)$ ist ein Radikal -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<5->{% -\begin{block}{Algebraische Formulierung} -Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$, -finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart, -dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{: -$\Bbbk = \mathbb{Q}[X]/(p)$} -\end{block}} -\uncover<7->{% -\begin{block}{Radikalerweiterung} -Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften -\begin{itemize} -\item<8-> $\alpha$ ist eine Einheitswurzel -\item<9-> $\alpha^k\in\Bbbk$ -\end{itemize} -\end{block}} -\vspace{-5pt} -\uncover<10->{% -\begin{block}{Lösbar durch Radikale} -Radikalerweiterungen -\[ -\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha -\] -\end{block}} -\end{column} -\end{columns} -\end{frame} +% +% radikale.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lösung durch Radikale} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Problemstellung} +Finde Nullstellen eines Polynomes +\[ +p(X) += +a_nX^n + a_{n-1}X^{n-1} ++\dots+ +a_1X+a_0 +\] +$p\in\mathbb{Q}[X]$ +\end{block} +\uncover<2->{% +\begin{block}{Radikale} +Geschachtelte Wurzelausdrücke +\[ +\sqrt[3]{ +-\frac{q}2 +\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} ++ +\sqrt[3]{ +-\frac{q}2 -\sqrt{\frac{q^2}{4}+\frac{p^3}{27}} +} +\] +\uncover<3->{(Lösung von $x^3+px+q=0$)} +\end{block}} +\uncover<4->{% +\begin{block}{Lösbar durch Radikale} +Nullstelle von $p(X)$ ist ein Radikal +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Algebraische Formulierung} +Gegeben ein irreduzibles Polynom $p\in\mathbb{Q}[X]$, +finde eine Körpererweiterung $\mathbb{Q}\subset\Bbbk$, derart, +dass $p$ in $\Bbbk$ eine Nullstelle hat\uncover<6->{: +$\Bbbk = \mathbb{Q}[X]/(p)$} +\end{block}} +\uncover<7->{% +\begin{block}{Radikalerweiterung} +Körpererweiterung $\Bbbk\subset\Bbbk'$ um $\alpha$ mit einer der Eigenschaften +\begin{itemize} +\item<8-> $\alpha$ ist eine Einheitswurzel +\item<9-> $\alpha^k\in\Bbbk$ +\end{itemize} +\end{block}} +\vspace{-5pt} +\uncover<10->{% +\begin{block}{Lösbar durch Radikale} +Radikalerweiterungen +\[ +\mathbb{Q} \subset \Bbbk \subset \Bbbk' \subset \dots \subset \Bbbk'' \ni \alpha +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/sn.tex b/vorlesungen/slides/4/galois/sn.tex index 1cae3fa..f340825 100644 --- a/vorlesungen/slides/4/galois/sn.tex +++ b/vorlesungen/slides/4/galois/sn.tex @@ -1,87 +1,87 @@ -% -% sn.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Nichtauflösbarkeit von $S_n$} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Die symmetrische Gruppe $S_n$} -Permutationen auf $n$ Elementen -\[ -\sigma -= -\begin{pmatrix} -1&2&3&\dots&n\\ -\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n) -\end{pmatrix} -\] -\end{block} -\vspace{-10pt} -\uncover<2->{% -\begin{block}{Signum} -$t(\sigma)=\mathstrut$ Anzahl Transpositionen -\[ -\operatorname{sgn}(\sigma) -= -(-1)^{t(\sigma)} -= -\begin{cases} -\phantom{-}1&\text{$t(\sigma)$ gerade} -\\ --1&\text{$t(\sigma)$ ungerade} -\end{cases} -\] -Homomorphismus! -\end{block}} -\uncover<3->{% -\begin{block}{Die alternierende Gruppe $A_n$} -\vspace{-12pt} -\[ -A_n = \ker \operatorname{sgn} -= -\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\} -\] -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<4->{% -\begin{block}{Normale Untergruppe} -\begin{itemize} -\item -$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$ -\item -$G/N$ ist wohldefiniert -\end{itemize} -\end{block}} -\vspace{-10pt} -\uncover<5->{% -\begin{block}{Einfache Gruppe} -$G$ einfach $\Leftrightarrow$ -\[ -H\triangleleft G -\; -\Rightarrow -\; -\text{$H=\{e\}$ oder $H=G$} -\] -\end{block}} -\vspace{-10pt} -\uncover<6->{% -\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$} -\begin{enumerate} -\item<7-> Zeigen, dass $A_5$ einfach ist -\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach -\end{enumerate} -\uncover<9->{% -$\Rightarrow$ i.~A.~keine Lösung der -einer Polynomgleichung vom Grad $\ge 5$ durch Radikale -} -\end{block}} -\end{column} -\end{columns} -\end{frame} +% +% sn.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Nichtauflösbarkeit von $S_n$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Die symmetrische Gruppe $S_n$} +Permutationen auf $n$ Elementen +\[ +\sigma += +\begin{pmatrix} +1&2&3&\dots&n\\ +\sigma(1)&\sigma(2)&\sigma(3)&\dots&\sigma(n) +\end{pmatrix} +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Signum} +$t(\sigma)=\mathstrut$ Anzahl Transpositionen +\[ +\operatorname{sgn}(\sigma) += +(-1)^{t(\sigma)} += +\begin{cases} +\phantom{-}1&\text{$t(\sigma)$ gerade} +\\ +-1&\text{$t(\sigma)$ ungerade} +\end{cases} +\] +Homomorphismus! +\end{block}} +\uncover<3->{% +\begin{block}{Die alternierende Gruppe $A_n$} +\vspace{-12pt} +\[ +A_n = \ker \operatorname{sgn} += +\{\sigma\in S_n\;|\;\operatorname{sgn}(\sigma)=1\} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Normale Untergruppe} +\begin{itemize} +\item +$H\triangleleft G$ wenn $gHg^{-1}\subset G\;\forall g\in G$ +\item +$G/N$ ist wohldefiniert +\end{itemize} +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Einfache Gruppe} +$G$ einfach $\Leftrightarrow$ +\[ +H\triangleleft G +\; +\Rightarrow +\; +\text{$H=\{e\}$ oder $H=G$} +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{$n\ge 5 \Rightarrow A_n \text{ einfach}$} +\begin{enumerate} +\item<7-> Zeigen, dass $A_5$ einfach ist +\item<8-> Vollständige Induktion: $A_n$ einfach $\Rightarrow A_{n+1}$ einfach +\end{enumerate} +\uncover<9->{% +$\Rightarrow$ i.~A.~keine Lösung der +einer Polynomgleichung vom Grad $\ge 5$ durch Radikale +} +\end{block}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/winkeldreiteilung.tex b/vorlesungen/slides/4/galois/winkeldreiteilung.tex index 54b941b..28c07fe 100644 --- a/vorlesungen/slides/4/galois/winkeldreiteilung.tex +++ b/vorlesungen/slides/4/galois/winkeldreiteilung.tex @@ -1,94 +1,94 @@ -% -% winkeldreiteilung.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Winkeldreiteilung} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.43\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\r{5} -\def\a{25} - -\uncover<3->{ - \draw[line width=0.7pt] (\r,0) arc (0:90:\r); -} - -\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle; -\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$}; - -\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r); - -\uncover<2->{ - \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle; - \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r); - \node[color=red] at ({0.5*\a}:{0.7*\r}) - {$\displaystyle\frac{\alpha}{3}$}; -} - -\uncover<3->{ - \fill[color=blue] ({3*\a}:\r) circle[radius=0.05]; - \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1); - - \fill[color=red] ({\a}:\r) circle[radius=0.05]; - \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1); - - \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}]; - \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}]; -} - - -\uncover<4->{ -\node at ({0.5*\r},-0.5) [below] {$\displaystyle -\cos{\color{blue}\alpha} -= -4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3} -$}; -} - -\uncover<5->{ - \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$}; - \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$}; -} - -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.53\textwidth} -\begin{block}{Aufgabe} -Teile einen Winkel in drei gleiche Teile -\end{block} -\vspace{-2pt} -\uncover<6->{% -\begin{block}{Algebraisierte Aufgabe} -Konstruiere $x$ aus $a$ derart, dass -\[ -p(x) -= -x^3-\frac34 x -a = 0 -\] -\uncover<7->{% -$a=0$:} -\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$} -\end{block}} -\vspace{-2pt} -\uncover<10->{% -\begin{proof}[Unmöglichkeitsbeweis] -\begin{itemize} -\item<11-> -$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel -\item<12-> -$p(x)$ definiert eine Körpererweiterung vom Grad $3$ -\item<13-> -Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$ -\qedhere -\end{itemize} -\end{proof}} -\end{column} -\end{columns} -\end{frame} +% +% winkeldreiteilung.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Winkeldreiteilung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.43\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\r{5} +\def\a{25} + +\uncover<3->{ + \draw[line width=0.7pt] (\r,0) arc (0:90:\r); +} + +\fill[color=blue!20] (0,0) -- (\r,0) arc(0:{3*\a}:\r) -- cycle; +\node[color=blue] at ({1.5*\a}:{1.05*\r}) {$\alpha$}; + +\draw[color=blue,line width=1.3pt] (\r,0) arc (0:{3*\a}:\r); + +\uncover<2->{ + \fill[color=red!40,opacity=0.5] (0,0) -- (\r,0) arc(0:\a:\r) -- cycle; + \draw[color=red,line width=1.4pt] (\r,0) arc (0:\a:\r); + \node[color=red] at ({0.5*\a}:{0.7*\r}) + {$\displaystyle\frac{\alpha}{3}$}; +} + +\uncover<3->{ + \fill[color=blue] ({3*\a}:\r) circle[radius=0.05]; + \draw[color=blue] ({3*\a}:\r) -- ({\r*cos(3*\a)},-0.1); + + \fill[color=red] ({\a}:\r) circle[radius=0.05]; + \draw[color=red] ({\a}:\r) -- ({\r*cos(\a)},-0.1); + + \draw[->] (-0.1,0) -- ({\r+0.4},0) coordinate[label={$x$}]; + \draw[->] (0,-0.1) -- (0,{\r+0.4}) coordinate[label={right:$y$}]; +} + + +\uncover<4->{ +\node at ({0.5*\r},-0.5) [below] {$\displaystyle +\cos{\color{blue}\alpha} += +4\cos^3{\color{red}\frac{\alpha}3} -3 \cos {\color{red}\frac{\alpha}3} +$}; +} + +\uncover<5->{ + \node[color=blue] at ({\r*cos(3*\a)},0) [below] {$a\mathstrut$}; + \node[color=red] at ({\r*cos(\a)},0) [below] {$x\mathstrut$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.53\textwidth} +\begin{block}{Aufgabe} +Teile einen Winkel in drei gleiche Teile +\end{block} +\vspace{-2pt} +\uncover<6->{% +\begin{block}{Algebraisierte Aufgabe} +Konstruiere $x$ aus $a$ derart, dass +\[ +p(x) += +x^3-\frac34 x -a = 0 +\] +\uncover<7->{% +$a=0$:} +\uncover<8->{$p(x) = x(x^2-\frac{3}{4})\uncover<9->{\Rightarrow x = \frac{\sqrt{3}}2}$} +\end{block}} +\vspace{-2pt} +\uncover<10->{% +\begin{proof}[Unmöglichkeitsbeweis] +\begin{itemize} +\item<11-> +$a\ne 0$ $\Rightarrow$ $p(x)$ irreduzibel +\item<12-> +$p(x)$ definiert eine Körpererweiterung vom Grad $3$ +\item<13-> +Konstruierbar sind nur Körpererweiterungen vom Grad $2^l$ +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/4/galois/wuerfel.tex b/vorlesungen/slides/4/galois/wuerfel.tex index ada6079..907d60a 100644 --- a/vorlesungen/slides/4/galois/wuerfel.tex +++ b/vorlesungen/slides/4/galois/wuerfel.tex @@ -1,64 +1,64 @@ -% -% wuerfel.tex -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\begin{frame}[t] -\frametitle{Würfelverdoppelung} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}}; -\uncover<2->{ -\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}}; -} - -\uncover<3->{ - \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25); - \node[color=blue] at (0.75,-2.3) [above] {$a$}; -} - -\uncover<4->{ - \begin{scope}[yshift=0.03cm] - \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19); - \draw[color=red] (2.85,-2.7) -- (2.85,-3.0); - \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9); - \end{scope} - \node[color=red] at (0.36,-2.9) [below] {$b$}; -} - -\uncover<5->{ -\node at (0,-4) {$ - 2{\color{blue}a}^3={\color{red}b}^3 - \uncover<6->{\;\Rightarrow\; - \frac{b}{a} = \sqrt[3]{2}}$}; -} - -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.52\textwidth} -\begin{block}{Aufgabe} -Konstruiere einen Würfel mit doppeltem Volumen -\end{block} -\uncover<7->{% -\begin{block}{Algebraisierte Aufgabe} -Konstruiere eine Nullstelle von $p(x)=x^3-2$ -\end{block}} -\uncover<8->{% -\begin{proof}[Unmöglichkeitsbeweis] -\begin{itemize} -\item<9-> -$p(x)$ irreduzibel -\item<10-> -$p(x)$ definiert eine Körpererweiterung vom Grad $3$ -\item<11-> -Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar -\qedhere -\end{itemize} -\end{proof}} -\end{column} -\end{columns} -\end{frame} +% +% wuerfel.