update to current state of book

1 files changed, 145 insertions, 145 deletions

diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex
index 35d62d8..8931a24 100644
--- a/vorlesungen/slides/7/symmetrien.tex
+++ b/vorlesungen/slides/7/symmetrien.tex
@@ -1,145 +1,145 @@
-%
-% symmetrien.tex -- Symmetrien
-%
-% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
-%
-\bgroup
-\definecolor{darkgreen}{rgb}{0,0.6,0}
-\begin{frame}[t]
-\setlength{\abovedisplayskip}{5pt}
-\setlength{\belowdisplayskip}{5pt}
-\frametitle{Symmetrien}
-\vspace{-20pt}
-\begin{columns}[t,onlytextwidth]
-\begin{column}{0.48\textwidth}
-\begin{block}{Diskrete Symmetrien}
-\begin{itemize}
-\item<2->
-Ebenen-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
-{\tiny
-\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\uncover<4->{\!,\;
-\vec{x}
-\mapsto
-\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}
-}
-\]
-\vspace{-10pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\a{10}
-\def\b{50}
-\def\r{2}
-\coordinate (O) at (0,0);
-\coordinate (A) at (\b:\r);
-\coordinate (B) at ({180+2*\a-\b}:\r);
-\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});
-\coordinate (N) at (\a:2);
-\coordinate (D) at (\a:{\r*cos(\b-\a)});
-\uncover<3->{
-\clip (-2.5,-0.45) rectangle (2.5,1.95);
-
-	\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)
-		-- cycle;
-	\draw[->,color=darkgreen] (O) -- (N);
-	\node[color=darkgreen] at (N) [above] {$\vec{n}$};
-
-
-	\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)
-		-- cycle;
-	\fill[color=red] (O) circle[radius=0.06];
-	\draw[color=red] ({\a-90}:2) -- ({\a+90}:2);
-	\fill[color=blue] (C) circle[radius=0.06];
-	\draw[color=blue,line width=0.1pt] (A) -- (D);
-	\node[color=darkgreen] at (D) [below,rotate=\a]
-		{$(\vec{n}\cdot\vec{x})\vec{n}$};
-	\draw[color=blue,line width=0.5pt] (A)--(B);
-
-	\node[color=blue] at (A) [above right] {$\vec{x}$};
-	\node[color=blue] at (B) [above left] {$\vec{x}'$};
-
-	\node[color=red] at (O) [below left] {$O$};
-
-	\draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);
-	\draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);
-}
-
-\end{tikzpicture}
-\end{center}
-\vspace{-5pt}
-$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$
-\item<5->
-Punkt-Spiegelung:
-\[
-{\tiny
-\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\mapsto
--
-{\tiny
-\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}
-}
-\]
-\end{itemize}
-\end{block}
-\end{column}
-\begin{column}{0.48\textwidth}
-\uncover<6->{%
-\begin{block}{Kontinuierliche Symmetrien}
-\begin{itemize}
-\item<7-> Translation:
-\(
-\vec{x} \mapsto \vec{x} + \vec{t}
-\)
-\item<8-> Drehung:
-\vspace{-3pt}
-\begin{center}
-\begin{tikzpicture}[>=latex,thick]
-\def\a{25}
-\def\r{1.3}
-\coordinate (O) at (0,0);
-\begin{scope}
-\clip (-1.1,-0.1) rectangle (2.3,2.3);
-\draw[color=red] (O) circle[radius=2];
-\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;
-\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;
-\node at ({0.5*\a}:1) {$\alpha$};
-\node at ({90+0.5*\a}:1) {$\alpha$};
-\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);
-\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);
-\end{scope}
-\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];
-\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];
-\end{tikzpicture}
-\end{center}
-\[
-\uncover<9->{%
-\begin{pmatrix}x\\y\end{pmatrix}
-\mapsto
-\begin{pmatrix}
-{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\
-{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}
-\end{pmatrix}
-\begin{pmatrix}x\\y\end{pmatrix}
-}
-\]
-\end{itemize}
-\end{block}}
-\vspace{-10pt}
-\uncover<10->{%
-\begin{block}{Definition}
-Längen/Winkel bleiben erhalten
-\\
-\uncover<11->{%
-$\Rightarrow$ $\exists$ Erhaltungsgrösse}
-\end{block}}
-\end{column}
-\end{columns}
-\end{frame}
-\egroup
+%

+% symmetrien.tex -- Symmetrien

+%

+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule

+%

+\bgroup

+\definecolor{darkgreen}{rgb}{0,0.6,0}

+\begin{frame}[t]

