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Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/7/kurven.tex | 208 |
1 files changed, 104 insertions, 104 deletions
diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex index e0690eb..bca8417 100644 --- a/vorlesungen/slides/7/kurven.tex +++ b/vorlesungen/slides/7/kurven.tex @@ -1,104 +1,104 @@ -% -% kurven.tex -- slide template -% -% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -% -\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 +%
+% kurven.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\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
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