diff options
Diffstat (limited to 'vorlesungen/slides/7')
40 files changed, 3528 insertions, 2093 deletions
diff --git a/vorlesungen/slides/7/Makefile.inc b/vorlesungen/slides/7/Makefile.inc index 2391099..ffd5091 100644 --- a/vorlesungen/slides/7/Makefile.inc +++ b/vorlesungen/slides/7/Makefile.inc @@ -1,22 +1,35 @@ -#
-# 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/liealgbeispiel.tex \ + ../slides/7/vektorlie.tex \ + ../slides/7/kommutator.tex \ + ../slides/7/bch.tex \ + ../slides/7/dg.tex \ + ../slides/7/interpolation.tex \ + ../slides/7/exponentialreihe.tex \ + ../slides/7/logarithmus.tex \ + ../slides/7/zusammenhang.tex \ + ../slides/7/quaternionen.tex \ + ../slides/7/qdreh.tex \ + ../slides/7/ueberlagerung.tex \ + ../slides/7/hopf.tex \ + ../slides/7/haar.tex \ + ../slides/7/integration.tex \ + ../slides/7/chapter.tex + diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex index 5a4b94e..12f9084 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 fba42cf..31d209a 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/bch.tex b/vorlesungen/slides/7/bch.tex new file mode 100644 index 0000000..0148dc4 --- /dev/null +++ b/vorlesungen/slides/7/bch.tex @@ -0,0 +1,76 @@ +% +% bch.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Baker-Campbell-Hausdorff-Formel} +$g(t),h(t)\in G +\uncover<2->{\Rightarrow \exists A,B\in LG\text{ mit } +g(t)=\exp At, h(t)=\exp Bt}$ +\uncover<3->{% +\begin{align*} +g(t) +&= +I + At + \frac{A^2t^2}{2!} + \frac{A^3t^3}{3!} + \dots, +& +h(t) +&= +I + Bt + \frac{B^2t^2}{2!} + \frac{B^3t^3}{3!} + \dots +\end{align*}} +\uncover<5->{% +\begin{block}{Kommutator in G: $c(t) = g(t)h(t)g(t)^{-1}h(t)^{-1}$} +\begin{align*} +\uncover<6->{c(t) +&= +\biggl( + {\color<7,9-11,13-15,19-21>{red}I} + + {\color<8,16-19>{red}A}t + + \frac{{\color<12>{red}A^2}t^2}{2!} + + \dots +\biggr) +\biggl( + {\color<7,8,10-12,14-15,17-18,21>{red}I} + + {\color<9,16,19-20>{red}B}t + + \frac{{\color<13>{red}B^2}t^2}{2!} + + \dots +\biggr) +\exp(-{\color<10,14,17,19,21>{red}A}t) +\exp(-{\color<11,15,18,20-21>{red}B}t) +} +\\ +&\uncover<7->{={\color<7>{red}I}} +\uncover<8->{+t( + \uncover<8->{ {\color<8>{red}A}} + \uncover<9->{+ {\color<9>{red}B}} + \uncover<10->{- {\color<10>{red}A}} + \uncover<11->{- {\color<11>{red}B}} +)} +\uncover<12->{+\frac{t^2}{2!}( + \uncover<12->{ {\color<12>{red}A^2}} + \uncover<13->{+ {\color<13>{red}B^2}} + \uncover<14->{+ {\color<14>{red}A^2}} + \uncover<15->{+ {\color<15>{red}B^2}} +)} +\\ +&\phantom{\mathstrut=I} +\uncover<12->{+t^2( + \uncover<16->{ {\color<16>{red}AB}} + \uncover<17->{- {\color<17>{red}A^2}} + \uncover<18->{- {\color<18>{red}AB}} + \uncover<19->{- {\color<19>{red}BA}} + \uncover<20->{- {\color<20>{red}B^2}} + \uncover<21->{+ {\color<21>{red}AB}} +)} +\uncover<22->{+t^3(\dots)+\dots} +\\ +&\uncover<23->{= +I + \frac{t^2}{2}[A,B] + o(t^3) +} +\end{align*}} +\end{block} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/chapter.tex b/vorlesungen/slides/7/chapter.tex index 0f14a9a..3736e0f 100644 --- a/vorlesungen/slides/7/chapter.tex +++ b/vorlesungen/slides/7/chapter.tex @@ -1,19 +1,32 @@ -%
-% 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/liealgbeispiel.tex} +\folie{7/vektorlie.tex} +\folie{7/kommutator.tex} +\folie{7/bch.tex} +\folie{7/dg.tex} +\folie{7/interpolation.tex} +\folie{7/exponentialreihe.tex} +\folie{7/logarithmus.tex} +\folie{7/zusammenhang.tex} +\folie{7/quaternionen.tex} +\folie{7/qdreh.tex} +\folie{7/ueberlagerung.tex} +\folie{7/hopf.tex} +\folie{7/haar.tex} +\folie{7/integration.tex} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 446b2ab..f9528a4 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 776617f..ac136f1 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 e7b4a92..02201d4 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 e9699a6..a32affd 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/exponentialreihe.tex b/vorlesungen/slides/7/exponentialreihe.tex new file mode 100644 index 0000000..b1aeda6 --- /dev/null +++ b/vorlesungen/slides/7/exponentialreihe.tex @@ -0,0 +1,24 @@ +% +% exponentialreihe.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Exponentialreihe} +\begin{align*} +h(s) &= \exp(tA_0 + sB) = \sum_{k=0}^\infty \frac{1}{k!} (tA_0 + sB)^k +\\ +&= +I + (tA_0 + sB) + \frac{1}{2!