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\frametitle{Würfelverdoppelung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel.png}}; +\uncover<2->{ +\node at (0,0) {\includegraphics[width=6.0cm]{../slides/4/galois/images/wuerfel2.png}}; +} + +\uncover<3->{ + \draw[<->,color=blue] (-1.25,-2.4) -- (2.55,-2.25); + \node[color=blue] at (0.75,-2.3) [above] {$a$}; +} + +\uncover<4->{ + \begin{scope}[yshift=0.03cm] + \draw[color=red] (-2.13,-2.89) -- (-2.13,-3.19); + \draw[color=red] (2.85,-2.7) -- (2.85,-3.0); + \draw[<->,color=red] (-2.13,-3.09) -- (2.85,-2.9); + \end{scope} + \node[color=red] at (0.36,-2.9) [below] {$b$}; +} + +\uncover<5->{ +\node at (0,-4) {$ + 2{\color{blue}a}^3={\color{red}b}^3 + \uncover<6->{\;\Rightarrow\; + \frac{b}{a} = \sqrt[3]{2}}$}; +} + +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Aufgabe} +Konstruiere einen Würfel mit doppeltem Volumen +\end{block} +\uncover<7->{% +\begin{block}{Algebraisierte Aufgabe} +Konstruiere eine Nullstelle von $p(x)=x^3-2$ +\end{block}} +\uncover<8->{% +\begin{proof}[Unmöglichkeitsbeweis] +\begin{itemize} +\item<9-> +$p(x)$ irreduzibel +\item<10-> +$p(x)$ definiert eine Körpererweiterung vom Grad $3$ +\item<11-> +Nur Körpererweiterungen vom Grad $2^l$ sind konstruierbar +\qedhere +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 7afeea1..2391099 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -1,22 +1,22 @@ -# -# Makefile.inc -- additional depencencies -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# -chapter5 = \ - ../slides/7/symmetrien.tex \ - ../slides/7/algebraisch.tex \ - ../slides/7/parameter.tex \ - ../slides/7/mannigfaltigkeit.tex \ - ../slides/7/sl2.tex \ - ../slides/7/drehung.tex \ - ../slides/7/drehanim.tex \ - ../slides/7/semi.tex \ - ../slides/7/kurven.tex \ - ../slides/7/einparameter.tex \ - ../slides/7/ableitung.tex \ - ../slides/7/liealgebra.tex \ - ../slides/7/kommutator.tex \ - ../slides/7/dg.tex \ - ../slides/7/chapter.tex - +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +chapter5 = \ + ../slides/7/symmetrien.tex \ + ../slides/7/algebraisch.tex \ + ../slides/7/parameter.tex \ + ../slides/7/mannigfaltigkeit.tex \ + ../slides/7/sl2.tex \ + ../slides/7/drehung.tex \ + ../slides/7/drehanim.tex \ + ../slides/7/semi.tex \ + ../slides/7/kurven.tex \ + ../slides/7/einparameter.tex \ + ../slides/7/ableitung.tex \ + ../slides/7/liealgebra.tex \ + ../slides/7/kommutator.tex \ + ../slides/7/dg.tex \ + ../slides/7/chapter.tex + diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex index 12f9084..5a4b94e 100644 --- a/vorlesungen/slides/7/ableitung.tex +++ b/vorlesungen/slides/7/ableitung.tex @@ -1,68 +1,68 @@ -% -% ableitung.tex -- Ableitung in der Lie-Gruppe -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Ableitung in der Matrix-Gruppe} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Ableitung in $\operatorname{O}(n)$} -\uncover<2->{% -$s \mapsto A(s)\in\operatorname{O}(n)$ -} -\begin{align*} -\uncover<3->{I -&= -A(s)^tA(s)} -\\ -\uncover<4->{0 -= -\frac{d}{ds} I -&= -\frac{d}{ds} (A(s)^t A(s))} -\\ -&\uncover<5->{= -\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)} -\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}} -\uncover<7->{0 -&= -\dot{A}(0)^t -+ -\dot{A}(0)} -\\ -\uncover<8->{\Leftrightarrow -\qquad -\dot{A}(0)^t &= -\dot{A}(0)} -\end{align*} -\uncover<9->{% -``Tangentialvektoren'' sind antisymmetrische Matrizen -} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} -\uncover<2->{% -$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ -} -\begin{align*} -\uncover<3->{1 &= \det A(t)} -\\ -\uncover<10->{0 -= -\frac{d}{dt}1 -&= -\frac{d}{dt} \det A(t)} -\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}} -\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)} -\end{align*} -\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} -\end{block} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% ableitung.tex -- Ableitung in der Lie-Gruppe +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ableitung in der Matrix-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{O}(n)$} +\uncover<2->{% +$s \mapsto A(s)\in\operatorname{O}(n)$ +} +\begin{align*} +\uncover<3->{I +&= +A(s)^tA(s)} +\\ +\uncover<4->{0 += +\frac{d}{ds} I +&= +\frac{d}{ds} (A(s)^t A(s))} +\\ +&\uncover<5->{= +\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)} +\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}} +\uncover<7->{0 +&= +\dot{A}(0)^t ++ +\dot{A}(0)} +\\ +\uncover<8->{\Leftrightarrow +\qquad +\dot{A}(0)^t &= -\dot{A}(0)} +\end{align*} +\uncover<9->{% +``Tangentialvektoren'' sind antisymmetrische Matrizen +} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} +\uncover<2->{% +$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ +} +\begin{align*} +\uncover<3->{1 &= \det A(t)} +\\ +\uncover<10->{0 += +\frac{d}{dt}1 +&= +\frac{d}{dt} \det A(t)} +\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}} +\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)} +\end{align*} +\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex index 31d209a..fba42cf 100644 --- a/vorlesungen/slides/7/algebraisch.tex +++ b/vorlesungen/slides/7/algebraisch.tex @@ -1,115 +1,115 @@ -% -% algebraisch.tex -- algebraische Definition der Symmetrien -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Erhaltungsgrössen und Algebra} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Längen und Winkel} -Längenmessung mit Skalarprodukt -\begin{align*} -\|\vec{v}\|^2 -&= -\langle \vec{v},\vec{v}\rangle -= -\vec{v}\cdot \vec{v} -\uncover<2->{= -\vec{v}^t\vec{v}} -\end{align*} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<3->{% -\begin{block}{Flächeninhalt/Volumen} -$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ -\\ -Volumen des Parallelepipeds: $\det V$ -\end{block}} -\end{column} -\end{columns} -% -\vspace{-7pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<4->{% -\begin{block}{Längenerhaltende Transformationen} -$A\in\operatorname{GL}_n(\mathbb{R})$ -\begin{align*} -\vec{x}^t\vec{y} -&= -(A\vec{x}) -\cdot -(A\vec{y}) -\uncover<5->{= -(A\vec{x})^t -(A\vec{y})} -\\ -\uncover<6->{ -\vec{x}^tI\vec{y} -&= -\vec{x}^tA^tA\vec{y}} -\uncover<7->{ -\Rightarrow I=A^tA} -\end{align*} -\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<9->{% -\begin{block}{Volumenerhaltende Transformationen} -$A\in\operatorname{GL}_n(\mathbb{R})$ -\begin{align*} -\det(V) -&= -\det(AV) -\uncover<10->{= -\det(A)\det(V)} -\\ -\uncover<11->{ -1&=\det(A)} -\end{align*} -\uncover<10->{ -(Produktsatz für Determinante) -} -\end{block}} -\end{column} -\end{columns} -% -\vspace{-3pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<12->{% -\begin{block}{Orthogonale Matrizen} -Längentreue Abbildungen = orthogonale Matrizen: -\[ -O(n) -= -\{ -A \in \operatorname{GL}_n(\mathbb{R}) -\;|\; -A^tA=I -\} -\] -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<13->{% -\begin{block}{``Spezielle'' Matrizen} -Volumen-/Orientierungserhaltende Transformationen: -\[ -\operatorname{SL}_n(\mathbb R) -= -\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} -\] -\end{block}} -\end{column} -\end{columns} - -\end{frame} -\egroup +% +% algebraisch.tex -- algebraische Definition der Symmetrien +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Erhaltungsgrössen und Algebra} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Längen und Winkel} +Längenmessung mit Skalarprodukt +\begin{align*} +\|\vec{v}\|^2 +&= +\langle \vec{v},\vec{v}\rangle += +\vec{v}\cdot \vec{v} +\uncover<2->{= +\vec{v}^t\vec{v}} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Flächeninhalt/Volumen} +$n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ +\\ +Volumen des Parallelepipeds: $\det V$ +\end{block}} +\end{column} +\end{columns} +% +\vspace{-7pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Längenerhaltende Transformationen} +$A\in\operatorname{GL}_n(\mathbb{R})$ +\begin{align*} +\vec{x}^t\vec{y} +&= +(A\vec{x}) +\cdot +(A\vec{y}) +\uncover<5->{= +(A\vec{x})^t +(A\vec{y})} +\\ +\uncover<6->{ +\vec{x}^tI\vec{y} +&= +\vec{x}^tA^tA\vec{y}} +\uncover<7->{ +\Rightarrow I=A^tA} +\end{align*} +\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<9->{% +\begin{block}{Volumenerhaltende Transformationen} +$A\in\operatorname{GL}_n(\mathbb{R})$ +\begin{align*} +\det(V) +&= +\det(AV) +\uncover<10->{= +\det(A)\det(V)} +\\ +\uncover<11->{ +1&=\det(A)} +\end{align*} +\uncover<10->{ +(Produktsatz für Determinante) +} +\end{block}} +\end{column} +\end{columns} +% +\vspace{-3pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Orthogonale Matrizen} +Längentreue Abbildungen = orthogonale Matrizen: +\[ +O(n) += +\{ +A \in \operatorname{GL}_n(\mathbb{R}) +\;|\; +A^tA=I +\} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<13->{% +\begin{block}{``Spezielle'' Matrizen} +Volumen-/Orientierungserhaltende Transformationen: +\[ +\operatorname{SL}_n(\mathbb R) += +\{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} +\] +\end{block}} +\end{column} +\end{columns} + +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 079cf16..0f14a9a 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -1,19 +1,19 @@ -% -% chapter.tex -% -% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi -% -\folie{7/symmetrien.tex} -\folie{7/algebraisch.tex} -\folie{7/parameter.tex} -\folie{7/mannigfaltigkeit.tex} -\folie{7/sl2.tex} -\folie{7/drehung.tex} -\folie{7/drehanim.tex} -\folie{7/semi.tex} -\folie{7/kurven.tex} -\folie{7/einparameter.tex} -\folie{7/ableitung.tex} -\folie{7/liealgebra.tex} -\folie{7/kommutator.tex} -\folie{7/dg.tex} +% +% chapter.tex +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi +% +\folie{7/symmetrien.tex} +\folie{7/algebraisch.tex} +\folie{7/parameter.tex} +\folie{7/mannigfaltigkeit.tex} +\folie{7/sl2.tex} +\folie{7/drehung.tex} +\folie{7/drehanim.tex} +\folie{7/semi.tex} +\folie{7/kurven.tex} +\folie{7/einparameter.tex} +\folie{7/ableitung.tex} +\folie{7/liealgebra.tex} +\folie{7/kommutator.tex} +\folie{7/dg.tex} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 4447bac..446b2ab 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -1,92 +1,92 @@ -% -% dg.tex -- Differentialgleichung für die Exponentialabbildung -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Zurück zur Lie-Gruppe} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Tangentialvektor im Punkt $\gamma(t)$} -Ableitung von $\gamma(t)$ an der Stelle $t$: -\begin{align*} -\dot{\gamma}(t) -&\uncover<2->{= -\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} -} -\\ -&\uncover<3->{= -\frac{d}{ds} -\gamma(t+s) -\bigg|_{s=0} -} -\\ -&\uncover<4->{= -\frac{d}{ds} -\gamma(t)\gamma(s) -\bigg|_{s=0} -} -\\ -&\uncover<5->{= -\gamma(t) -\frac{d}{ds} -\gamma(s) -\bigg|_{s=0} -} -\uncover<6->{= -\gamma(t) \dot{\gamma}(0) -} -\end{align*} -\end{block} -\vspace{-10pt} -\uncover<7->{% -\begin{block}{Differentialgleichung} -\vspace{-10pt} -\[ -\dot{\gamma}(t) = \gamma(t) A -\quad -\text{mit} -\quad -A=\dot{\gamma}(0)\in LG -\] -\end{block}} -\end{column} -\begin{column}{0.50\textwidth} -\uncover<8->{% -\begin{block}{Lösung} -Exponentialfunktion -\[ -\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k -\] -\end{block}} -\vspace{-5pt} -\uncover<9->{% -\begin{block}{Kontrolle: Tangentialvektor berechnen} -\vspace{-10pt} -\begin{align*} -\frac{d}{dt}e^{At} -&\uncover<10->{= -\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!} -} -\\ -&\uncover<11->{= -\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A -} -\\ -&\uncover<12->{= -\sum_{k=0} A^k\frac{t^k}{k!