+\setlength{\abovedisplayskip}{5pt}

+\setlength{\belowdisplayskip}{5pt}

+\frametitle{Symmetrien}

+\vspace{-20pt}

+\begin{columns}[t,onlytextwidth]

+\begin{column}{0.48\textwidth}

+\begin{block}{Diskrete Symmetrien}

+\begin{itemize}

+\item<2->

+Ebenen-Spiegelung:

+\[

+{\tiny

+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}

+}

+\mapsto

+{\tiny

+\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}

+}

+\uncover<4->{\!,\;

+\vec{x}

+\mapsto

+\vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n}

+}

+\]

+\vspace{-10pt}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\a{10}

+\def\b{50}

+\def\r{2}

+\coordinate (O) at (0,0);

+\coordinate (A) at (\b:\r);

+\coordinate (B) at ({180+2*\a-\b}:\r);

+\coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)});

+\coordinate (N) at (\a:2);

+\coordinate (D) at (\a:{\r*cos(\b-\a)});

+\uncover<3->{

+\clip (-2.5,-0.45) rectangle (2.5,1.95);

+

+	\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2)

+		-- cycle;

+	\draw[->,color=darkgreen] (O) -- (N);

+	\node[color=darkgreen] at (N) [above] {$\vec{n}$};

+

+

+	\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2)

+		-- cycle;

+	\fill[color=red] (O) circle[radius=0.06];

+	\draw[color=red] ({\a-90}:2) -- ({\a+90}:2);

+	\fill[color=blue] (C) circle[radius=0.06];

+	\draw[color=blue,line width=0.1pt] (A) -- (D);

+	\node[color=darkgreen] at (D) [below,rotate=\a]

+		{$(\vec{n}\cdot\vec{x})\vec{n}$};

+	\draw[color=blue,line width=0.5pt] (A)--(B);

+

+	\node[color=blue] at (A) [above right] {$\vec{x}$};

+	\node[color=blue] at (B) [above left] {$\vec{x}'$};

+

+	\node[color=red] at (O) [below left] {$O$};

+

+	\draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A);

+	\draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B);

+}

+

+\end{tikzpicture}

+\end{center}

+\vspace{-5pt}

+$\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$

+\item<5->

+Punkt-Spiegelung:

+\[

+{\tiny

+\begin{pmatrix*}[r] x_1\\x_2\\x_3 \end{pmatrix*}

+}

+\mapsto

+-

+{\tiny

+\begin{pmatrix*}[r]x_1\\x_2\\x_3 \end{pmatrix*}

+}

+\]

+\end{itemize}

+\end{block}

+\end{column}

+\begin{column}{0.48\textwidth}

+\uncover<6->{%

+\begin{block}{Kontinuierliche Symmetrien}

+\begin{itemize}

+\item<7-> Translation:

+\(

+\vec{x} \mapsto \vec{x} + \vec{t}

+\)

+\item<8-> Drehung:

+\vspace{-3pt}

+\begin{center}

+\begin{tikzpicture}[>=latex,thick]

+\def\a{25}

+\def\r{1.3}

+\coordinate (O) at (0,0);

+\begin{scope}

+\clip (-1.1,-0.1) rectangle (2.3,2.3);

+\draw[color=red] (O) circle[radius=2];

+\fill[color=blue!20] (O) -- (0:\r) arc (0:\a:\r) -- cycle;

+\fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle;

+\node at ({0.5*\a}:1) {$\alpha$};

+\node at ({90+0.5*\a}:1) {$\alpha$};

+\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2);

+\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2);

+\end{scope}

+\draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}];

+\draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}];

+\end{tikzpicture}

+\end{center}

+\[

+\uncover<9->{%

+\begin{pmatrix}x\\y\end{pmatrix}

+\mapsto

+\begin{pmatrix}

+{\color{blue}\cos\alpha}&{\color{darkgreen}-\sin\alpha}\\

+{\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha}

+\end{pmatrix}

+\begin{pmatrix}x\\y\end{pmatrix}

+}

+\]

+\end{itemize}

+\end{block}}

+\vspace{-10pt}

+\uncover<10->{%

+\begin{block}{Definition}

+Längen/Winkel bleiben erhalten

+\\

+\uncover<11->{%

+$\Rightarrow$ $\exists$ Erhaltungsgrösse}

+\end{block}}

+\end{column}

+\end{columns}

+\end{frame}

+\egroup