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Diffstat (limited to 'vorlesungen/slides/7')
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diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc new file mode 100644 index 0000000..7afeea1 --- /dev/null +++ b/vorlesungen/slides/7/Makefile.inc @@ -0,0 +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 + diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex new file mode 100644 index 0000000..12f9084 --- /dev/null +++ b/vorlesungen/slides/7/ableitung.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex new file mode 100644 index 0000000..31d209a --- /dev/null +++ b/vorlesungen/slides/7/algebraisch.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex new file mode 100644 index 0000000..079cf16 --- /dev/null +++ b/vorlesungen/slides/7/chapter.tex @@ -0,0 +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} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex new file mode 100644 index 0000000..4447bac --- /dev/null +++ b/vorlesungen/slides/7/dg.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/drehanim.tex b/vorlesungen/slides/7/drehanim.tex new file mode 100644 index 0000000..ac136f1 --- /dev/null +++ b/vorlesungen/slides/7/drehanim.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex new file mode 100644 index 0000000..2d7b317 --- /dev/null +++ b/vorlesungen/slides/7/drehung.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex new file mode 100644 index 0000000..5171085 --- /dev/null +++ b/vorlesungen/slides/7/einparameter.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile new file mode 100644 index 0000000..cc67c8a --- /dev/null +++ b/vorlesungen/slides/7/images/Makefile @@ -0,0 +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 + + diff --git a/vorlesungen/slides/7/images/c/c01.jpg b/vorlesungen/slides/7/images/c/c01.jpg Binary files differnew file mode 100644 index 0000000..b2dbdb2 --- /dev/null +++ b/vorlesungen/slides/7/images/c/c01.jpg diff --git a/vorlesungen/slides/7/images/c/c02.jpg b/vorlesungen/slides/7/images/c/c02.jpg Binary files differnew file mode 100644 index 0000000..9b45ba3 --- /dev/null +++ b/vorlesungen/slides/7/images/c/c02.jpg diff --git a/vorlesungen/slides/7/images/c/c03.jpg b/vorlesungen/slides/7/images/c/c03.jpg Binary files differnew file mode 100644 index 0000000..882be40 --- /dev/null +++ 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/dev/null +++ b/vorlesungen/slides/7/images/c/c59.jpg diff --git a/vorlesungen/slides/7/images/c/c60.jpg b/vorlesungen/slides/7/images/c/c60.jpg Binary files differnew file mode 100644 index 0000000..2bc77e7 --- /dev/null +++ b/vorlesungen/slides/7/images/c/c60.jpg diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc new file mode 100644 index 0000000..0e27c9a --- /dev/null +++ b/vorlesungen/slides/7/images/common.inc @@ -0,0 +1,70 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.025; +#declare O = <0, 0, 0>; +#declare at = 0.015; + +camera { + location <18, 15, -50> + look_at <0.0, 0.5, 0> + right 16/9 * x * imagescale + up y * imagescale +} + +light_source { + <-40, 30, -50> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.1 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end + +#declare l = 1.2; + +arrow(< -l, 0, 0 >, < l, 0, 0 >, at, White) +arrow(< 0, 0, -l >, < 0, 0, l >, at, White) +arrow(< 0, -l, 0 >, < 0, l, 0 >, at, White) + diff --git a/vorlesungen/slides/7/images/commutator.ini b/vorlesungen/slides/7/images/commutator.ini new file mode 100644 index 0000000..8c2211e --- /dev/null +++ b/vorlesungen/slides/7/images/commutator.ini @@ -0,0 +1,8 @@ +Input_File_Name=commutator.pov +Initial_Frame=1 +Final_Frame=60 +Initial_Clock=1 +Final_Clock=60 +Cyclic_Animation=off +Pause_when_Done=off + diff --git a/vorlesungen/slides/7/images/commutator.m b/vorlesungen/slides/7/images/commutator.m new file mode 100644 index 0000000..5a448db --- /dev/null +++ b/vorlesungen/slides/7/images/commutator.m @@ -0,0 +1,111 @@ +# +# commutator.