} -A -} -\uncover<13->{= -e^{At} A -} -\end{align*} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% dg.tex -- Differentialgleichung für die Exponentialabbildung +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zurück zur Lie-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Tangentialvektor im Punkt $\gamma(t)$} +Ableitung von $\gamma(t)$ an der Stelle $t$: +\begin{align*} +\dot{\gamma}(t) +&\uncover<2->{= +\frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} +} +\\ +&\uncover<3->{= +\frac{d}{ds} +\gamma(t+s) +\bigg|_{s=0} +} +\\ +&\uncover<4->{= +\frac{d}{ds} +\gamma(t)\gamma(s) +\bigg|_{s=0} +} +\\ +&\uncover<5->{= +\gamma(t) +\frac{d}{ds} +\gamma(s) +\bigg|_{s=0} +} +\uncover<6->{= +\gamma(t) \dot{\gamma}(0) +} +\end{align*} +\end{block} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Differentialgleichung} +\vspace{-10pt} +\[ +\dot{\gamma}(t) = \gamma(t) A +\quad +\text{mit} +\quad +A=\dot{\gamma}(0)\in LG +\] +\end{block}} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<8->{% +\begin{block}{Lösung} +Exponentialfunktion +\[ +\exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k +\] +\end{block}} +\vspace{-5pt} +\uncover<9->{% +\begin{block}{Kontrolle: Tangentialvektor berechnen} +\vspace{-10pt} +\begin{align*} +\frac{d}{dt}e^{At} +&\uncover<10->{= +\sum_{k=1}^\infty A^k \frac{d}{dt} \frac{t^k}{k!} +} +\\ +&\uncover<11->{= +\sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A +} +\\ +&\uncover<12->{= +\sum_{k=0} A^k\frac{t^k}{k!} +A +} +\uncover<13->{= +e^{At} A +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex index ac136f1..776617f 100644 --- a/vorlesungen/slides/7/drehanim.tex +++ b/vorlesungen/slides/7/drehanim.tex @@ -1,155 +1,155 @@ -% -% template.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup - -\definecolor{darkgreen}{rgb}{0,0.6,0} -\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) } - -\makeatletter -\hoffset=-2cm -\advance\textwidth2cm -\hsize\textwidth -\columnwidth\textwidth -\makeatother - -\begin{frame}[t,plain] -\vspace{-5pt} -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] - -\fill[color=white] (-4,-4) rectangle (9,4.5); - -\def\a{60} - -\pgfmathparse{tan(\a)} -\xdef\T{\pgfmathresult} - -\pgfmathparse{-sin(\a)*cos(\a)} -\xdef\S{\pgfmathresult} - -\pgfmathparse{1/cos(\a)} -\xdef\E{\pgfmathresult} - -\def\N{20} -\pgfmathparse{2*\N} -\xdef\Nzwei{\pgfmathresult} -\pgfmathparse{3*\N} -\xdef\Ndrei{\pgfmathresult} - -\node at (4.2,4.2) [below right] {\begin{minipage}{7cm} -\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$} -\begin{itemize} -\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized -as shear mappings and axis rescalings -\[ -A= -\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix} -\begin{pmatrix}1&s\\0&1\end{pmatrix} -\begin{pmatrix}1&0\\t&1\end{pmatrix} -\] -\item Most rotations can be decomposed into a product of -shear mappings and axis rescalings -\end{itemize} -\end{block} -\end{minipage}}; - -\foreach \d in {1,2,...,\Ndrei}{ - % Scherung in Y-Richtung - \ifnum \d>\N - \pgfmathparse{\T} - \else - \pgfmathparse{\T*(\d-1)/(\N-1)} - \fi - \xdef\t{\pgfmathresult} - - % Scherung in X-Richtung - \ifnum \d>\Nzwei - \xdef\s{\S} - \else - \ifnum \d<\N - \xdef\s{0} - \else - \ifnum \d=\N - \xdef\s{0} - \else - \pgfmathparse{\S*(\d-\N-1)/(\N-1)} - \xdef\s{\pgfmathresult} - \fi - \fi - \fi - - % Reskalierung der Achsen - \ifnum \d>\Nzwei - \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))} - \else - \pgfmathparse{1} - \fi - \xdef\e{\pgfmathresult} - - % Matrixelemente - \pgfmathparse{(\e)*((\s)*(\t)+1)} - \xdef\A{\pgfmathresult} - - \pgfmathparse{(\e)*(\s)} - \xdef\B{\pgfmathresult} - - \pgfmathparse{(\t)/(\e)} - \xdef\C{\pgfmathresult} - - \pgfmathparse{1/(\e)} - \xdef\D{\pgfmathresult} - - \only<\d>{ - \node at (5.0,-0.9) [below right] {$ - \begin{aligned} - t &= \t \\ - s &= \s \\ - d &= \e \\ - D &= \begin{pmatrix} - \A&\B\\ - \C&\D - \end{pmatrix} - \qquad - \only<60>{\checkmark} - \end{aligned} - $}; - } - - \begin{scope} - \clip (-4.05,-4.05) rectangle (4.05,4.05); - \only<\d>{ - \foreach \x in {-6,...,6}{ - \draw[color=blue,line width=0.5pt] - \punkt{\x}{-12} -- \punkt{\x}{12}; - } - \foreach \y in {-12,...,12}{ - \draw[color=darkgreen,line width=0.5pt] - \punkt{-6}{\y} -- \punkt{6}{\y}; - } - - \foreach \r in {1,2,3,4}{ - \draw[color=red] plot[domain=0:359,samples=360] - ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))}) - -- - cycle; - } - } - \end{scope} - -% \uncover<\d>{ -% \node at (5,4) {\d}; -% } -} - -\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}]; -\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}]; - -\end{tikzpicture} -\end{center} -\end{frame} -\egroup +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup + +\definecolor{darkgreen}{rgb}{0,0.6,0} +\def\punkt#1#2{ ({\A*(#1)+\B*(#2)},{\C*(#1)+\D*(#2)}) } + +\makeatletter +\hoffset=-2cm +\advance\textwidth2cm +\hsize\textwidth +\columnwidth\textwidth +\makeatother + +\begin{frame}[t,plain] +\vspace{-5pt} +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\fill[color=white] (-4,-4) rectangle (9,4.5); + +\def\a{60} + +\pgfmathparse{tan(\a)} +\xdef\T{\pgfmathresult} + +\pgfmathparse{-sin(\a)*cos(\a)} +\xdef\S{\pgfmathresult} + +\pgfmathparse{1/cos(\a)} +\xdef\E{\pgfmathresult} + +\def\N{20} +\pgfmathparse{2*\N} +\xdef\Nzwei{\pgfmathresult} +\pgfmathparse{3*\N} +\xdef\Ndrei{\pgfmathresult} + +\node at (4.2,4.2) [below right] {\begin{minipage}{7cm} +\begin{block}{$\operatorname{SO}(2)\subset\operatorname{SL}_2(\mathbb{R})$} +\begin{itemize} +\item Thus most $A\in\operatorname{SL}_2(\mathbb{R})$ can be parametrized +as shear mappings and axis rescalings +\[ +A= +\begin{pmatrix}d&0\\0&d^{-1}\end{pmatrix} +\begin{pmatrix}1&s\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\t&1\end{pmatrix} +\] +\item Most rotations can be decomposed into a product of +shear mappings and axis rescalings +\end{itemize} +\end{block} +\end{minipage}}; + +\foreach \d in {1,2,...,\Ndrei}{ + % Scherung in Y-Richtung + \ifnum \d>\N + \pgfmathparse{\T} + \else + \pgfmathparse{\T*(\d-1)/(\N-1)} + \fi + \xdef\t{\pgfmathresult} + + % Scherung in X-Richtung + \ifnum \d>\Nzwei + \xdef\s{\S} + \else + \ifnum \d<\N + \xdef\s{0} + \else + \ifnum \d=\N + \xdef\s{0} + \else + \pgfmathparse{\S*(\d-\N-1)/(\N-1)} + \xdef\s{\pgfmathresult} + \fi + \fi + \fi + + % Reskalierung der Achsen + \ifnum \d>\Nzwei + \pgfmathparse{exp(ln(\E)*(\d-2*\N-1)/(\N-1))} + \else + \pgfmathparse{1} + \fi + \xdef\e{\pgfmathresult} + + % Matrixelemente + \pgfmathparse{(\e)*((\s)*(\t)+1)} + \xdef\A{\pgfmathresult} + + \pgfmathparse{(\e)*(\s)} + \xdef\B{\pgfmathresult} + + \pgfmathparse{(\t)/(\e)} + \xdef\C{\pgfmathresult} + + \pgfmathparse{1/(\e)} + \xdef\D{\pgfmathresult} + + \only<\d>{ + \node at (5.0,-0.9) [below right] {$ + \begin{aligned} + t &= \t \\ + s &= \s \\ + d &= \e \\ + D &= \begin{pmatrix} + \A&\B\\ + \C&\D + \end{pmatrix} + \qquad + \only<60>{\checkmark} + \end{aligned} + $}; + } + + \begin{scope} + \clip (-4.05,-4.05) rectangle (4.05,4.05); + \only<\d>{ + \foreach \x in {-6,...,6}{ + \draw[color=blue,line width=0.5pt] + \punkt{\x}{-12} -- \punkt{\x}{12}; + } + \foreach \y in {-12,...,12}{ + \draw[color=darkgreen,line width=0.5pt] + \punkt{-6}{\y} -- \punkt{6}{\y}; + } + + \foreach \r in {1,2,3,4}{ + \draw[color=red] plot[domain=0:359,samples=360] + ({\r*(\A*cos(\x)+\B*sin(\x))},{\r*(\C*cos(\x)+\D*sin(\x))}) + -- + cycle; + } + } + \end{scope} + +% \uncover<\d>{ +% \node at (5,4) {\d}; +% } +} + +\draw[->] (-4,0) -- (4.2,0) coordinate[label={$x$}]; +\draw[->] (0,-4) -- (0,4.2) coordinate[label={right:$y$}]; + +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index 2d7b317..e7b4a92 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -1,132 +1,132 @@ -% -% drehung.tex -- Drehung aus streckungen -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\definecolor{darkgreen}{rgb}{0,0.6,0} -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Drehung aus Streckungen und Scherungen} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.38\textwidth} -\begin{block}{Drehung} -{\color{blue}Längen}, {\color<2->{blue}Winkel}, -{\color<2->{darkgreen}Orientierung} -erhalten -\uncover<2->{ -\[ -\operatorname{SO}(2) -= -{\color{blue}\operatorname{O}(2)} -\cap -{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})} -\]} -\vspace{-20pt} -\end{block} -\uncover<3->{% -\begin{block}{Zusammensetzung} -Eine Drehung muss als Zusammensetzung geschrieben werden können: -\[ -D_{\alpha} -= -\begin{pmatrix} -\cos\alpha & -\sin\alpha\\ -\sin\alpha &\phantom{-}\cos\alpha -\end{pmatrix} -= -DST -\] -\end{block}} -\vspace{-10pt} -\uncover<12->{% -\begin{block}{Beispiel} -\vspace{-12pt} -\[ -D_{60^\circ} -= -{\tiny -\begin{pmatrix}2&0\\0&\frac12\end{pmatrix} -\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} -\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix} -} -\] -\end{block}} -\end{column} -\begin{column}{0.58\textwidth} -\uncover<4->{% -\begin{block}{Ansatz} -\vspace{-12pt} -\begin{align*} -DST -&= -\begin{pmatrix} -c^{-1}&0\\ - 0 &c -\end{pmatrix} -\begin{pmatrix} -1&-s\\ -0&1 -\end{pmatrix} -\begin{pmatrix} -1&0\\ -t&1 -\end{pmatrix} -\\ -&\uncover<5->{= -\begin{pmatrix} -c^{-1}&0\\ - 0 &c -\end{pmatrix} -\begin{pmatrix} -1-st&-s\\ - t& 1 -\end{pmatrix} -} -\\ -&\uncover<6->{= -\begin{pmatrix} -{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ -{\color<9->{blue}ct}&{\color<8->{red}c} -\end{pmatrix}} -\uncover<7->{= -\begin{pmatrix} -{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\ -{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha} -\end{pmatrix}} -\end{align*} -\end{block}} -\vspace{-10pt} -\uncover<7->{% -\begin{block}{Koeffizientenvergleich} -\vspace{-15pt} -\begin{align*} -\uncover<8->{ -{\color{red} c} -&= -{\color{red}\cos\alpha }} -&& -& -\uncover<9->{ -{\color{blue} -t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ -\uncover<10->{ -{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} -& -&\Rightarrow& -{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha -} -\\ -\uncover<11->{ -{\color{orange} (1-st)c^{-t}} -&= -\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $} -} -\end{align*} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% drehung.tex -- Drehung aus streckungen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehung aus Streckungen und Scherungen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Drehung} +{\color{blue}Längen}, {\color<2->{blue}Winkel}, +{\color<2->{darkgreen}Orientierung} +erhalten +\uncover<2->{ +\[ +\operatorname{SO}(2) += +{\color{blue}\operatorname{O}(2)} +\cap +{\color{darkgreen}\operatorname{SL}_2(\mathbb{R})} +\]} +\vspace{-20pt} +\end{block} +\uncover<3->{% +\begin{block}{Zusammensetzung} +Eine Drehung muss als Zusammensetzung geschrieben werden können: +\[ +D_{\alpha} += +\begin{pmatrix} +\cos\alpha & -\sin\alpha\\ +\sin\alpha &\phantom{-}\cos\alpha +\end{pmatrix} += +DST +\] +\end{block}} +\vspace{-10pt} +\uncover<12->{% +\begin{block}{Beispiel} +\vspace{-12pt} +\[ +D_{60^\circ} += +{\tiny +\begin{pmatrix}2&0\\0&\frac12\end{pmatrix} +\begin{pmatrix}1&-\frac{\sqrt{3}}4\\0&1\end{pmatrix} +\begin{pmatrix}1&0\\\sqrt{3}&1\end{pmatrix} +} +\] +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<4->{% +\begin{block}{Ansatz} +\vspace{-12pt} +\begin{align*} +DST +&= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +1&-s\\ +0&1 +\end{pmatrix} +\begin{pmatrix} +1&0\\ +t&1 +\end{pmatrix} +\\ +&\uncover<5->{= +\begin{pmatrix} +c^{-1}&0\\ + 0 &c +\end{pmatrix} +\begin{pmatrix} +1-st&-s\\ + t& 1 +\end{pmatrix} +} +\\ +&\uncover<6->{= +\begin{pmatrix} +{\color<11->{orange}(1-st)c^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<9->{blue}ct}&{\color<8->{red}c} +\end{pmatrix}} +\uncover<7->{= +\begin{pmatrix} +{\color<11->{orange}\cos\alpha} & {\color<10->{darkgreen}- \sin\alpha} \\ +{\color<9->{blue}\sin\alpha} & \phantom{-} {\color<8->{red}\cos\alpha} +\end{pmatrix}} +\end{align*} +\end{block}} +\vspace{-10pt} +\uncover<7->{% +\begin{block}{Koeffizientenvergleich} +\vspace{-15pt} +\begin{align*} +\uncover<8->{ +{\color{red} c} +&= +{\color{red}\cos\alpha }} +&& +& +\uncover<9->{ +{\color{blue} +t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$}}\\ +\uncover<10->{ +{\color{darkgreen}sc^{-1}}&={\color{darkgreen}-\sin\alpha} +& +&\Rightarrow& +{\color{darkgreen}s}&={\color{darkgreen}-\sin\alpha}\cos\alpha +} +\\ +\uncover<11->{ +{\color{orange} (1-st)c^{-t}} +&= +\rlap{$\displaystyle\frac{(1-\sin^2\alpha)}{\cos\alpha} = \cos\alpha $} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex index 5171085..e9699a6 100644 --- a/vorlesungen/slides/7/einparameter.tex +++ b/vorlesungen/slides/7/einparameter.tex @@ -1,93 +1,93 @@ -% -% einparameter.tex -- Einparameter Untergruppen -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Einparameter-Untergruppen} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Definition} -Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$, -die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit} -\[ -\uncover<3->{ -\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R} -} -\] -\end{block} -\uncover<4->{% -\begin{block}{Drehungen} -Drehmatrizen bilden Einparameter- Untergruppen -\begin{align*} -t \mapsto D_{x,t} -&= -\begin{pmatrix} -1&0&0\\ -0&\cos t&-\sin t\\ -0&\sin t& \cos t -\end{pmatrix} -\\ -D_{x,t}D_{x,s} -&= -D_{x,t+s} -\end{align*} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<5->{% -\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} -\vspace{-12pt} -\[ -\begin{pmatrix} -1&s\\ -0&1 -\end{pmatrix} -\begin{pmatrix} -1&t\\ -0&1 -\end{pmatrix} -= -\begin{pmatrix} -1&s+t\\ -0&1 -\end{pmatrix} -\] -\end{block}} -\vspace{-12pt} -\uncover<6->{% -\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} -\vspace{-12pt} -\[ -\begin{pmatrix} -e^s&0\\0&e^{-s} -\end{pmatrix} -\begin{pmatrix} -e^t&0\\0&e^{-t} -\end{pmatrix} -= -\begin{pmatrix} -e^{t+s}&0\\0&e^{-(t+s)} -\end{pmatrix} -\] -\end{block}} -\vspace{-12pt} -\uncover<7->{% -\begin{block}{Gemischt} -\vspace{-12pt} -\begin{gather*} -A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t -\\ -\text{dank}\quad -\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 -=I -\end{gather*} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% einparameter.tex -- Einparameter Untergruppen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Einparameter-Untergruppen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +Eine Kurve $\gamma\colon \mathbb{R}\to G\subset\operatorname{GL}_n(\mathbb{R})$, +die {\color<2->{red}gleichzeitig eine Untergruppe von $G$} ist \uncover<3->{mit} +\[ +\uncover<3->{ +\gamma(t+s) = \gamma(t)\gamma(s)\quad\forall t,s\in\mathbb{R} +} +\] +\end{block} +\uncover<4->{% +\begin{block}{Drehungen} +Drehmatrizen bilden Einparameter- Untergruppen +\begin{align*} +t \mapsto D_{x,t} +&= +\begin{pmatrix} +1&0&0\\ +0&\cos t&-\sin t\\ +0&\sin t& \cos t +\end{pmatrix} +\\ +D_{x,t}D_{x,s} +&= +D_{x,t+s} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} +\vspace{-12pt} +\[ +\begin{pmatrix} +1&s\\ +0&1 +\end{pmatrix} +\begin{pmatrix} +1&t\\ +0&1 +\end{pmatrix} += +\begin{pmatrix} +1&s+t\\ +0&1 +\end{pmatrix} +\] +\end{block}} +\vspace{-12pt} +\uncover<6->{% +\begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} +\vspace{-12pt} +\[ +\begin{pmatrix} +e^s&0\\0&e^{-s} +\end{pmatrix} +\begin{pmatrix} +e^t&0\\0&e^{-t} +\end{pmatrix} += +\begin{pmatrix} +e^{t+s}&0\\0&e^{-(t+s)} +\end{pmatrix} +\] +\end{block}} +\vspace{-12pt} +\uncover<7->{% +\begin{block}{Gemischt} +\vspace{-12pt} +\begin{gather*} +A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t +\\ +\text{dank}\quad +\begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 +=I +\end{gather*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile index cc67c8a..9de1c34 100644 --- a/vorlesungen/slides/7/images/Makefile +++ b/vorlesungen/slides/7/images/Makefile @@ -1,19 +1,19 @@ -# -# Makefile -- Illustrationen zu -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -#  -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 - - +# +# Makefile -- Illustrationen zu +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +#  +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/common.inc b/vorlesungen/slides/7/images/common.inc index 0e27c9a..b028956 100644 --- a/vorlesungen/slides/7/images/common.inc +++ b/vorlesungen/slides/7/images/common.inc @@ -1,70 +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) - +// +// 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 index 8c2211e..44a5ac5 100644 --- a/vorlesungen/slides/7/images/commutator.ini +++ b/vorlesungen/slides/7/images/commutator.ini @@ -1,8 +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 - +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 index 5a448db..3f5ea17 100644 --- a/vorlesungen/slides/7/images/commutator.m +++ b/vorlesungen/slides/7/images/commutator.m @@ -1,111 +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 +# +# 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 index 9ae11b9..8229a06 100644 --- a/vorlesungen/slides/7/images/commutator.pov +++ b/vorlesungen/slides/7/images/commutator.pov @@ -1,59 +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 - } -} - +// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +sphere { O, 0.99 + pigment { + color rgbt<1,1,1,0.5> + } + finish { + specular 0.9 + metallic + } +} + +#declare filename = concat("f/", str(clock, -4, 0), ".inc"); + +#include filename + +#declare n1 = vcross(<0,1,0>, finalXY); +#declare n2 = vcross(<0,1,0>, finalYX); + +intersection { + sphere { O, 1 } + plane { -n1, 0 } + plane { n2, 0 } + pigment { + color rgb<0,0.4,0.1> + } + finish { + specular 0.9 + metallic + } +} + +union { + XYkurve() + pigment { + color Red + } + finish { + specular 0.9 + metallic + } +} + +union { + YXkurve() + pigment { + color Blue + } + finish { + specular 0.9 + metallic + } +} + diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov index 07aec19..62306f8 100644 --- a/vorlesungen/slides/7/images/rodriguez.pov +++ b/vorlesungen/slides/7/images/rodriguez.pov @@ -1,118 +1,118 @@ -// -// rodriguez.pov -// -// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -// -#version 3.7; -#include "colors.inc" - -global_settings { - assumed_gamma 1 -} - -#declare imagescale = 0.020; -#declare O = <0, 0, 0>; -#declare at = 0.015; - -camera { - location <8, 15, -50> - look_at <0.1, 0.475, 0> - right 16/9 * x * imagescale - up y * imagescale -} - -light_source { - <-4, 20, -50> color White - area_light <1,0,0> <0,0,1>, 10, 10 - adaptive 1 - jitter -} - -sky_sphere { - pigment { - color rgb<1,1,1> - } -} - -#macro arrow(from, to, arrowthickness, c) -#declare arrowdirection = vnormalize(to - from); -#declare arrowlength = vlength(to - from); -union { - sphere { - from, 1.1 * arrowthickness - } - cylinder { - from, - from + (arrowlength - 5 * arrowthickness) * arrowdirection, - arrowthickness - } - cone { - from + (arrowlength - 5 * arrowthickness) * arrowdirection, - 2 * arrowthickness, - to, - 0 - } - pigment { - color c - } - finish { - specular 0.9 - metallic - } -} -#end - -#declare K = vnormalize(<0.2,1,0.1>); -#declare X = vnormalize(<1.1,1,-1.2>); -#declare O = <0,0,0>; - -#declare r = vlength(vcross(K, X)) / vlength(K); - -#declare l = 1.0; - -arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) -arrow(< 0, 0, -l >, < 0, 0, l >, at, White) -arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) - -arrow(O, X, at, Red) -arrow(O, K, at, Blue) - -#macro punkt(H,phi) - ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K) -#end - -cone { vdot(K, X) * K, r, O, 0 - pigment { - color rgbt<0.6,0.6,0.6,0.5> - } - finish { - specular 0.9 - metallic - } -} - - -union { - #declare phistep = pi / 100; - #declare phi = 0; - #while (phi < 2 * pi - phistep/2) - sphere { punkt(K, phi), at/2 } - cylinder { - punkt(X, phi), - punkt(X, phi + phistep), - at/2 - } - #declare phi = phi + phistep; - #end - pigment { - color Orange - } - finish { - specular 0.9 - metallic - } -} - -arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow) -#declare Darkgreen = rgb<0,0.5,0>; -arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen) +// +// rodriguez.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.020; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <8, 15, -50> + look_at <0.1, 0.475, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-4, 20, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#declare K = vnormalize(<0.2,1,0.1>); +#declare X = vnormalize(<1.1,1,-1.2>); +#declare O = <0,0,0>; + +#declare r = vlength(vcross(K, X)) / vlength(K); + +#declare l = 1.0; + +arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) +arrow(< 0, 0, -l >, < 0, 0, l >, at, White) +arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) + +arrow(O, X, at, Red) +arrow(O, K, at, Blue) + +#macro punkt(H,phi) + ((H-vdot(K,H)*K)*cos(phi) + vcross(K,H)*sin(phi) + vdot(K,X)*K) +#end + +cone { vdot(K, X) * K, r, O, 0 + pigment { + color rgbt<0.6,0.6,0.6,0.5> + } + finish { + specular 0.9 + metallic + } +} + + +union { + #declare phistep = pi / 100; + #declare phi = 0; + #while (phi < 2 * pi - phistep/2) + sphere { punkt(K, phi), at/2 } + cylinder { + punkt(X, phi), + punkt(X, phi + phistep), + at/2 + } + #declare phi = phi + phistep; + #end + pigment { + color Orange + } + finish { + specular 0.9 + metallic + } +} + +arrow(vdot(K,X)*K, punkt(X, 0), at, Yellow) +#declare Darkgreen = rgb<0,0.5,0>; +arrow(vdot(K,X)*K, punkt(X, pi/2), at, Darkgreen) diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index 84bf034..9000160 100644 --- a/vorlesungen/slides/7/kommutator.tex +++ b/vorlesungen/slides/7/kommutator.tex @@ -1,166 +1,166 @@ -% -% template.