}(t^2A_0^2 + ts(A_0B + BA_0) + s^2B^2) ++ \frac{1}{3!}(t^3A_0^3 + t^2s(A_0^2B + A_0BA_0 + BA_0^2) + \dots) ++ \dots +\\ +\frac{dg(s)}{ds} +&= +B + \frac1{2!}t(A_0B+BA_0) + \frac{1}{3!}t^2(A_0^2B+A_0BA_0+BA_0^2) + \dots +\end{align*} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/haar.tex b/vorlesungen/slides/7/haar.tex new file mode 100644 index 0000000..454dd69 --- /dev/null +++ b/vorlesungen/slides/7/haar.tex @@ -0,0 +1,84 @@ +% +% haar.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Haar-Mass} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Invariantes Mass} +Auf jeder lokalkompakten Gruppe $G$ gibt es ein \only<2->{invariantes }% +Integral +\begin{align*} +\uncover<2->{\text{rechts:}}&& +\int_G f(g)\,d\mu(g) +&\uncover<2->{= +\int_G f(gh)\,d\mu(g)} +\\ +\uncover<3->{ +\text{links:}&& +\int_G f(g)\,d\mu(g) +&= +\int_G f(hg)\,d\mu(g)} +\end{align*} + +\end{block} +\uncover<7->{% +\begin{block}{Modulus-Funktion} +$\mu$ linksinvariant, dann ist die Rechtsverschiebung ebenfalls +linksinvariant +\[ +\int_G f(gh) \, d\mu(g) +\uncover<8->{ += +\int_G f(g) \Delta(h)\, d\mu(g) +} +\] +\uncover<9->{$\Delta(h)$ heisst Modulus-Funktion} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<4->{% +\begin{block}{Beispiel: $G=\mathbb{R}$} +\[ +\int_Gf(g)\,d\mu(g) += +\int_{-\infty}^{\infty} f(x)\,dx +\] +\end{block}} +\vspace{-10pt} +\uncover<5->{% +\begin{block}{Beispiel: $\operatorname{SO}(2)$} +\[ +\int_{\operatorname{SO}(2)} +f(g)\,d\mu(g) += +\frac{1}{2\pi} +\int_{0}^{2\pi} f(D_{\alpha})\,d\alpha +\] +\end{block}} +\vspace{-10pt} +\uncover<6->{% +\begin{block}{Beispiel: $G$ endlich} +\[ +\int_G f(g)\,d\mu(g) = \frac{1}{|G|}\sum_{g\in G}f(g) +\] +\end{block}} +\vspace{-10pt} +\uncover<10->{% +\begin{block}{Unimodular} +$\Delta(h)=1$ heisst rechtsinvariant = linksinvariant +\\ +\uncover<11->{% +$G$ kompakt $\Rightarrow$ unimodular +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/hopf.tex b/vorlesungen/slides/7/hopf.tex new file mode 100644 index 0000000..a90737f --- /dev/null +++ b/vorlesungen/slides/7/hopf.tex @@ -0,0 +1,69 @@ +% +% hopf.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Orbit-Räume} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aktion von $\operatorname{SO}(3)$ auf $S^2$} +\begin{align*} +S^2 &= \{x\in\mathbb{R}^3\;|\; |x|=1\} +\\ +\operatorname{SO}(3) \times S^2 &\to S^2: (g, x) \mapsto gx +\end{align*} +\uncover<2->{% +Allgemein: Aktion von $G$ auf $X$ +\begin{align*} +\text{links:}&& +G\times X \to X &: (g,x) \mapsto gx +\\ +\text{rechts:}&& +X\times G \to X &: (x,g) \mapsto xg +\end{align*}} +\end{block} +\vspace{-10pt} +\uncover<3->{% +\begin{block}{Stabilisator} +Zu $x\in X$ gibt es eine Untergruppe +\begin{align*} +G_x = \{g\in G\;|\; gx=x\}, +\end{align*} +der {\em Stabilisator} von $x$. + +\uncover<4->{% +Der Stabilisator von $v\in S^2$ ist die Gruppe der Drehungen um +die Achse $v$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<5->{% +\begin{block}{Quotient} +$G$ operiert von rechts auf $X$ +\[ +X/G = \{ xG \;|\; x\in X\} +\] +heisst Quotient +\end{block}} +\uncover<6->{ +\begin{block}{$\operatorname{SO}(3)/\operatorname{SO}(2)$} +Wähle $\operatorname{SO}(2)$ als Drehungen um die $z$-Achse: +\[ +\operatorname{SO}(3) \to S^2 +: +g \mapsto \text{letzte Spalte von $g$} +\] +\uncover<7->{Daher +\[ +S^2 \cong \operatorname{SO}(3) / \operatorname{SO}(2) +\]} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/images/Makefile b/vorlesungen/slides/7/images/Makefile index 9de1c34..6f99bc3 100644 --- a/vorlesungen/slides/7/images/Makefile +++ b/vorlesungen/slides/7/images/Makefile @@ -1,19 +1,29 @@ -#
-# 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 test.png + +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 + +dreibein/timestamp: interpolation.m + octave interpolation.m + touch dreibein/timestamp + +test.png: test.pov drehung.inc dreibein/d025.inc dreibein/timestamp + povray +A0.1 -W1080 -H1080 -Otest.png test.pov + +dreibein/d025.inc: dreibein/timestamp + +animation: + povray +A0.1 -W1080 -H1080 -Ointerpolation/i.png interpolation.ini diff --git a/vorlesungen/slides/7/images/common.inc b/vorlesungen/slides/7/images/common.inc index b028956..0e27c9a 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 44a5ac5..8c2211e 