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% kurven.tex -- slide template
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% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
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\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