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\definecolor{darkgreen}{rgb}{0,0.6,0} -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Kommutator in $\operatorname{SO}(3)$} -\vspace{-20pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\t{14.0cm} -\ifthenelse{\boolean{presentation}}{ -\only<1>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};} -\only<2>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};} -\only<3>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};} -\only<4>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};} -\only<5>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};} -\only<6>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};} -\only<7>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c07.jpg}};} -\only<8>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};} -\only<9>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c09.jpg}};} -\only<10>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c10.jpg}};} -\only<11>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c11.jpg}};} -\only<12>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c12.jpg}};} -\only<13>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c13.jpg}};} -\only<14>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c14.jpg}};} -\only<15>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c15.jpg}};} -\only<16>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c16.jpg}};} -\only<17>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c17.jpg}};} -\only<18>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c18.jpg}};} -\only<19>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c19.jpg}};} -\only<20>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c20.jpg}};} -\only<21>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c21.jpg}};} -\only<22>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c22.jpg}};} -\only<23>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c23.jpg}};} -\only<24>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c24.jpg}};} -\only<25>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c25.jpg}};} -\only<26>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c26.jpg}};} -\only<27>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c27.jpg}};} -\only<28>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c28.jpg}};} -\only<29>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c29.jpg}};} -\only<30>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c30.jpg}};} -\only<31>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c31.jpg}};} -\only<32>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c32.jpg}};} -\only<33>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c33.jpg}};} -\only<34>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c34.jpg}};} -\only<35>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c35.jpg}};} -\only<36>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c36.jpg}};} -\only<37>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c37.jpg}};} -\only<38>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c38.jpg}};} -\only<39>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c39.jpg}};} -\only<40>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c40.jpg}};} -\only<41>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c41.jpg}};} -\only<42>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c42.jpg}};} -\only<43>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c43.jpg}};} -\only<44>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c44.jpg}};} -\only<45>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c45.jpg}};} -\only<46>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c46.jpg}};} -\only<47>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c47.jpg}};} -\only<48>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c48.jpg}};} -\only<49>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c49.jpg}};} -\only<50>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c50.jpg}};} -\only<51>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};} -\only<52>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};} -\only<53>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};} -\only<54>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};} -\only<55>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};} -\only<56>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};} -\only<57>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};} -\only<58>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};} -\only<59>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};} -}{} -\only<60>{\node at (0,0) { -\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};} -\coordinate (A) at (-0.3,3); -\coordinate (B) at (-1.1,2); -\coordinate (C) at (-2.1,-1.2); -\draw[->,color=red,line width=1.4pt] - (A) - to[out=-143,in=60] - (B) - to[out=-120,in=80] - (C); -%\fill[color=red] (B) circle[radius=0.08]; -\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$}; -\coordinate (D) at (0.3,3.2); -\coordinate (E) at (1.8,2.8); -\coordinate (F) at (5.2,-0.3); -\draw[->,color=blue,line width=1.4pt] - (D) - to[out=-10,in=157] - (E) - to[out=-23,in=120] - (F); -\fill[color=blue] (E) circle[radius=0.08]; -\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$}; -\draw[->,color=darkgreen,line width=1.4pt] - (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6); -\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$}; -\end{tikzpicture} -\end{center} -\end{frame} -\egroup +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kommutator in $\operatorname{SO}(3)$} +\vspace{-20pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\t{14.0cm} +\ifthenelse{\boolean{presentation}}{ +\only<1>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c01.jpg}};} +\only<2>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c02.jpg}};} +\only<3>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c03.jpg}};} +\only<4>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c04.jpg}};} +\only<5>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c05.jpg}};} +\only<6>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c06.jpg}};} +\only<7>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c07.jpg}};} +\only<8>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c08.jpg}};} +\only<9>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c09.jpg}};} +\only<10>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c10.jpg}};} +\only<11>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c11.jpg}};} +\only<12>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c12.jpg}};} +\only<13>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c13.jpg}};} +\only<14>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c14.jpg}};} +\only<15>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c15.jpg}};} +\only<16>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c16.jpg}};} +\only<17>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c17.jpg}};} +\only<18>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c18.jpg}};} +\only<19>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c19.jpg}};} +\only<20>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c20.jpg}};} +\only<21>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c21.jpg}};} +\only<22>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c22.jpg}};} +\only<23>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c23.jpg}};} +\only<24>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c24.jpg}};} +\only<25>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c25.jpg}};} +\only<26>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c26.jpg}};} +\only<27>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c27.jpg}};} +\only<28>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c28.jpg}};} +\only<29>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c29.jpg}};} +\only<30>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c30.jpg}};} +\only<31>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c31.jpg}};} +\only<32>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c32.jpg}};} +\only<33>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c33.jpg}};} +\only<34>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c34.jpg}};} +\only<35>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c35.jpg}};} +\only<36>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c36.jpg}};} +\only<37>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c37.jpg}};} +\only<38>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c38.jpg}};} +\only<39>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c39.jpg}};} +\only<40>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c40.jpg}};} +\only<41>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c41.jpg}};} +\only<42>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c42.jpg}};} +\only<43>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c43.jpg}};} +\only<44>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c44.jpg}};} +\only<45>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c45.jpg}};} +\only<46>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c46.jpg}};} +\only<47>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c47.jpg}};} +\only<48>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c48.jpg}};} +\only<49>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c49.jpg}};} +\only<50>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c50.jpg}};} +\only<51>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c51.jpg}};} +\only<52>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c52.jpg}};} +\only<53>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c53.jpg}};} +\only<54>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c54.jpg}};} +\only<55>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};} +\only<56>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};} +\only<57>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};} +\only<58>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};} +\only<59>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};} +}{} +\only<60>{\node at (0,0) { +\includegraphics[width=\t]{../slides/7/images/c/c60.jpg}};} +\coordinate (A) at (-0.3,3); +\coordinate (B) at (-1.1,2); +\coordinate (C) at (-2.1,-1.2); +\draw[->,color=red,line width=1.4pt] + (A) + to[out=-143,in=60] + (B) + to[out=-120,in=80] + (C); +%\fill[color=red] (B) circle[radius=0.08]; +\node[color=red] at (-1.2,1.5) [above left] {$D_{x,\alpha}$}; +\coordinate (D) at (0.3,3.2); +\coordinate (E) at (1.8,2.8); +\coordinate (F) at (5.2,-0.3); +\draw[->,color=blue,line width=1.4pt] + (D) + to[out=-10,in=157] + (E) + to[out=-23,in=120] + (F); +\fill[color=blue] (E) circle[radius=0.08]; +\node[color=blue] at (2.4,2.4) [above right] {$D_{y,\beta}$}; +\draw[->,color=darkgreen,line width=1.4pt] + (0.7,-3.1) to[out=1,in=-160] (3.9,-2.6); +\node[color=darkgreen] at (2.5,-3.4) {$D_{z,\gamma}$}; +\end{tikzpicture} +\end{center} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex index e0690eb..bca8417 100644 --- a/vorlesungen/slides/7/kurven.tex +++ b/vorlesungen/slides/7/kurven.tex @@ -1,104 +1,104 @@ -% -% kurven.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Kurven und Tangenten} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Kurven} -Kurve in $\mathbb{R}^n$: -\vspace{-12pt} -\[ -\gamma -\colon -I=[a,b] \to \mathbb{R}^n -: -t\mapsto \gamma(t) -\uncover<2->{ -= -\begin{pmatrix} -x_1(t)\\ -x_2(t)\\ -\vdots\\ -x_n(t) -\end{pmatrix} -} -\] -\vspace{-15pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\coordinate (A) at (1,0.5); -\coordinate (B) at (4,0.5); -\coordinate (C) at (2,2.2); -\coordinate (D) at (5,2); -\coordinate (E) at ($(C)+(80:2)$); - -\draw[color=red,line width=1.4pt] - (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D); -\fill[color=red] (C) circle[radius=0.06]; -\node[color=red] at (C) [left] {$\gamma(t)$}; - -\uncover<4->{ - \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); - \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; -} - -\uncover<2->{ - \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; - \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; -} -\end{tikzpicture} -\end{center} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<4->{% -\begin{block}{Tangenten} -Ableitung -\[ -\frac{d}{dt}\gamma(t) -= -\dot{\gamma}(t) -= -\begin{pmatrix} -\dot{x}_1(t)\\ -\dot{x}_2(t)\\ -\vdots\\ -\dot{x}_n(t) -\end{pmatrix} -\] -\uncover<5->{% -Lineare Approximation: -\[ -\gamma(t+h) -= -\gamma(t) -+ -\dot{\gamma}(t) \cdot h -+ -o(h) -\]}% -\vspace{-10pt} -\begin{itemize} -\item<6-> -Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ -in $\mathbb{R}^n$ liegen -\item<7-> -Gilt auch für -\[ -\operatorname{GL}_n(\mathbb{R}) -\uncover<8->{\subset M_n(\mathbb{R})} -\uncover<9->{ = \mathbb{R}^{n\times n}} -\] -\end{itemize} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% kurven.