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 3f5ea17..5a448db 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 8229a06..9ae11b9 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/drehung.inc b/vorlesungen/slides/7/images/drehung.inc new file mode 100644 index 0000000..c9b4bb7 --- /dev/null +++ b/vorlesungen/slides/7/images/drehung.inc @@ -0,0 +1,142 @@ +// +// common.inc +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#version 3.7; +#include "colors.inc" + +global_settings { + assumed_gamma 1 +} + +#declare imagescale = 0.23; +#declare O = <0, 0, 0>; +#declare at = 0.02; + +camera { + location <8.5, 2, 6.5> + look_at <0, 0, 0> + right x * imagescale + up y * imagescale +} + +//light_source { +// <-14, 20, -50> color White +// area_light <1,0,0> <0,0,1>, 10, 10 +// adaptive 1 +// jitter +//} + +light_source { + <41, 20, 10> color White + area_light <1,0,0> <0,0,1>, 10, 10 + adaptive 1 + jitter +} + +sky_sphere { + pigment { + color rgb<1,1,1> + } +} + +#macro arrow(from, to, arrowthickness, c) +#declare arrowdirection = vnormalize(to - from); +#declare arrowlength = vlength(to - from); +union { + sphere { + from, 1.0 * arrowthickness + } + cylinder { + from, + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + arrowthickness + } + cone { + from + (arrowlength - 5 * arrowthickness) * arrowdirection, + 2 * arrowthickness, + to, + 0 + } + pigment { + color c + } + finish { + specular 0.9 + metallic + } +} +#end +#declare r = 1.0; + +arrow(< -r-0.2, 0.0, 0 >, < r+0.2, 0.0, 0.0 >, at, Gray) +arrow(< 0.0, 0.0, -r-0.2>, < 0.0, 0.0, r+0.2 >, at, Gray) +arrow(< 0.0, -r-0.2, 0 >, < 0.0, r+0.2, 0.0 >, at, Gray) + +#declare farbeX = rgb<1.0,0.2,0.6>; +#declare farbeY = rgb<0.0,0.8,0.4>; +#declare farbeZ = rgb<0.4,0.6,1.0>; + +#declare farbex = rgb<1.0,0.0,0.0>; +#declare farbey = rgb<0.0,0.6,0.0>; +#declare farbez = rgb<0.0,0.0,1.0>; + +#macro quadrant(X, Y, Z) + intersection { + sphere { O, 0.5 } + plane { -X, 0 } + plane { -Y, 0 } + plane { -Z, 0 } + pigment { + color rgb<1.0,0.6,0.2> + } + finish { + specular 0.95 + metallic + } + } + arrow(O, X, 1.1*at, farbex) + arrow(O, Y, 1.1*at, farbey) + arrow(O, Z, 1.1*at, farbez) +#end + +#macro drehung(X, Y, Z) +// intersection { +// sphere { O, 0.5 } +// plane { -X, 0 } +// plane { -Y, 0 } +// plane { -Z, 0 } +// pigment { +// color Gray +// } +// finish { +// specular 0.95 +// metallic +// } +// } + arrow(O, 1.1*X, 0.9*at, farbeX) + arrow(O, 1.1*Y, 0.9*at, farbeY) + arrow(O, 1.1*Z, 0.9*at, farbeZ) +#end + +#macro achse(H) + cylinder { H, -H, at + pigment { + color rgb<0.6,0.4,0.2> + } + finish { + specular 0.95 + metallic + } + } + cylinder { 0.003 * H, -0.003 * H, 1 + pigment { + color rgbt<0.6,0.4,0.2,0.5> + } + finish { + specular 0.95 + metallic + } + } +#end diff --git a/vorlesungen/slides/7/images/interpolation.ini b/vorlesungen/slides/7/images/interpolation.ini new file mode 100644 index 0000000..f07c079 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.ini @@ -0,0 +1,8 @@ +Input_File_Name=interpolation.pov +Initial_Frame=0 +Final_Frame=50 +Initial_Clock=0 +Final_Clock=50 +Cyclic_Animation=off +Pause_when_Done=off + diff --git a/vorlesungen/slides/7/images/interpolation.m b/vorlesungen/slides/7/images/interpolation.m new file mode 100644 index 0000000..31554e8 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.m @@ -0,0 +1,54 @@ +# +# interpolation.m +# +# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global N; +N = 50; +global A; +global B; + +A = (pi / 2) * [ + 0, 0, 0; + 0, 0, -1; + 0, 1, 0 +]; +g0 = expm(A) + +B = (pi / 2) * [ + 0, 0, 1; + 0, 0, 0; + -1, 0, 0 +]; +g1 = expm(B) + +function retval = g(t) + global A; + global B; + retval = expm((1-t)*A+t*B); +endfunction + +function dreibein(fn, M, funktion) + fprintf(fn, "%s(<%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>, <%.4f,%.4f,%.4f>)\n", + funktion, + M(1,1), M(3,1), M(2,1), + M(1,2), M(3,2), M(2,2), + M(1,3), M(3,3), M(2,3)); +endfunction + +G = g1 * inverse(g0); +[V, lambda] = eig(G); +H = real(V(:,3)); + +D = logm(g1*inverse(g0)); + +for i = (0:N) + filename = sprintf("dreibein/d%03d.inc", i); + fn = fopen(filename, "w"); + t = i/N; + dreibein(fn, g(t), "quadrant"); + dreibein(fn, expm(t*D)*g0, "drehung"); + fprintf(fn, "achse(<%.4f,%.4f,%.4f>)\n", H(1,1), H(3,1), H(2,1)); + fclose(fn); +endfor + diff --git a/vorlesungen/slides/7/images/interpolation.pov b/vorlesungen/slides/7/images/interpolation.pov new file mode 100644 index 0000000..71e0257 --- /dev/null +++ b/vorlesungen/slides/7/images/interpolation.pov @@ -0,0 +1,10 @@ +// +// commutator.