phases

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 @@
 \vspace{-20pt}
 \begin{center}
 \begin{tikzpicture}[>=latex,thick]
+\def\t{14.0cm}
 \ifthenelse{\boolean{presentation}}{
 \only<1>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c01.jpg}};}
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 \only<55>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c55.jpg}};}
+\includegraphics[width=\t]{../slides/7/images/c/c55.jpg}};}
 \only<56>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c56.jpg}};}
+\includegraphics[width=\t]{../slides/7/images/c/c56.jpg}};}
 \only<57>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c57.jpg}};}
+\includegraphics[width=\t]{../slides/7/images/c/c57.jpg}};}
 \only<58>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c58.jpg}};}
+\includegraphics[width=\t]{../slides/7/images/c/c58.jpg}};}
 \only<59>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c59.jpg}};}
+\includegraphics[width=\t]{../slides/7/images/c/c59.jpg}};}
 }{}
 \only<60>{\node at (0,0) {
-\includegraphics[width=\textwidth]{../slides/7/images/c/c60.jpg}};}
+\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}
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}