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Kurven und Tangenten} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Kurven} +Kurve in $\mathbb{R}^n$: +\vspace{-12pt} +\[ +\gamma +\colon +I=[a,b] \to \mathbb{R}^n +: +t\mapsto \gamma(t) +\uncover<2->{ += +\begin{pmatrix} +x_1(t)\\ +x_2(t)\\ +\vdots\\ +x_n(t) +\end{pmatrix} +} +\] +\vspace{-15pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (1,0.5); +\coordinate (B) at (4,0.5); +\coordinate (C) at (2,2.2); +\coordinate (D) at (5,2); +\coordinate (E) at ($(C)+(80:2)$); + +\draw[color=red,line width=1.4pt] + (A) to[in=-160] (B) to[out=20,in=-100] (C) to[out=80] (D); +\fill[color=red] (C) circle[radius=0.06]; +\node[color=red] at (C) [left] {$\gamma(t)$}; + +\uncover<4->{ + \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); + \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; +} + +\uncover<2->{ + \draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; + \draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; +} +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Tangenten} +Ableitung +\[ +\frac{d}{dt}\gamma(t) += +\dot{\gamma}(t) += +\begin{pmatrix} +\dot{x}_1(t)\\ +\dot{x}_2(t)\\ +\vdots\\ +\dot{x}_n(t) +\end{pmatrix} +\] +\uncover<5->{% +Lineare Approximation: +\[ +\gamma(t+h) += +\gamma(t) ++ +\dot{\gamma}(t) \cdot h ++ +o(h) +\]}% +\vspace{-10pt} +\begin{itemize} +\item<6-> +Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ +in $\mathbb{R}^n$ liegen +\item<7-> +Gilt auch für +\[ +\operatorname{GL}_n(\mathbb{R}) +\uncover<8->{\subset M_n(\mathbb{R})} +\uncover<9->{ = \mathbb{R}^{n\times n}} +\] +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 574467b..59c9121 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -1,85 +1,85 @@ -% -% liealgebra.tex -- Lie-Algebra -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Lie-Algebra} -\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}} -\begin{block}{Vektorraum} -Tangentialvektoren im Punkt $I$: -\begin{center} -\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} -\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ -\hline -\uncover<2->{ -\operatorname{GL}_n(\mathbb{R}) -& beliebige Matrizen -& M_n(\mathbb{R}) -} -\\ -\uncover<3->{ -\operatorname{O(n)} -& antisymmetrische Matrizen -& \operatorname{o}(n) -} -\\ -\uncover<4->{ -\operatorname{SL}_n(\mathbb{R}) -& spurlose Matrizen -& \operatorname{sl}_2(\mathbb{R}) -} -\\ -\uncover<5->{ -\operatorname{U(n)} -& antihermitesche Matrizen -& \operatorname{u}(n) -} -\\ -\uncover<6->{ -\operatorname{SU(n)} -& spurlose, antihermitesche Matrizen -& \operatorname{su}(n) -} -\end{tabular} -\end{center} -\end{block} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.40\textwidth} -\uncover<7->{% -\begin{block}{Lie-Klammer} -Kommutator: $[A,B] = AB-BA$ -\end{block}} -\uncover<8->{% -\begin{block}{Nachprüfen} -$[A,B]\in LG$ -für $A,B\in LG$ -\end{block}} -\end{column} -\begin{column}{0.56\textwidth} -\uncover<9->{% -\begin{block}{Algebraische Eigenschaften} -\begin{itemize} -\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ -\item<11-> Jacobi-Identität -\[ -[A,[B,C]]+ -[B,[C,A]]+ -[C,[A,B]] -= 0 -\] -\end{itemize} -\vspace{-13pt} -\uncover<12->{% -{\usebeamercolor[fg]{title} -Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ -} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% liealgebra.tex -- Lie-Algebra +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra} +\ifthenelse{\boolean{presentation}}{\vspace{-15pt}}{\vspace{-8pt}} +\begin{block}{Vektorraum} +Tangentialvektoren im Punkt $I$: +\begin{center} +\begin{tabular}{>{$}c<{$}|p{6cm}|>{$}c<{$}} +\text{Lie-Gruppe $G$}&Tangentialvektoren&\text{Lie-Algebra $LG$} \\ +\hline +\uncover<2->{ +\operatorname{GL}_n(\mathbb{R}) +& beliebige Matrizen +& M_n(\mathbb{R}) +} +\\ +\uncover<3->{ +\operatorname{O(n)} +& antisymmetrische Matrizen +& \operatorname{o}(n) +} +\\ +\uncover<4->{ +\operatorname{SL}_n(\mathbb{R}) +& spurlose Matrizen +& \operatorname{sl}_2(\mathbb{R}) +} +\\ +\uncover<5->{ +\operatorname{U(n)} +& antihermitesche Matrizen +& \operatorname{u}(n) +} +\\ +\uncover<6->{ +\operatorname{SU(n)} +& spurlose, antihermitesche Matrizen +& \operatorname{su}(n) +} +\end{tabular} +\end{center} +\end{block} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\uncover<7->{% +\begin{block}{Lie-Klammer} +Kommutator: $[A,B] = AB-BA$ +\end{block}} +\uncover<8->{% +\begin{block}{Nachprüfen} +$[A,B]\in LG$ +für $A,B\in LG$ +\end{block}} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<9->{% +\begin{block}{Algebraische Eigenschaften} +\begin{itemize} +\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ +\item<11-> Jacobi-Identität +\[ +[A,[B,C]]+ +[B,[C,A]]+ +[C,[A,B]] += 0 +\] +\end{itemize} +\vspace{-13pt} +\uncover<12->{% +{\usebeamercolor[fg]{title} +Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex index 077dc9d..f88042a 100644 --- a/vorlesungen/slides/7/mannigfaltigkeit.tex +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -1,46 +1,46 @@ -% -% mannigfaltigkeit.tex -- Mannigfaltigkeit -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Mannigfaltigkeit} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{center} -\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf} -\end{center} -\end{column} -\begin{column}{0.48\textwidth} -\begin{block}{Definition} -\begin{itemize} -\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ -\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet -\[ -\varphi_\beta\circ\varphi_\alpha^{-1} -\colon -\varphi_\alpha(U_\alpha\cap U_\beta) -\to -\varphi_\beta(U_\alpha\cap U_\beta) -\] -\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken -\end{itemize} -\end{block} -\vspace{-7pt} -\uncover<5->{% -\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$} -Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus -\end{block}} -\vspace{-3pt} -\uncover<6->{% -\begin{block}{Lie-Gruppe} -Gruppe und Mannigfaltigkeit -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% mannigfaltigkeit.tex -- Mannigfaltigkeit +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Mannigfaltigkeit} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\includegraphics[width=\textwidth]{../../buch/chapters/60-gruppen/images/karten.pdf} +\end{center} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Definition} +\begin{itemize} +\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ +\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet +\[ +\varphi_\beta\circ\varphi_\alpha^{-1} +\colon +\varphi_\alpha(U_\alpha\cap U_\beta) +\to +\varphi_\beta(U_\alpha\cap U_\beta) +\] +\item<4-> Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken +\end{itemize} +\end{block} +\vspace{-7pt} +\uncover<5->{% +\begin{block}{Lokal$\mathstrut\cong\mathbb{R}^n$} +Differenzierbare Mannigfaltigkeiten sehen lokal wie $\mathbb{R}^n$ aus +\end{block}} +\vspace{-3pt} +\uncover<6->{% +\begin{block}{Lie-Gruppe} +Gruppe und Mannigfaltigkeit +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index 52c8e4a..afc67c5 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -1,107 +1,107 @@ -% -% parameter.tex -- Parametrisierung der Matrizen -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\definecolor{darkgreen}{rgb}{0,0.6,0} -\definecolor{darkyellow}{rgb}{1,0.8,0} -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Drehungen Parametrisieren} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.4\textwidth} -\begin{block}{Drehung um Achsen} -\vspace{-12pt} -\begin{align*} -\uncover<2->{ -D_{x,\alpha} -&= -\begin{pmatrix} -1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha -\end{pmatrix} -} -\\ -\uncover<3->{ -D_{y,\beta} -&= -\begin{pmatrix} -\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta -\end{pmatrix} -} -\\ -\uncover<4->{ -D_{z,\gamma} -&= -\begin{pmatrix} -\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 -\end{pmatrix} -} -\intertext{\uncover<5->{beliebige Drehung:}} -\uncover<5->{ -D -&= -D_{x,\alpha} -D_{y,\beta} -D_{z,\gamma} -} -\end{align*} -\end{block} -\end{column} -\begin{column}{0.56\textwidth} -\uncover<6->{% -\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional} -\uncover<7->{% -$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel -} -\\ -\uncover<8->{% -$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse -} -\[ -\uncover<9->{ -{\color{red}\vec{x}} -\mapsto -} -\uncover<10->{ -({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) -\cos\omega -+ -} -\uncover<11->{ -({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega -+ -} -\uncover<9->{ -{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) -} -\] -\vspace{-40pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\uncover<9->{ - \node at (0,0) - {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; - \node[color=red] at (1.6,-0.9) {$\vec{x}$}; - \node[color=blue] at (0.5,2) {$\vec{k}$}; -} -\uncover<11->{ - \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; -} -\uncover<10->{ - \node[color=yellow] at (2.2,-0.2) - {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; -} -\end{tikzpicture} -\end{center} -\end{block}} -\end{column} -\end{columns} -\vspace{-15pt} -\uncover<5->{% -{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine -dreidimensionale Gruppe} -\end{frame} -\egroup +% +% parameter.tex -- Parametrisierung der Matrizen +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkyellow}{rgb}{1,0.8,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen Parametrisieren} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.4\textwidth} +\begin{block}{Drehung um Achsen} +\vspace{-12pt} +\begin{align*} +\uncover<2->{ +D_{x,\alpha} +&= +\begin{pmatrix} +1&0&0\\0&\cos\alpha&-\sin\alpha\\0&\sin\alpha&\cos\alpha +\end{pmatrix} +} +\\ +\uncover<3->{ +D_{y,\beta} +&= +\begin{pmatrix} +\cos\beta&0&\sin\beta\\0&1&0\\-\sin\beta&0&\cos\beta +\end{pmatrix} +} +\\ +\uncover<4->{ +D_{z,\gamma} +&= +\begin{pmatrix} +\cos\gamma&-\sin\gamma&0\\\sin\gamma&\cos\gamma&0\\0&0&1 +\end{pmatrix} +} +\intertext{\uncover<5->{beliebige Drehung:}} +\uncover<5->{ +D +&= +D_{x,\alpha} +D_{y,\beta} +D_{z,\gamma} +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.56\textwidth} +\uncover<6->{% +\begin{block}{Drehung um $\vec{\omega}\in\mathbb{R}^3$: 3-dimensional} +\uncover<7->{% +$\omega=|\vec{\omega}|=\mathstrut$Drehwinkel +} +\\ +\uncover<8->{% +$\vec{k}=\vec{\omega}^0=\mathstrut$Drehachse +} +\[ +\uncover<9->{ +{\color{red}\vec{x}} +\mapsto +} +\uncover<10->{ +({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) +\cos\omega ++ +} +\uncover<11->{ +({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega ++ +} +\uncover<9->{ +{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) +} +\] +\vspace{-40pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \node at (0,0) + {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; + \node[color=red] at (1.6,-0.9) {$\vec{x}$}; + \node[color=blue] at (0.5,2) {$\vec{k}$}; +} +\uncover<11->{ + \node[color=darkgreen] at (-3,1.1) {$\vec{x}\times\vec{k}$}; +} +\uncover<10->{ + \node[color=yellow] at (2.2,-0.2) + {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +} +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\vspace{-15pt} +\uncover<5->{% +{\usebeamercolor[fg]{title}Dimension:} $\operatorname{SO}(3)$ ist eine +dreidimensionale Gruppe} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 66b8d27..d74b7d0 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -1,117 +1,117 @@ -% -% semi.tex -- Beispiele: semidirekte Produkte -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Drehung/Skalierung und Verschiebung} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Skalierung und Verschiebung} -Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$ -\\ -Wirkung auf $\mathbb{R}$: -\[ -x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t -\] -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<2->{% -\begin{block}{Drehung und Verschiebung} -Gruppe -$G= -\{ (\alpha,\vec{t}) -\;|\; -\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2 -\}$ -Wirkung auf $\mathbb{R}^2$: -\[ -\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} -\] -\end{block}} -\end{column} -\end{columns} -\vspace{-15pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<3->{% -\begin{block}{Verknüpfung} -\vspace{-15pt} -\begin{align*} -(e^{s_1},t_1)(e^{s_2},t_2)x -&\uncover<4->{= -(e^{s_1},t_1)(e^{s_2}x+t_2)} -\\ -&\uncover<5->{= -e^{s_1+s_2}x + e^{s_1}t_2+t_1} -\\ -\uncover<6->{ -(e^{s_1},t_1)(e^{s_2},t_2) -&= -(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} -\end{align*} -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<7->{% -\begin{block}{Verknüpfung} -\vspace{-15pt} -\begin{align*} -(\alpha_1,\vec{t}_1) -(\alpha_2,\vec{t}_2) -\vec{x} -&\uncover<8->{= -(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} -\\ -&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1} -\\ -\uncover<10->{ -(\alpha_1,\vec{t}_1) -(\alpha_2,\vec{t}_2) -&= -(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) -} -\end{align*} -\end{block}} -\end{column} -\end{columns} -\vspace{-10pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\uncover<11->{% -\begin{block}{Matrixschreibweise} -\vspace{-12pt} -\[ -g=(e^s,t) = -\begin{pmatrix} -e^s&t\\ -0&1 -\end{pmatrix} -\quad\text{auf}\quad -\begin{pmatrix}x\\1\end{pmatrix} -\] -\end{block}} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<12->{% -\begin{block}{Matrixschreibweise} -\vspace{-12pt} -\[ -g=(\alpha,\vec{t}) = -\begin{pmatrix} -D_{\alpha}&\vec{t}\\ -0&1 -\end{pmatrix} -\quad\text{auf}\quad -\begin{pmatrix}\vec{x}\\1\end{pmatrix} -\] -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% semi.tex -- Beispiele: semidirekte Produkte +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehung/Skalierung und Verschiebung} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Skalierung und Verschiebung} +Gruppe $G=\{(e^s,t)\;|\;s,t\in\mathbb{R}\}$ +\\ +Wirkung auf $\mathbb{R}$: +\[ +x\mapsto \underbrace{e^s\cdot x}_{\text{Skalierung}} \mathstrut+ t +\] +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Drehung und Verschiebung} +Gruppe +$G= +\{ (\alpha,\vec{t}) +\;|\; +\alpha\in\mathbb{R},\vec{t}\in\mathbb{R}^2 +\}$ +Wirkung auf $\mathbb{R}^2$: +\[ +\vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} +\] +\end{block}} +\end{column} +\end{columns} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(e^{s_1},t_1)(e^{s_2},t_2)x +&\uncover<4->{= +(e^{s_1},t_1)(e^{s_2}x+t_2)} +\\ +&\uncover<5->{= +e^{s_1+s_2}x + e^{s_1}t_2+t_1} +\\ +\uncover<6->{ +(e^{s_1},t_1)(e^{s_2},t_2) +&= +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<7->{% +\begin{block}{Verknüpfung} +\vspace{-15pt} +\begin{align*} +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +\vec{x} +&\uncover<8->{= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} +\\ +&\uncover<9->{=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1} +\\ +\uncover<10->{ +(\alpha_1,\vec{t}_1) +(\alpha_2,\vec{t}_2) +&= +(\alpha_1+\alpha_2, D_{\alpha_1}\vec{t}_2+\vec{t}_1) +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\vspace{-10pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Matrixschreibweise} +\vspace{-12pt} +\[ +g=(e^s,t) = +\begin{pmatrix} +e^s&t\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}x\\1\end{pmatrix} +\] +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<12->{% +\begin{block}{Matrixschreibweise} +\vspace{-12pt} +\[ +g=(\alpha,\vec{t}) = +\begin{pmatrix} +D_{\alpha}&\vec{t}\\ +0&1 +\end{pmatrix} +\quad\text{auf}\quad +\begin{pmatrix}\vec{x}\\1\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex index a65b4f6..58e87a1 100644 --- a/vorlesungen/slides/7/sl2.tex +++ b/vorlesungen/slides/7/sl2.tex @@ -1,242 +1,242 @@ -% -% sl2.tex -- Beispiel: Parametrisierung von SL_2(R) -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\begin{frame}[t,fragile] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.44\textwidth} -\begin{block}{Determinante} -\[ -A=\begin{pmatrix} -a&b\\ -c&d -\end{pmatrix} -\;\Rightarrow\; -\det A = ad-bc -\] -\end{block} -\end{column} -\begin{column}{0.52\textwidth} -\begin{block}{Dimension} -\[ -4\; \text{Variablen} -- -1\; \text{Bedingung} -= -3\; \text{Dimensionen} -\] -\end{block} -\end{column} -\end{columns} -\vspace{-10pt} -\uncover<3->{% -\begin{columns}[t,onlytextwidth] -\def\s{0.94} -\begin{column}{0.33\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick,scale=\s] -\begin{scope} - \clip (-2.1,-2.1) rectangle (2.3,2.3); - \fill[color=blue!20] (-1,-1) rectangle (1,1); - \foreach \x in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); - } - \foreach \y in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); - } - \ifthenelse{\boolean{presentation}}{ - \foreach \d in {4,...,10}{ - \only<\d>{ - \pgfmathparse{1+(\d-4)/10} - \xdef\t{\pgfmathresult} - \fill[color=red!40,opacity=0.5] - ({-\t},{-1/\t}) rectangle (\t,{1/\t}); - \foreach \x in {-2,...,2}{ - \draw[color=red,line width=0.3pt] - ({\x*\t},-3) -- ({\x*\t},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{\y/\t}) -- (3,{\y/\t}); - } - } - } - }{} - \uncover<11->{ - \xdef\t{1.6} - \fill[color=red!40,opacity=0.5] - ({-\t},{-1/\t}) rectangle (\t,{1/\t}); - \foreach \x in {-2,...,2}{ - \draw[color=red,line width=0.3pt] - ({\x*\t},-3) -- ({\x*\t},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{\y/\t}) -- (3,{\y/\t}); - } - } -\end{scope} -\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; -\uncover<3->{% - \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); - \node at (0,-2.1) {$ - D - = - \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} - $}; -} -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.33\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick,scale=\s] -\fill[color=blue!20] (-1,-1) rectangle (1,1); -\begin{scope} - \clip (-2.1,-2.1) rectangle (2.3,2.3); - \foreach \x in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); - } - \foreach \y in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); - } - \ifthenelse{\boolean{presentation}}{ - \foreach \d in {11,...,17}{ - \only<\d>{ - \pgfmathparse{(\d-11)/10} - \xdef\t{\pgfmathresult} - \fill[color=red!40,opacity=0.5] - ({-1+\t*(-1)},{-1}) - -- - ({1+\t*(-1)},{-1}) - -- - ({1+\t},{1}) - -- - ({-1+\t},{1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({-3+\t*\y},\y) -- ({3+\t*\y},\y); - } - } - } - }{} - \uncover<18->{ - \xdef\t{0.6} - \fill[color=red!40,opacity=0.5] - ({-1+\t*(-1)},{-1}) - -- - ({1+\t*(-1)},{-1}) - -- - ({1+\t},{1}) - -- - ({-1+\t},{1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - ({-3+\t*\y},\y) -- ({3+\t*\y},\y); - } - } -\end{scope} -\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; -\uncover<11->{ - \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); - \node at (0,-2.1) {$ - S - = - \begin{pmatrix} 1&s\\ 0&1\end{pmatrix} - $}; -} -\end{tikzpicture} -\end{center} -\end{column} -\begin{column}{0.33\textwidth} -\begin{center} -\begin{tikzpicture}[>=latex,thick,scale=\s] -\fill[color=blue!20] (-1,-1) rectangle (1,1); -\begin{scope} - \clip (-2.1,-2.1) rectangle (2.3,2.3); - \foreach \x in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); - } - \foreach \y in {-2,...,2}{ - \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); - } - \ifthenelse{\boolean{presentation}}{ - \foreach \d in {18,...,24}{ - \only<\d>{ - \pgfmathparse{(\d-18)/10} - \xdef\t{\pgfmathresult} - \fill[color=red!40,opacity=0.5] - (-1,{\t*(-1)-1}) - -- - (1,{\t*1-1}) - -- - (1,{\t*1+1}) - -- - (-1,{\t*(-1)+1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); - } - } - } - }{} - \uncover<25->{ - \xdef\t{0.6} - \fill[color=red!40,opacity=0.5] - (-1,{\t*(-1)-1}) - -- - (1,{\t*1-1}) - -- - (1,{\t*1+1}) - -- - (-1,{\t*(-1)+1}) - -- cycle; - \foreach \x in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); - } - \foreach \y in {-3,...,3}{ - \draw[color=red,line width=0.3pt] - (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); - } - } -\end{scope} -\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; -\uncover<18->{% -\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); - \node at (0,-2.1) {$ - T - = - \begin{pmatrix} 1&0\\t&1\end{pmatrix} - $}; -} -\end{tikzpicture} -\end{center} -\end{column} -\end{columns}} -\end{frame} -\egroup +% +% sl2.tex -- Beispiel: Parametrisierung von SL_2(R) +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t,fragile] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$\operatorname{SL}_2(\mathbb{R})\subset\operatorname{GL}_n(\mathbb{R})$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.44\textwidth} +\begin{block}{Determinante} +\[ +A=\begin{pmatrix} +a&b\\ +c&d +\end{pmatrix} +\;\Rightarrow\; +\det A = ad-bc +\] +\end{block} +\end{column} +\begin{column}{0.52\textwidth} +\begin{block}{Dimension} +\[ +4\; \text{Variablen} +- +1\; \text{Bedingung} += +3\; \text{Dimensionen} +\] +\end{block} +\end{column} +\end{columns} +\vspace{-10pt} +\uncover<3->{% +\begin{columns}[t,onlytextwidth] +\def\s{0.94} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \fill[color=blue!20] (-1,-1) rectangle (1,1); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \ifthenelse{\boolean{presentation}}{ + \foreach \d in {4,...,10}{ + \only<\d>{ + \pgfmathparse{1+(\d-4)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + ({-\t},{-1/\t}) rectangle (\t,{1/\t}); + \foreach \x in {-2,...,2}{ + \draw[color=red,line width=0.3pt] + ({\x*\t},-3) -- ({\x*\t},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{\y/\t}) -- (3,{\y/\t}); + } + } + } + }{} + \uncover<11->{ + \xdef\t{1.6} + \fill[color=red!40,opacity=0.5] + ({-\t},{-1/\t}) rectangle (\t,{1/\t}); + \foreach \x in {-2,...,2}{ + \draw[color=red,line width=0.3pt] + ({\x*\t},-3) -- ({\x*\t},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{\y/\t}) -- (3,{\y/\t}); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<3->{% + \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + D + = + \begin{pmatrix} e^t & 0 \\ 0 & e^{-t} \end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\fill[color=blue!20] (-1,-1) rectangle (1,1); +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \ifthenelse{\boolean{presentation}}{ + \foreach \d in {11,...,17}{ + \only<\d>{ + \pgfmathparse{(\d-11)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + ({-1+\t*(-1)},{-1}) + -- + ({1+\t*(-1)},{-1}) + -- + ({1+\t},{1}) + -- + ({-1+\t},{1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({-3+\t*\y},\y) -- ({3+\t*\y},\y); + } + } + } + }{} + \uncover<18->{ + \xdef\t{0.6} + \fill[color=red!40,opacity=0.5] + ({-1+\t*(-1)},{-1}) + -- + ({1+\t*(-1)},{-1}) + -- + ({1+\t},{1}) + -- + ({-1+\t},{1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({\x+\t*(-3)},-3) -- ({\x+\t*(3)},3); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + ({-3+\t*\y},\y) -- ({3+\t*\y},\y); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<11->{ + \fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + S + = + \begin{pmatrix} 1&s\\ 0&1\end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\begin{column}{0.33\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick,scale=\s] +\fill[color=blue!20] (-1,-1) rectangle (1,1); +\begin{scope} + \clip (-2.1,-2.1) rectangle (2.3,2.3); + \foreach \x in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (\x,-3) -- (\x,3); + } + \foreach \y in {-2,...,2}{ + \draw[color=blue,line width=0.3pt] (-3,\y) -- (3,\y); + } + \ifthenelse{\boolean{presentation}}{ + \foreach \d in {18,...,24}{ + \only<\d>{ + \pgfmathparse{(\d-18)/10} + \xdef\t{\pgfmathresult} + \fill[color=red!40,opacity=0.5] + (-1,{\t*(-1)-1}) + -- + (1,{\t*1-1}) + -- + (1,{\t*1+1}) + -- + (-1,{\t*(-1)+1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); + } + } + } + }{} + \uncover<25->{ + \xdef\t{0.6} + \fill[color=red!40,opacity=0.5] + (-1,{\t*(-1)-1}) + -- + (1,{\t*1-1}) + -- + (1,{\t*1+1}) + -- + (-1,{\t*(-1)+1}) + -- cycle; + \foreach \x in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (\x,{\x*\t-3}) -- (\x,{\x*\t+3}); + } + \foreach \y in {-3,...,3}{ + \draw[color=red,line width=0.3pt] + (-3,{-3*\t+\y}) -- (3,{3*\t+\y}); + } + } +\end{scope} +\draw[->] (-2.