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "drehung.inc" + +#declare filename = concat("dreibein/d", str(clock, -3, 0), ".inc"); +#include filename + diff --git a/vorlesungen/slides/7/images/rodriguez.pov b/vorlesungen/slides/7/images/rodriguez.pov index 62306f8..07aec19 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/images/test.pov b/vorlesungen/slides/7/images/test.pov new file mode 100644 index 0000000..5707be1 --- /dev/null +++ b/vorlesungen/slides/7/images/test.pov @@ -0,0 +1,7 @@ +// +// test.pov +// +// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +// +#include "drehung.inc" +#include "dreibein/d025.inc" diff --git a/vorlesungen/slides/7/integration.tex b/vorlesungen/slides/7/integration.tex new file mode 100644 index 0000000..525e6de --- /dev/null +++ b/vorlesungen/slides/7/integration.tex @@ -0,0 +1,66 @@ +% +% integration.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Invariante Integration} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Koordinatenwechsel} +Die Koordinatentransformation +$f\colon\mathbb{R}^n\to\mathbb{R}^n:x\to y$ +hat die Ableitungsmatrix +\[ +t_{ij} += +\frac{\partial y_i}{\partial x_j} +\] +\uncover<2->{% +$n$-faches Integral +\begin{gather*} +\int\dots\int +h(f(x)) +\det +\biggl( +\frac{\partial y_i}{\partial x_j} +\biggr) +\,dx_1\,\dots dx_n +\\ += +\int\dots\int +h(y) +\,dy_1\,\dots dy_n +\end{gather*}} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{auf einer Lie-Gruppe} +Koordinatenwechsel sind Multiplikationen mit einer +Matrix $g\in G$ +\end{block}} +\uncover<4->{% +\begin{block}{Volumenelement in $I$} +Man muss nur das Volumenelement in $I$ in einem beliebigen +Koordinatensystem definieren: +\[ +dV = dy_1\,\dots\,dy_n +\] +\end{block}} +\uncover<5->{% +\begin{block}{Volumenelement in $g$} +\[ +\text{``\strut}g\cdot dV\text{\strut''} += +\det(g) \, dy_1\,\dots\,dy_n +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/interpolation.tex b/vorlesungen/slides/7/interpolation.tex new file mode 100644 index 0000000..249ee26 --- /dev/null +++ b/vorlesungen/slides/7/interpolation.tex @@ -0,0 +1,112 @@ +% +% interpolation.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\def\bild#1#2{\only<#1|handout:0>{\includegraphics[width=\textwidth]{../slides/7/images/interpolation/#2.png}}} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Interpolation} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Aufgabe} +Finde einen Weg $g(t)\in \operatorname{SO}(3)$ zwischen +$g_0\in\operatorname{SO}(3)$ +und +$g_1\in\operatorname{SO}(3)$: +\[ +g_0=g(0) +\quad\wedge\quad +g_1=g(1) +\] +\end{block} +\vspace{-10pt} +\uncover<2->{% +\begin{block}{Lösung} +$g_i=\exp(A_i) \uncover<3->{\Rightarrow A_i^t=-A_i}$ +\begin{align*} +\uncover<4->{A(t) &= (1-t)A_0 + tA_1}\uncover<8->{ \in \operatorname{so}(3)} +\\ +\uncover<5->{A(t)^t +&=(1-t)A_0^t + tA_1^t} +\\ +&\uncover<6->{= +-(1-t)A_0 - t A_1} +\uncover<7->{= +-A(t)} +\\ +\uncover<9->{\Rightarrow +g(t) &= \exp A(t) \in \operatorname{SO}(3)} +\\ +&\uncover<10->{\ne +\exp (\log(g_1g_0^{-1})t) g_0} +\end{align*} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Animation} +\centering +\ifthenelse{\boolean{presentation}}{ +\bild{12}{i00} +\bild{13}{i01} +\bild{14}{i02} +\bild{15}{i03} +\bild{16}{i04} +\bild{17}{i05} +\bild{18}{i06} +\bild{19}{i07} +\bild{20}{i08} +\bild{21}{i09} +\bild{22}{i10} +\bild{23}{i11} +\bild{24}{i12} +\bild{25}{i13} +\bild{26}{i14} +\bild{27}{i15} +\bild{28}{i16} +\bild{29}{i17} +\bild{30}{i18} +\bild{31}{i19} +\bild{32}{i20} +\bild{33}{i21} +\bild{34}{i22} +\bild{35}{i23} +\bild{36}{i24} +\bild{37}{i25} +\bild{38}{i26} +\bild{39}{i27} +\bild{40}{i28} +\bild{41}{i29} +\bild{42}{i30} +\bild{43}{i31} +\bild{44}{i32} +\bild{45}{i33} +\bild{46}{i34} +\bild{47}{i35} +\bild{48}{i36} +\bild{49}{i37} +\bild{50}{i38} +\bild{51}{i39} +\bild{52}{i40} +\bild{53}{i41} +\bild{54}{i42} +\bild{55}{i43} +\bild{56}{i44} +\bild{57}{i45} +\bild{58}{i46} +\bild{59}{i47} +\bild{60}{i48} +\bild{61}{i49} +\bild{62}{i50} +}{ +\includegraphics[width=\textwidth]{../slides/7/images/interpolation/i25.png} +} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index 9000160..84bf034 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}}{
