diff options
Diffstat (limited to 'vorlesungen/slides/7')
| -rw-r--r-- | vorlesungen/slides/7/ableitung.tex | 40 | ||||
| -rw-r--r-- | vorlesungen/slides/7/dg.tex | 36 | ||||
| -rw-r--r-- | vorlesungen/slides/7/drehung.tex | 53 | ||||
| -rw-r--r-- | vorlesungen/slides/7/einparameter.tex | 18 | ||||
| -rw-r--r-- | vorlesungen/slides/7/kommutator.tex | 147 | ||||
| -rw-r--r-- | vorlesungen/slides/7/kurven.tex | 31 | ||||
| -rw-r--r-- | vorlesungen/slides/7/liealgebra.tex | 27 | ||||
| -rw-r--r-- | vorlesungen/slides/7/mannigfaltigkeit.tex | 12 | ||||
| -rw-r--r-- | vorlesungen/slides/7/parameter.tex | 51 | ||||
| -rw-r--r-- | vorlesungen/slides/7/semi.tex | 34 |
10 files changed, 299 insertions, 150 deletions
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex index b061b9a..12f9084 100644 --- a/vorlesungen/slides/7/ableitung.tex +++ b/vorlesungen/slides/7/ableitung.tex @@ -12,49 +12,55 @@ \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*} -I +\uncover<3->{I &= -A(s)^tA(s) +A(s)^tA(s)} \\ -0 +\uncover<4->{0 = \frac{d}{ds} I &= -\frac{d}{ds} (A(s)^t A(s)) +\frac{d}{ds} (A(s)^t A(s))} \\ -&= -\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s) -\intertext{An der Stelle $s=0$, d.~h.~$A(0)=I$} -0 +&\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) +\dot{A}(0)} \\ -\Leftrightarrow +\uncover<8->{\Leftrightarrow \qquad -\dot{A}(0)^t &= -\dot{A}(0) +\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*} -1 &= \det A(t) +\uncover<3->{1 &= \det A(t)} \\ -0 +\uncover<10->{0 = \frac{d}{dt}1 &= -\frac{d}{dt} \det A(t) -\intertext{mit dem Entwicklungssatz kann man nachrechnen:} -0&=\operatorname{Spur}\dot{A}(0) +\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*} -``Tangentialvektoren'' sind spurlose Matrizen +\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} \end{block} \end{column} \end{columns} diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 36b1ade..a99cebb 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -15,29 +15,35 @@ 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} \[ @@ -47,33 +53,39 @@ Ableitung von $\gamma(t)$ an der Stelle $t$: \quad A=\dot{\gamma}(0)\in LG \] -\end{block} +\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} +\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} 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{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/drehung.tex b/vorlesungen/slides/7/drehung.tex index ae0dbe3..7744e99 100644 --- a/vorlesungen/slides/7/drehung.tex +++ b/vorlesungen/slides/7/drehung.tex @@ -13,12 +13,20 @@ \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->{ \[ \operatorname{SO}(2) = -\operatorname{SL}_2(\mathbb{R}) \cap \operatorname{O}(2) -\] +{\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: \[ @@ -31,7 +39,9 @@ D_{\alpha} = DST \] -\end{block} +\end{block}} +\vspace{-10pt} +\uncover<12->{% \begin{block}{Beispiel} \vspace{-12pt} \[ @@ -43,9 +53,10 @@ D_{60^\circ} \begin{pmatrix}1&0\\\frac{\sqrt{3}}2&1\end{pmatrix} } \] -\end{block} +\end{block}} \end{column} \begin{column}{0.58\textwidth} +\uncover<4->{% \begin{block}{Ansatz} \vspace{-12pt} \begin{align*} @@ -64,7 +75,7 @@ c^{-1}&0\\ t&1 \end{pmatrix} \\ -&= +&\uncover<5->{= \begin{pmatrix} c^{-1}&0\\ 0 &c @@ -73,40 +84,48 @@ c^{-1}&0\\ -st&-s\\ t& 1 \end{pmatrix} +} \\ -&= +&\uncover<6->{= \begin{pmatrix} --stc^{-1}&{\color{darkgreen}sc^{-1}}\\ -{\color{blue}ct}&{\color{red}c} -\end{pmatrix} -= +{\color<11->{orange}-stc^{-1}}&{\color<10->{darkgreen}sc^{-1}}\\ +{\color<9->{blue}ct}&{\color<8->{red}c} +\end{pmatrix}} +\uncover<7->{= \begin{pmatrix} -\cos\alpha & {\color{darkgreen}- \sin\alpha} \\ -{\color{blue}\sin\alpha} & \phantom{-} {\color{red}\cos\alpha} -\end{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} +\end{block}} \vspace{-10pt} +\uncover<7->{% \begin{block}{Koeffizientenvergleich} \vspace{-15pt} \begin{align*} +\uncover<8->{ {\color{red} c} &= -{\color{red}\cos\alpha } +{\color{red}\cos\alpha }} && & +\uncover<9->{ {\color{blue} -t}&=\rlap{$\displaystyle\frac{\sin\alpha}{c} = \tan\alpha$} \\ +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} -stc^{-t}} &= \rlap{$\sin\alpha\tan\alpha = \cos\alpha \quad $} +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/einparameter.tex b/vorlesungen/slides/7/einparameter.tex index 52924bf..5171085 100644 --- a/vorlesungen/slides/7/einparameter.tex +++ b/vorlesungen/slides/7/einparameter.tex @@ -7,17 +7,20 @@ \begin{frame}[t] \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} -\frametitle{Einparameter Untergruppen} +\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 gleichzeitig eine Untergruppe von $G$ ist mit +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*} @@ -33,9 +36,10 @@ D_{x,t}D_{x,s} &= D_{x,t+s} \end{align*} -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<5->{% \begin{block}{Scherungen in $\operatorname{SL}_2(\mathbb{R})$} \vspace{-12pt} \[ @@ -53,8 +57,9 @@ D_{x,t+s} 0&1 \end{pmatrix} \] -\end{block} +\end{block}} \vspace{-12pt} +\uncover<6->{% \begin{block}{Skalierungen in $\operatorname{SL}_2(\mathbb{R})$} \vspace{-12pt} \[ @@ -69,8 +74,9 @@ e^t&0\\0&e^{-t} e^{t+s}&0\\0&e^{-(t+s)} \end{pmatrix} \] -\end{block} +\end{block}} \vspace{-12pt} +\uncover<7->{% \begin{block}{Gemischt} \vspace{-12pt} \begin{gather*} @@ -80,7 +86,7 @@ A_t = I \cosh t + \begin{pmatrix}1&a\\0&-1\end{pmatrix}\sinh t \begin{pmatrix}1&s\\0&-1\end{pmatrix}^2 =I \end{gather*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/kommutator.tex b/vorlesungen/slides/7/kommutator.tex index f9004df..84bf034 100644 --- a/vorlesungen/slides/7/kommutator.tex +++ b/vorlesungen/slides/7/kommutator.tex @@ -4,6 +4,7 @@ % (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} @@ -11,128 +12,154 @@ 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(-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} diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex index 196fa2a..e0690eb 100644 --- a/vorlesungen/slides/7/kurven.tex +++ b/vorlesungen/slides/7/kurven.tex @@ -20,6 +20,7 @@ Kurve in $\mathbb{R}^n$: I=[a,b] \to \mathbb{R}^n : t\mapsto \gamma(t) +\uncover<2->{ = \begin{pmatrix} x_1(t)\\ @@ -27,6 +28,7 @@ x_2(t)\\ \vdots\\ x_n(t) \end{pmatrix} +} \] \vspace{-15pt} \begin{center} @@ -42,16 +44,21 @@ x_n(t) \fill[color=red] (C) circle[radius=0.06]; \node[color=red] at (C) [left] {$\gamma(t)$}; -\draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); -\node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; +\uncover<4->{ + \draw[->,color=blue,line width=1.4pt,shorten <= 0.06cm] (C) -- (E); + \node[color=blue] at (E) [right] {$\dot{\gamma}(t)$}; +} -\draw[->] (-0.1,0) -- (5.9,0) coordinate[label={$x_1$}]; -\draw[->] (0,-0.1) -- (0,4.3) coordinate[label={right:$x_2$}]; +\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 \[ @@ -66,6 +73,7 @@ Ableitung \dot{x}_n(t) \end{pmatrix} \] +\uncover<5->{% Lineare Approximation: \[ \gamma(t+h) @@ -75,10 +83,21 @@ Lineare Approximation: \dot{\gamma}(t) \cdot h + o(h) -\] +\]}% +\vspace{-10pt} +\begin{itemize} +\item<6-> Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ in $\mathbb{R}^n$ liegen -\end{block} +\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} diff --git a/vorlesungen/slides/7/liealgebra.tex b/vorlesungen/slides/7/liealgebra.tex index 892216e..574467b 100644 --- a/vorlesungen/slides/7/liealgebra.tex +++ b/vorlesungen/slides/7/liealgebra.tex @@ -8,50 +8,64 @@ \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} +\end{block}} +\uncover<8->{% \begin{block}{Nachprüfen} $[A,B]\in LG$ für $A,B\in LG$ -\end{block} +\end{block}} \end{column} \begin{column}{0.56\textwidth} +\uncover<9->{% \begin{block}{Algebraische Eigenschaften} \begin{itemize} -\item antisymmetrisch: $[A,B]=-[B,A]$ -\item Jacobi-Identität +\item<10-> antisymmetrisch: $[A,B]=-[B,A]$ +\item<11-> Jacobi-Identität \[ [A,[B,C]]+ [B,[C,A]]+ @@ -59,9 +73,12 @@ für $A,B\in LG$ = 0 \] \end{itemize} +\vspace{-13pt} +\uncover<12->{% {\usebeamercolor[fg]{title} Beispiel:} $\mathbb{R}^3$ mit Vektorprodukt $\mathstrut = \operatorname{so}(3)$ -\end{block} +} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/mannigfaltigkeit.tex b/vorlesungen/slides/7/mannigfaltigkeit.tex index 7809ea5..077dc9d 100644 --- a/vorlesungen/slides/7/mannigfaltigkeit.tex +++ b/vorlesungen/slides/7/mannigfaltigkeit.tex @@ -18,8 +18,8 @@ \begin{column}{0.48\textwidth} \begin{block}{Definition} \begin{itemize} -\item Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ -\item differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet +\item<2-> Karte: Abbildung $\varphi_\alpha\colon U_\alpha\to\mathbb{R}^n$ +\item<3-> differenzierbare Kartenwechsel: Koordinatenumrechnung im Überschneidungsgebiet \[ \varphi_\beta\circ\varphi_\alpha^{-1} \colon @@ -27,17 +27,19 @@ \to \varphi_\beta(U_\alpha\cap U_\beta) \] -\item Atlas: Menge von Karten, die die ganze Mannigfaltigkeit überdecken +\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} +\end{block}} \vspace{-3pt} +\uncover<6->{% \begin{block}{Lie-Gruppe} Gruppe und Mannigfaltigkeit -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} diff --git a/vorlesungen/slides/7/parameter.tex b/vorlesungen/slides/7/parameter.tex index b719207..1e8549c 100644 --- a/vorlesungen/slides/7/parameter.tex +++ b/vorlesungen/slides/7/parameter.tex @@ -5,6 +5,7 @@ % \bgroup \definecolor{darkgreen}{rgb}{0,0.6,0} +\definecolor{darkyellow}{rgb}{1,0.8,0} \begin{frame}[t] \setlength{\abovedisplayskip}{5pt} \setlength{\belowdisplayskip}{5pt} @@ -13,53 +14,85 @@ \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} -\begin{block}{Drehung um $\vec{\omega}$} +\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 +} \[ -\vec{x} +{\color{red}\vec{x}} \mapsto -(\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}) +\uncover<10->{ +({\color{darkyellow}\vec{x} -(\vec{k}\cdot\vec{x})\vec{k}}) \cos\omega + -(\vec{k}\times\vec{x})\sin\omega +} +\uncover<11->{ +({\color{darkgreen}\vec{x}\times\vec{k}}) \sin\omega + -\vec{k}(\vec{k}\cdot\vec{x}) +} +\uncover<9->{ +{\color{blue}\vec{k}} (\vec{k}\cdot\vec{x}) +} \] \vspace{-40pt} \begin{center} \begin{tikzpicture}[>=latex,thick] -\node at (0,0) {\includegraphics[width=\textwidth]{../slides/7/images/rodriguez.jpg}}; +\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) {$\vec{k}$}; -\node[color=darkgreen] at (-3,1.1) {$\vec{k}\times\vec{x}$}; -\node[color=yellow] at (2.2,-0.2) {$\vec{x}-(\vec{x}\cdot\vec{k})\vec{k}$}; +\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{block}} \end{column} \end{columns} \vspace{-15pt} diff --git a/vorlesungen/slides/7/semi.tex b/vorlesungen/slides/7/semi.tex index 46f6d03..66b8d27 100644 --- a/vorlesungen/slides/7/semi.tex +++ b/vorlesungen/slides/7/semi.tex @@ -21,6 +21,7 @@ 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= @@ -32,51 +33,57 @@ Wirkung auf $\mathbb{R}^2$: \[ \vec{x}\mapsto \underbrace{D_\alpha \vec{x}}_{\text{Drehung}} \mathstrut+ \vec{t} \] -\end{block} +\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 -&= -(e^{s_1},t_1)(e^{s_2}x+t_2) +&\uncover<4->{= +(e^{s_1},t_1)(e^{s_2}x+t_2)} \\ -&= -e^{s_1+s_2}x + e^{s_1}t_2+t_1 +&\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) +(e^{s_1}e^{s_2},t_1+e^{s_1}t_2)} \end{align*} -\end{block} +\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} -&= -(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2) +&\uncover<8->{= +(\alpha_1,\vec{t}_1)(D_{\alpha_2}\vec{x}+\vec{t}_2)} \\ -&=D_{\alpha_1+\alpha_2}\vec{x} + D_{\alpha_1}\vec{t}_2+\vec{t}_1 +&\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{block}} \end{column} \end{columns} \vspace{-10pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<11->{% \begin{block}{Matrixschreibweise} \vspace{-12pt} \[ @@ -88,9 +95,10 @@ e^s&t\\ \quad\text{auf}\quad \begin{pmatrix}x\\1\end{pmatrix} \] -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<12->{% \begin{block}{Matrixschreibweise} \vspace{-12pt} \[ @@ -102,7 +110,7 @@ D_{\alpha}&\vec{t}\\ \quad\text{auf}\quad \begin{pmatrix}\vec{x}\\1\end{pmatrix} \] -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} |