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-2.1) -- (0,2.3) coordinate[label={right:$y$}]; +\uncover<18->{% +\fill[color=white,opacity=0.8] (-1.5,-2.8) rectangle (1.5,-1.3); + \node at (0,-2.1) {$ + T + = + \begin{pmatrix} 1&0\\t&1\end{pmatrix} + $}; +} +\end{tikzpicture} +\end{center} +\end{column} +\end{columns}} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex index 35d62d8..8931a24 100644 --- a/vorlesungen/slides/7/symmetrien.tex +++ b/vorlesungen/slides/7/symmetrien.tex @@ -1,145 +1,145 @@ -% -% symmetrien.tex -- Symmetrien -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\bgroup -\definecolor{darkgreen}{rgb}{0,0.6,0} -\begin{frame}[t] -\setlength{\abovedisplayskip}{5pt} -\setlength{\belowdisplayskip}{5pt} -\frametitle{Symmetrien} -\vspace{-20pt} -\begin{columns}[t,onlytextwidth] -\begin{column}{0.48\textwidth} -\begin{block}{Diskrete Symmetrien} -\begin{itemize} -\item<2-> -Ebenen-Spiegelung: -\[ -{\tiny -\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} -} -\mapsto -{\tiny -\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*} -} -\uncover<4->{\!,\; -\vec{x} -\mapsto -\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} -} -\] -\vspace{-10pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\a{10} -\def\b{50} -\def\r{2} -\coordinate (O) at (0,0); -\coordinate (A) at (\b:\r); -\coordinate (B) at ({180+2*\a-\b}:\r); -\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); -\coordinate (N) at (\a:2); -\coordinate (D) at (\a:{\r*cos(\b-\a)}); -\uncover<3->{ -\clip (-2.5,-0.45) rectangle (2.5,1.95); - - \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) - -- cycle; - \draw[->,color=darkgreen] (O) -- (N); - \node[color=darkgreen] at (N) [above] {$\vec{n}$}; - - - \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) - -- cycle; - \fill[color=red] (O) circle[radius=0.06]; - \draw[color=red] ({\a-90}:2) -- ({\a+90}:2); - \fill[color=blue] (C) circle[radius=0.06]; - \draw[color=blue,line width=0.1pt] (A) -- (D); - \node[color=darkgreen] at (D) [below,rotate=\a] - {$(\vec{n}\cdot\vec{x})\vec{n}$}; - \draw[color=blue,line width=0.5pt] (A)--(B); - - \node[color=blue] at (A) [above right] {$\vec{x}$}; - \node[color=blue] at (B) [above left] {$\vec{x}'$}; - - \node[color=red] at (O) [below left] {$O$}; - - \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A); - \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B); -} - -\end{tikzpicture} -\end{center} -\vspace{-5pt} -$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ -\item<5-> -Punkt-Spiegelung: -\[ -{\tiny -\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} -} -\mapsto -- -{\tiny -\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*} -} -\] -\end{itemize} -\end{block} -\end{column} -\begin{column}{0.48\textwidth} -\uncover<6->{% -\begin{block}{Kontinuierliche Symmetrien} -\begin{itemize} -\item<7-> Translation: -\( -\vec{x} \mapsto \vec{x} + \vec{t} -\) -\item<8-> Drehung: -\vspace{-3pt} -\begin{center} -\begin{tikzpicture}[>=latex,thick] -\def\a{25} -\def\r{1.3} -\coordinate (O) at (0,0); -\begin{scope} -\clip (-1.1,-0.1) rectangle (2.3,2.3); -\draw[color=red] (O) circle[radius=2]; -\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle; -\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; -\node at ({0.5*\a}:1) {$\alpha$}; -\node at ({90+0.5*\a}:1) {$\alpha$}; -\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2); -\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2); -\end{scope} -\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; -\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; -\end{tikzpicture} -\end{center} -\[ -\uncover<9->{% -\begin{pmatrix}x\\y\end{pmatrix} -\mapsto -\begin{pmatrix} -{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\ -{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} -\end{pmatrix} -\begin{pmatrix}x\\y\end{pmatrix} -} -\] -\end{itemize} -\end{block}} -\vspace{-10pt} -\uncover<10->{% -\begin{block}{Definition} -Längen/Winkel bleiben erhalten -\\ -\uncover<11->{% -$\Rightarrow$ $\exists$ Erhaltungsgrösse} -\end{block}} -\end{column} -\end{columns} -\end{frame} -\egroup +% +% symmetrien.tex -- Symmetrien +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Symmetrien} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Diskrete Symmetrien} +\begin{itemize} +\item<2-> +Ebenen-Spiegelung: +\[ +{\tiny +\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} +} +\mapsto +{\tiny +\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*} +} +\uncover<4->{\!,\; +\vec{x} +\mapsto +\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} +} +\] +\vspace{-10pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{10} +\def\b{50} +\def\r{2} +\coordinate (O) at (0,0); +\coordinate (A) at (\b:\r); +\coordinate (B) at ({180+2*\a-\b}:\r); +\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); +\coordinate (N) at (\a:2); +\coordinate (D) at (\a:{\r*cos(\b-\a)}); +\uncover<3->{ +\clip (-2.5,-0.45) rectangle (2.5,1.95); + + \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) + -- cycle; + \draw[->,color=darkgreen] (O) -- (N); + \node[color=darkgreen] at (N) [above] {$\vec{n}$}; + + + \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) + -- cycle; + \fill[color=red] (O) circle[radius=0.06]; + \draw[color=red] ({\a-90}:2) -- ({\a+90}:2); + \fill[color=blue] (C) circle[radius=0.06]; + \draw[color=blue,line width=0.1pt] (A) -- (D); + \node[color=darkgreen] at (D) [below,rotate=\a] + {$(\vec{n}\cdot\vec{x})\vec{n}$}; + \draw[color=blue,line width=0.5pt] (A)--(B); + + \node[color=blue] at (A) [above right] {$\vec{x}$}; + \node[color=blue] at (B) [above left] {$\vec{x}'$}; + + \node[color=red] at (O) [below left] {$O$}; + + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A); + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B); +} + +\end{tikzpicture} +\end{center} +\vspace{-5pt} +$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ +\item<5-> +Punkt-Spiegelung: +\[ +{\tiny +\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*} +} +\mapsto +- +{\tiny +\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*} +} +\] +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<6->{% +\begin{block}{Kontinuierliche Symmetrien} +\begin{itemize} +\item<7-> Translation: +\( +\vec{x} \mapsto \vec{x} + \vec{t} +\) +\item<8-> Drehung: +\vspace{-3pt} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\a{25} +\def\r{1.3} +\coordinate (O) at (0,0); +\begin{scope} +\clip (-1.1,-0.1) rectangle (2.3,2.3); +\draw[color=red] (O) circle[radius=2]; +\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle; +\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; +\node at ({0.5*\a}:1) {$\alpha$}; +\node at ({90+0.5*\a}:1) {$\alpha$}; +\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2); +\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2); +\end{scope} +\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; +\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; +\end{tikzpicture} +\end{center} +\[ +\uncover<9->{% +\begin{pmatrix}x\\y\end{pmatrix} +\mapsto +\begin{pmatrix} +{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\ +{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} +\end{pmatrix} +\begin{pmatrix}x\\y\end{pmatrix} +} +\] +\end{itemize} +\end{block}} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Definition} +Längen/Winkel bleiben erhalten +\\ +\uncover<11->{% +$\Rightarrow$ $\exists$ Erhaltungsgrösse} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/Makefile.inc b/vorlesungen/slides/Makefile.inc index e2271b8..6454463 100644 --- a/vorlesungen/slides/Makefile.inc +++ b/vorlesungen/slides/Makefile.inc @@ -1,18 +1,18 @@ -# -# Makefile.inc -- additional depencencies -# -# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil -# -include ../slides/0/Makefile.inc -include ../slides/1/Makefile.inc -include ../slides/2/Makefile.inc -include ../slides/3/Makefile.inc -include ../slides/4/Makefile.inc -include ../slides/5/Makefile.inc -include ../slides/7/Makefile.inc -include ../slides/8/Makefile.inc -include ../slides/9/Makefile.inc - -slides = \ - $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ - $(chapter5) $(chapter7) $(chapter8) $(chapter9) +# +# Makefile.inc -- additional depencencies +# +# (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +# +include ../slides/0/Makefile.inc +include ../slides/1/Makefile.inc +include ../slides/2/Makefile.inc +include ../slides/3/Makefile.inc +include ../slides/4/Makefile.inc +include ../slides/5/Makefile.inc +include ../slides/7/Makefile.inc +include ../slides/8/Makefile.inc +include ../slides/9/Makefile.inc + +slides = \ + $(chapter0) $(chapter1) $(chapter2) $(chapter3) $(chapter4) \ + $(chapter5) $(chapter7) $(chapter8) $(chapter9) diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex index 4673f76..6c102f2 100644 --- a/vorlesungen/slides/test.tex +++ b/vorlesungen/slides/test.tex @@ -1,39 +1,39 @@ -% -% test.tex collection of all slides -% -% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil -% - -\section{Matrizen-Gruppen} -% Was sind Symmetrien -%\folie{7/symmetrien.tex} -% Algebraische Bedingungen für Matrixgruppen -%\folie{7/algebraisch.tex} -% Parametrisierung, Beispiel SO(3) -%\folie{7/parameter.tex} -% Mannigfaltigkeiten -%\folie{7/mannigfaltigkeit.tex} -% Weitere Beispiele -% SL_2(R) -%\folie{7/sl2.tex} -\folie{7/drehung.tex} -%\folie{7/drehanim.tex} -% Semidirekte Produkte SO(2) x R^2, R^+ x R -%\folie{7/semi.tex} - -\section{Ableitungen} -% Kurven in einer Gruppe -%\folie{7/kurven.tex} -% Einparameter-Gruppen -%\folie{7/einparameter.tex} -% Ableitung einer Einparameter-Gruppe -%\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} - +% +% test.tex collection of all slides +% +% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{Matrizen-Gruppen} +% Was sind Symmetrien +%\folie{7/symmetrien.tex} +% Algebraische Bedingungen für Matrixgruppen +%\folie{7/algebraisch.tex} +% Parametrisierung, Beispiel SO(3) +%\folie{7/parameter.tex} +% Mannigfaltigkeiten +%\folie{7/mannigfaltigkeit.tex} +% Weitere Beispiele +% SL_2(R) +%\folie{7/sl2.tex} +\folie{7/drehung.tex} +%\folie{7/drehanim.tex} +% Semidirekte Produkte SO(2) x R^2, R^+ x R +%\folie{7/semi.tex} + +\section{Ableitungen} +% Kurven in einer Gruppe +%\folie{7/kurven.tex} +% Einparameter-Gruppen +%\folie{7/einparameter.tex} +% Ableitung einer Einparameter-Gruppe +%\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} + diff --git a/vorlesungen/stream/countdown.html b/vorlesungen/stream/countdown.html index 940e269..739b39d 100644 --- a/vorlesungen/stream/countdown.html +++ b/vorlesungen/stream/countdown.html @@ -1,40 +1,40 @@ - - - - - - -
- - - - - + + + + + + +
+ + + + + diff --git a/vorlesungen/stream/ende.html b/vorlesungen/stream/ende.html index ee25dcf..cfd9e99 100644 --- a/vorlesungen/stream/ende.html +++ b/vorlesungen/stream/ende.html @@ -1,30 +1,30 @@ - - - - - - -
-

-Vielen Dank für Ihren Besuch. -

-
 
-

-Fortsetzung der Seminar-Sitzung in -der BBB-Konferenz in Moodle. -

-
- - - + + + + + + +
+

+Vielen Dank für Ihren Besuch. +

+
 
+

+Fortsetzung der Seminar-Sitzung in +der BBB-Konferenz in Moodle. +

+
+ + + -- cgit v1.2.1