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-\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) { 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+\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 bca8417..e0690eb 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/liealgbeispiel.tex b/vorlesungen/slides/7/liealgbeispiel.tex new file mode 100644 index 0000000..a17de40 --- /dev/null +++ b/vorlesungen/slides/7/liealgbeispiel.tex @@ -0,0 +1,78 @@ +% +% liealgbeispiel.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Lie-Algebra Beispiele} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{sl}_2(\mathbb{R})$} +Spurlose Matrizen: +\[ +\operatorname{sl}_2(\mathbb{R}) += +\{A\in M_n(\mathbb{R})\;|\; \operatorname{Spur}A=0\} +\] +\end{block} +\begin{block}{Lie-Algebra?} +Nachrechnen: $[A,B]\in \operatorname{sl}_2(\mathbb{R})$: +\begin{align*} +\operatorname{Spur}([A,B]) +&= +\operatorname{Spur}(AB-BA) +\\ +&= +\operatorname{Spur}(AB)-\operatorname{Spur}(BA) +\\ +&= +\operatorname{Spur}(AB)-\operatorname{Spur}(AB) +\\ +&=0 +\end{align*} +$\Rightarrow$ $\operatorname{sl}_2(\mathbb{R})$ ist eine Lie-Algebra +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{$\operatorname{so}(n)$} +Antisymmetrische Matrizen: +\[ +\operatorname{so}(n) += +\{A\in M_n(\mathbb{R}) +\;|\; +A=-A^t +\} +\] +\end{block} +\begin{block}{Lie-Algebra?} +Nachrechnen: $A,B\in \operatorname{so}(n)$ +\begin{align*} +[A,B]^t +&= +(AB-BA)^t +\\ +&= +B^tA^t - A^tB^t +\\ +&= +(-B)(-A)-(-A)(-B) +\\ +&= +BA-AB += +-(AB-BA) +\\ +&= +-[A,B] +\end{align*} +$\Rightarrow$ $\operatorname{so}(n)$ ist eine Lie-Algebra +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 59c9121..574467b 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/logarithmus.tex b/vorlesungen/slides/7/logarithmus.tex new file mode 100644 index 0000000..58065d7 --- /dev/null +++ b/vorlesungen/slides/7/logarithmus.tex @@ -0,0 +1,82 @@ +% +% logarithmus.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmus} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Taylor-Reihe} +\begin{align*} +\frac{d}{dx}\log(1+x) +&= \frac{1}{1+x} +\\ +\uncover<2->{ +\Rightarrow\quad +\log (1+x) +&= +\int_0^x \frac{1}{1+t}\,dt} +\end{align*} +\begin{align*} +\uncover<3->{\frac{1}{1+t} +&= +1-t+t^2-t^3+\dots} +\\ +\uncover<4->{\log(1+x) +&=\int_0^x +1-t+t^2-t^3+\dots +\,dt} +\\ +&\only<5>{= +x-\frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}4 + \dots} +\uncover<6->{= +\sum_{k=1}^\infty (-1)^{k-1}\frac{x^k}{k}} +\\ +\uncover<7->{\log (I+A) +&= +\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}A^k} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<8->{% +\begin{block}{Konvergenzradius} +Polstelle bei $x=-1$ +\( +\varrho =1 +\) +\end{block}} +\vspace{-5pt} +\begin{block}{\uncover<9->{Alternative: Spektraltheorie}} +\uncover<9->{ +Logarithmus $\log z$ in $\{z\in\mathbb{C}\;|\; \neg(\Re z\le 0\wedge\Im z=0)\}$ +definiert:} +\vspace{-15pt} +\uncover<8->{ +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\uncover<9->{ + \fill[color=red!20] (-2.1,-2.1) rectangle (2.5,2.1); +} +\draw[->] (-2.2,0) -- (2.9,0) coordinate[label={$\Re z$}]; +\draw[->] (0,-2.2) -- (0,2.4) coordinate[label={right:$\Im z$}]; +\fill[color=blue!40,opacity=0.5] (1,0) circle[radius=1]; +\draw[color=blue] (1,0) circle[radius=1]; +\uncover<9->{ + \draw[color=white,line width=5pt] (-2.2,0) -- (0.1,0); +} +\fill (1,0) circle[radius=0.08]; +\node at (2.3,1.9) {$\mathbb{C}$}; +\node at (1,0) [below] {$1$}; +\end{tikzpicture} +\end{center}} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex index f88042a..077dc9d 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 afc67c5..f3579a3 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/qdreh.tex b/vorlesungen/slides/7/qdreh.tex new file mode 100644 index 0000000..8ed512a --- /dev/null +++ b/vorlesungen/slides/7/qdreh.tex @@ -0,0 +1,110 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Drehungen mit Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Drehung?