%
% 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