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# + +X = [ + 0, 0, 0; + 0, 0, -1; + 0, 1, 0 +]; + +Y = [ + 0, 0, 1; + 0, 0, 0; + -1, 0, 0 +]; + +Z = [ + 0, -1, 0; + 1, 0, 0; + 0, 0, 0 +]; + +function retval = Dx(alpha) + retval = [ + 1, 0, 0 ; + 0, cos(alpha), -sin(alpha); + 0, sin(alpha), cos(alpha) + ]; +end + +function retval = Dy(beta) + retval = [ + cos(beta), 0, sin(beta); + 0, 1, 0 ; + -sin(beta), 0, cos(beta) + ]; +end + +t = 0.9; +P = Dx(t) * Dy(t) +Q = Dy(t) * Dx(t) +P - Q +(P - Q) * [0;0;1] + +function retval = kurven(filename, t) + retval = -1; + N = 20; + fn = fopen(filename, "w"); + fprintf(fn, "//\n"); + fprintf(fn, "// %s\n", filename); + fprintf(fn, "//\n"); + fprintf(fn, "#macro XYkurve()\n"); + for i = (0:N-1) + v1 = Dx(t * i / N) * [0;0;1]; + v2 = Dx(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + for i = (0:N-1) + v1 = Dx(t) * Dy(t * i / N) * [0;0;1]; + v2 = Dx(t) * Dy(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#end\n"); + fprintf(fn, "#declare finalXY = <%.4f, %.4f, %.4f>;\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#macro YXkurve()\n"); + for i = (0:N-1) + v1 = Dy(t * i / N) * [0;0;1]; + v2 = Dy(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + for i = (0:N-1) + v1 = Dy(t) * Dx(t * i / N) * [0;0;1]; + v2 = Dy(t) * Dx(t * (i+1) / N) * [0;0;1]; + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1)); + fprintf(fn, "cylinder { <%.4f,%.4f,%.4f>, <%.4f, %.4f, %.4f>, at }\n", + v1(1,1), v1(3,1), v1(2,1), v2(1,1), v2(3,1), v2(2,1)); + end + fprintf(fn, "sphere { <%.4f,%.4f,%.4f>, at }\n", + v2(1,1), v2(3,1), v2(2,1)); + fprintf(fn, "#end\n"); + fprintf(fn, "#declare finalYX = <%.4f, %.4f, %.4f>;\n", + v2(1,1), v2(3,1), v2(2,1)); + + fclose(fn); + retval = 0; +end + +function retval = kurve(i) + n = pi / 180; + filename = sprintf("f/%04d.inc", i); + kurven(filename, n * i); +end + +for i = (1:60) + kurve(i); +end diff --git a/vorlesungen/slides/7/images/commutator.pov b/vorlesungen/slides/7/images/commutator.pov new file mode 100644 index 0000000..9ae11b9 --- /dev/null +++ b/vorlesungen/slides/7/images/commutator.pov @@ -0,0 +1,59 @@ +// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "common.inc" + +sphere { O, 0.99 + pigment { + color rgbt<1,1,1,0.5> + } + finish { + specular 0.9 + metallic + } +} + +#declare filename = concat("f/", str(clock, -4, 0), ".inc"); + +#include filename + +#declare n1 = vcross(<0,1,0>, finalXY); +#declare n2 = vcross(<0,1,0>, finalYX); + +intersection { + sphere { O, 1 } + plane { -n1, 0 } + plane { n2, 0 } + pigment { + color rgb<0,0.4,0.1> + } + finish { + specular 0.9 + metallic + } +} + +union { + XYkurve() + pigment { + color Red + } + finish { + specular 0.9 + metallic + } +} + +union { + YXkurve() + pigment { + color Blue + } + finish { + specular 0.9 + metallic + } +} + diff --git a/vorlesungen/slides/7/images/rodriguez.jpg b/vorlesungen/slides/7/images/rodriguez.jpg Binary files differnew file mode 100644 index 0000000..5c49700 --- /dev/null +++ b/vorlesungen/slides/7/images/rodriguez.jpg diff --git a/vorlesungen/slides/7/images/rodriguez.png b/vorlesungen/slides/7/images/rodriguez.png Binary files differnew file mode 100644 index 0000000..6d9e9e4 --- /dev/null +++ b/vorlesungen/slides/7/images/rodriguez.png diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov new file mode 100644 index 0000000..07aec19 --- /dev/null +++ b/vorlesungen/slides/7/images/rodriguez.pov @@ -0,0 +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) diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex new file mode 100644 index 0000000..84bf034 --- /dev/null +++ b/vorlesungen/slides/7/kommutator.tex @@ -0,0 +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}};} 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+\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 new file mode 100644 index 0000000..e0690eb --- /dev/null +++ b/vorlesungen/slides/7/kurven.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex new file mode 100644 index 0000000..574467b --- /dev/null +++ b/vorlesungen/slides/7/liealgebra.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex new file mode 100644 index 0000000..077dc9d --- /dev/null +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex new file mode 100644 index 0000000..52c8e4a --- /dev/null +++ b/vorlesungen/slides/7/parameter.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex new file mode 100644 index 0000000..66b8d27 --- /dev/null +++ b/vorlesungen/slides/7/semi.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/sl2.tex b/vorlesungen/slides/7/sl2.tex new file mode 100644 index 0000000..a65b4f6 --- /dev/null +++ b/vorlesungen/slides/7/sl2.tex @@ -0,0 +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 diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex new file mode 100644 index 0000000..35d62d8 --- /dev/null +++ b/vorlesungen/slides/7/symmetrien.tex @@ -0,0 +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 |