} +Abbildung von $\vec{x}$ mit $\operatorname{Re}\vec{x}=0$: +\[ +\varrho_{q} +\colon +\vec{x}\mapsto q\vec{x}q^{-1} = q\vec{x}\overline{q} +\] +\end{block} +\uncover<2->{% +\begin{block}{Achse} +\begin{align*} +\varrho_q(q) +&= +qq\overline{q} +\uncover<3->{= +q(qq^{-1})} +\uncover<4->{= +q} +\end{align*} +\end{block}} +\uncover<4->{% +\begin{block}{Norm} +\begin{align*} +|\varrho_q(\vec{x})|^2 +&= +q\vec{x}\overline{q}\overline{(q\vec{x}\overline{q})} +\uncover<5->{= +q\vec{x}\overline{q}\overline{\overline{q}}\overline{\vec{x}}\overline{q} +} +\\ +&\uncover<6->{= +q\vec{x}(\overline{q}q)\overline{\vec{x}}\overline{q}} +\uncover<7->{= +q(\vec{x}\overline{\vec{x}})\overline{q}} +\uncover<8->{= +q\overline{q}|\vec{x}|^2} +\\ +&\uncover<9->{= +|\vec{x}|^2} +\end{align*} +\uncover<10->{% +$\Rightarrow$ $\varrho_q\in\operatorname{O}(3)$} +\end{block}} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<11->{% +\begin{block}{Drehung!} +$\vec{a},\vec{b},\vec{n}$ bilden ein on.~Rechtssystem +\begin{align*} +\uncover<12->{ +qa +&= +c\vec{a}+s\vec{n}\times \vec{a}} +\uncover<13->{= +c\vec{a} + s\vec{b}} +\\ +\uncover<14->{ +q\vec{a}\overline{q} +&= +(c\vec{a}+s\vec{b}) c +-(c\vec{a}+s\vec{b})\times s\vec{n}} +\\ +&\uncover<15->{= +c^2 \vec{a}+ sc\vec{b} ++sc\vec{b} - s^2 \vec{a}} +\\ +&\uncover<16->{= +\vec{a} \cos\alpha +\vec{b} \sin\alpha } +\end{align*} +\vspace{-5pt} +\uncover<17->{wegen +%\vspace{-5pt} +\[ +\begin{aligned} +\cos\alpha &= \cos^2\frac{\alpha}2 - \sin^2\frac{\alpha}2 &&=c^2-s^2 +\\ +\sin\alpha &= 2\cos\frac{\alpha}2\sin\frac{\alpha}2&&=2cs +\end{aligned}\]} +\end{block}} +\vspace{-18pt} +\uncover<18->{% +\begin{block}{Matrix} +\[ +D += +\tiny +\begin{pmatrix} +1-2(q_2^2+q_3^2)&-2q_0q_3+2q_1q_2&-2q_0q_2+2q_1q_3\\ + 2q_0q_3+2q_1q_2&1-2(q_1^2+q_3^2)&-2q_0q_1+2q_2q_3\\ +-2q_0q_2+2q_1q_3& 2q_0q_1+2q_2q_3&1-2(q_1^2+q_2^2) +\end{pmatrix} +\] +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/quaternionen.tex b/vorlesungen/slides/7/quaternionen.tex new file mode 100644 index 0000000..f526366 --- /dev/null +++ b/vorlesungen/slides/7/quaternionen.tex @@ -0,0 +1,74 @@ +% +% quaternionen.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quaternionen} +$4$-dimensionaler $\mathbb{R}$-Vektorraum +\[ +\mathbb{H} += +\langle 1,i,j,k\rangle_{\mathbb{R}} +\] +mit Rechenregeln +\[ +i^2=j^2=k^2=ijk=-1 +\] +$x=x_0+x_1i+x_2j+x_3k\in\mathbb{H}$ +\begin{itemize} +\item<2-> Realteil: $\operatorname{Re}x=x_0$ +\item<3-> Vektorteil: $\operatorname{Im}x=x_1i+x_2j+x_3k$ +\item<4-> Konjugation: $\overline{x}=\operatorname{Re}x-\operatorname{Im}x$ +\item<5-> Norm: $|x|^2 = x\overline{x} = x_0^2+x_1^2+x_2^2+x_3^2$ +\item<6-> Inverse: $x^{1}= \overline{x}/x\overline{x}$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<7->{% +\begin{block}{Skalarprodukt und Vektorprodukt} +\begin{align*} +pq +&= +\operatorname{Re}p \operatorname{Re}q +- +\operatorname{Im}p\cdot \operatorname{Im}q +\\ +&\phantom{=} ++ +\operatorname{Re}p\operatorname{Im}q ++ +\operatorname{Im}p\operatorname{Re}q ++ +\operatorname{Im}p\times\operatorname{Im}q +\end{align*} +\end{block}} +\uncover<8->{% +\begin{block}{Einheitsquaternionen} +$q\in \mathbb{H}$, $|q|=1, q^{-1}=\overline{q}$ +\end{block}} +\uncover<9->{% +\begin{block}{Polardarstellung} +\[ +q = \cos\frac{\alpha}2 + \vec{n} \sin\frac{\alpha}2 +\] +\vspace{-8pt} +\begin{itemize} +\item<10-> +Drehmatrix: 9 Parameter, 6 Bedingungen +\item<11-> +Quaternionen: 4 Parameter, 1 Bedingung +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index d74b7d0..cd974c9 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 58e87a1..a65b4f6 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 8931a24..35d62d8 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/7/ueberlagerung.tex b/vorlesungen/slides/7/ueberlagerung.tex new file mode 100644 index 0000000..426641a --- /dev/null +++ b/vorlesungen/slides/7/ueberlagerung.tex @@ -0,0 +1,98 @@ +% +% ueberlagerung.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{$S^3$, $\operatorname{SU}(2)$ und $\operatorname{SO}(3)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\uncover<6->{% +\begin{block}{Überlagerung} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (0,0); +\coordinate (B) at (2,0); +\coordinate (C) at (2,-2); +\coordinate (D) at (0,-2); + +\uncover<7->{ +\node at (A) {$\{\pm 1\}\mathstrut$}; +} +\uncover<6->{ +\node at (B) {$S^3\mathstrut$}; +\node at ($(B)+(0.1,0)$) [right] {$=\operatorname{SU}(2)\mathstrut$}; +} +\uncover<7->{ +\node at (C) {$\operatorname{SO}(3)\mathstrut$}; +\node at (D) {$\{I\}\mathstrut$}; +} + +\uncover<7->{ +\draw[->,shorten >= 0.3cm,shorten <= 0.5cm] (A) -- (B); +\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (A) -- (D); +\draw[->,shorten >= 0.3cm,shorten <= 0.3cm] (B) -- (C); +\draw[->,shorten >= 0.6cm,shorten <= 0.3cm] (D) -- (C); +} + +\end{tikzpicture} +\end{center} +\begin{itemize} +\item<7-> +$\pm q\in S^3$ $\Rightarrow$ $\varrho_{q}=\varrho_{-q}$ +\item<8-> +In der Nähe von $I$ sehen die Gruppen +$\operatorname{SO}(3)$ +und +$\operatorname{SU}(2)$ +``gleich'' aus +\item<9-> +$\operatorname{SU}(2)$ ist geometrisch ``einfacher'' +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\begin{block}{Pauli-Matrizen} +Quaternionen als $2\times 2$-Matrizen schreiben +\begin{align*} +1&=\begin{pmatrix}1&0\\0&1\end{pmatrix}=\sigma_0, +& +i&=\begin{pmatrix}0&i\\i&0\end{pmatrix}=-i\sigma_1 +\\ +j&=\begin{pmatrix}0&-1\\1&0\end{pmatrix}=-i\sigma_2, +& +k&=\begin{pmatrix}i&0\\0&-i\end{pmatrix}=-i\sigma_3 +\end{align*} +\uncover<2->{% +erfüllen $i^2=j^2=k^2=ijk=-1$.} +\end{block} +\uncover<3->{% +\begin{block}{$S^3 = \operatorname{SU}(2)$} +\[ +a+bi+cj+dk += +\begin{pmatrix} +a+id&-c+bi\\ +c+ib&a-id +\end{pmatrix} += +A +\] +\begin{align*} +\uncover<4->{ +\det A &= a^2 + b^2 + c^2 + d^2 = 1 +} +\\ +\uncover<5->{ +A^* &= a - ib - jc - kd +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/vektorlie.tex b/vorlesungen/slides/7/vektorlie.tex new file mode 100644 index 0000000..621a832 --- /dev/null +++ b/vorlesungen/slides/7/vektorlie.tex @@ -0,0 +1,206 @@ +% +% viktorlie.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Vektorprodukt als Lie-Algebra} +%\vspace{-10pt} +\centering +\begin{tikzpicture}[>=latex,thick] +\arraycolsep=2.4pt +\def\Ax{0} +\def\Ux{4.1} +\def\Kx{7.2} +\def\Rx{13.1} + +\def\Lx{2.2} +\def\Ly{0} +\def\Lz{-2.2} + +\fill[color=red!20] (\Ax,{\Lx-1.55}) rectangle ({\Ux-0.1},{\Lx+0.55}); +\fill[color=red!20] (\Ux,{\Lx-1.55}) rectangle ({\Kx-0.1},{\Lx+0.55}); +\fill[color=red!20] (\Kx,{\Lx-1.55}) rectangle ({\Rx},{\Lx+0.55}); + +\fill[color=darkgreen!20] (\Ax,{\Ly-1.55}) rectangle ({\Ux-0.1},{\Ly+0.55}); +\fill[color=darkgreen!20] (\Ux,{\Ly-1.55}) rectangle ({\Kx-0.1},{\Ly+0.55}); +\fill[color=darkgreen!20] (\Kx,{\Ly-1.55}) rectangle ({\Rx},{\Ly+0.55}); + +\fill[color=blue!20] (\Ax,{\Lz-1.55}) rectangle ({\Ux-0.1},{\Lz+0.55}); +\fill[color=blue!20] (\Ux,{\Lz-1.55}) rectangle ({\Kx-0.1},{\Lz+0.55}); +\fill[color=blue!20] (\Kx,{\Lz-1.55}) rectangle ({\Rx},{\Lz+0.55}); + +\coordinate (A) at (\Ax,3.2); +\coordinate (Ax) at (\Ax,\Lx); +\coordinate (Ay) at (\Ax,\Ly); +\coordinate (Az) at (\Ax,\Lz); + +\node at (A) [right] + {\usebeamercolor[fg]{title}Drehmatrix, $\operatorname{SO}(n)$\strut}; + +\node at (Ax) [right] {$\displaystyle\tiny +D_{x,\alpha}=\begin{pmatrix} +1&0&0\\ +0&\cos\alpha&-\sin\alpha\\ +0&\sin\alpha&\cos\alpha +\end{pmatrix}$}; + +\node at (Ay) [right] {$\displaystyle\tiny +D_{y,\alpha}=\begin{pmatrix} +\cos\alpha&0&\sin\alpha\\ +0&1&0\\ +-\sin\alpha&0&\cos\alpha +\end{pmatrix}$}; + +\node at (Az) [right] {$\displaystyle\tiny +D_{z,\alpha}=\begin{pmatrix} +\cos\alpha&-\sin\alpha&0\\ +\sin\alpha&\cos\alpha&0\\ +0&0&1 +\end{pmatrix}$}; + +\coordinate (U) at (\Ux,3.2); +\coordinate (Ux) at (\Ux,\Lx); +\coordinate (Uy) at (\Ux,\Ly); +\coordinate (Uz) at (\Ux,\Lz); +\coordinate (Ex) at (\Ux,{\Lx-1}); +\coordinate (Ey) at (\Ux,{\Ly-1}); +\coordinate (Ez) at (\Ux,{\Lz-1}); + +\uncover<2->{ +\node at (U) [right] + {\usebeamercolor[fg]{title}Ableitung, $\operatorname{so}(n)$\strut}; + +\node at (Ux) [right] {$\displaystyle\tiny +U_x=\begin{pmatrix*}[r] +0&0&0\\ +0&0&-1\\ +0&1&0 +\end{pmatrix*} +$}; + +\node at (Uy) [right] {$\displaystyle\tiny +U_y=\begin{pmatrix*}[r] +0&0&1\\ +0&0&0\\ +-1&0&0 +\end{pmatrix*} +$}; + +\node at (Uz) [right] {$\displaystyle\tiny +U_z=\begin{pmatrix*}[r] +0&-1&0\\ +1&0&0\\ +0&0&0 +\end{pmatrix*} +$}; +} + +\uncover<9->{ +\node at (Ex) [right] {$\displaystyle +\, e_x = \tiny\begin{pmatrix}1\\0\\0\end{pmatrix} +$}; + +\node at (Ey) [right] {$\displaystyle +\, e_y = \tiny\begin{pmatrix}0\\1\\0\end{pmatrix} +$}; + +\node at (Ez) [right] {$\displaystyle +\, e_z = \tiny\begin{pmatrix}0\\0\\1\end{pmatrix} +$}; +} + +\coordinate (K) at (\Kx,3.2); +\coordinate (Kx) at (\Kx,\Lx); +\coordinate (Ky) at (\Kx,\Ly); +\coordinate (Kz) at (\Kx,\Lz); +\coordinate (Vx) at (\Kx,{\Lx-1}); +\coordinate (Vy) at (\Kx,{\Ly-1}); +\coordinate (Vz) at (\Kx,{\Lz-1}); + +\uncover<3->{ +\node at (K) [right] + {\usebeamercolor[fg]{title}Kommutator\strut}; + +\node at (Kx) [right] {$\displaystyle +\begin{aligned} +[U_y,U_z] &\uncover<4->{= +{\tiny +\begin{pmatrix} +0&0&0\\ +0&0&0\\ +0&1&0 +\end{pmatrix}} +\uncover<5->{\mathstrut- +\tiny +\begin{pmatrix} +0&0&0\\ +0&0&1\\ +0&0&0 +\end{pmatrix}}} +\uncover<6->{=U_x} +\end{aligned} +$}; +} + +\uncover<7->{ +\node at (Ky) [right] {$\displaystyle +\begin{aligned} +[U_z,U_x] &= +{\tiny +\begin{pmatrix} +0&0&1\\ +0&0&0\\ +0&0&0 +\end{pmatrix} +- +\begin{pmatrix} +0&0&0\\ +0&0&0\\ +1&0&0 +\end{pmatrix}} +=U_y +\end{aligned} +$}; +} + +\uncover<8->{ +\node at (Kz) [right] {$\displaystyle +\begin{aligned} +[U_x,U_y] &= +{\tiny +\begin{pmatrix} +0&0&0\\ +1&0&0\\ +0&0&0 +\end{pmatrix} +- +\begin{pmatrix} +0&1&0\\ +0&0&0\\ +0&0&0 +\end{pmatrix}} +=U_z +\end{aligned} +$}; +} + +\uncover<10->{ +\node at (Vx) [right] {$\displaystyle \phantom{]}e_y\times e_z = e_x$}; +} + +\uncover<11->{ +\node at (Vy) [right] {$\displaystyle \phantom{]}e_z\times e_x = e_y$}; +} + +\uncover<12->{ +\node at (Vz) [right] {$\displaystyle \phantom{]}e_x\times e_y = e_z$}; +} + +\end{tikzpicture} +\end{frame} +\egroup diff --git a/vorlesungen/slides/7/zusammenhang.tex b/vorlesungen/slides/7/zusammenhang.tex new file mode 100644 index 0000000..6a43cd8 --- /dev/null +++ b/vorlesungen/slides/7/zusammenhang.tex @@ -0,0 +1,99 @@ +% +% template.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zusammenhang} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Zusammenhängend --- oder nicht} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\ds{2.4} +\coordinate (A) at (0,0); +\coordinate (B) at (\ds,0); +\coordinate (C) at ({2*\ds},0); + +\node at (A) {$\operatorname{SO}(n)$}; +\node at (B) {$\operatorname{O}(n)$}; +\node at (C) {$\{\pm 1\}$}; + +\draw[->,shorten <= 0.6cm,shorten >= 0.5cm] (A) -- (B); +\draw[->,shorten <= 0.5cm,shorten >= 0.5cm] (B) -- (C); +\node at ($0.5*(B)+0.5*(C)$) [above] {$\det$}; + +\coordinate (A2) at (0,-1.0); +\coordinate (B2) at (\ds,-1.0); +\coordinate (C2) at ({2*\ds},-1.0); + +\draw[color=blue] (A2) ellipse (1cm and 0.3cm); +\draw[color=blue] (B2) ellipse (1cm and 0.3cm); +\node[color=blue] at (C2) {$+1$}; + +\coordinate (A3) at (0,-1.7); +\coordinate (B3) at (\ds,-1.7); +\coordinate (C3) at ({2*\ds},-1.7); + +\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B2) -- (C2); +\draw[->,shorten <= 1.1cm,shorten >= 0.3cm] (B3) -- (C3); + +\draw[color=red] (B3) ellipse (1cm and 0.3cm); +\node[color=red] at (C3) {$-1$}; + +\end{tikzpicture} +\end{center} +\end{block} +\begin{block}{Zusammenhangskomponente von $e$} +$G_e\subset G$ grösste zusammenhängende Menge, die $e$ enthält: +\begin{align*} +\operatorname{SO}(n)&\subset \operatorname{O}(n) +\\ +\{A\in\operatorname{GL}_n(\mathbb{R})\,|\, \det A > 0\} + &\subset \operatorname{GL}_n(\mathbb{R}) +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Eigenschaften} +\begin{itemize} +\item +{\bf Untergruppe}: $\gamma_i(t)$ Weg von $e$ nach $g_i$, +dann ist +\begin{itemize} +\item +$\gamma_1(t)\gamma_2(t)$ ein Weg von $e$ nach $g_1g_2$ +\item +$\gamma_1(t)^{-1}$ Weg von $e$ nach $g_1^{-1}$ +\end{itemize} +\item +{\bf Normalteiler}: $\gamma(t)$ ein Weg von $e$ nach $g$, dann +ist $h\gamma(t)h^{-1}$ ein Weg von $h$ nach $hgh^{-1}$ +$\Rightarrow hG_eh^{-1}\subset G_e$ +\end{itemize} +\end{block} +\begin{block}{Quotient} +$G/G_e$ ist eine diskrete Gruppe +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\coordinate (A) at (0,0); +\coordinate (B) at (2,0); +\coordinate (C) at (4,0); +\node at (A) {$G_e$}; +\node at (B) {$G$}; +\node at (C) {$G/G_e$}; +\draw [->,shorten <= 0.3cm,shorten >= 0.3cm] (A) -- (B); +\draw [->,shorten <= 0.3cm,shorten >= 0.5cm] (B) -- (C); +\end{tikzpicture} +\end{center} +\vspace{-7pt} +$\Rightarrow$ $G_e$ und $G/G_e$ separat studieren +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |