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diff --git a/vorlesungen/slides/7/kurven.tex b/vorlesungen/slides/7/kurven.tex new file mode 100644 index 0000000..196fa2a --- /dev/null +++ b/vorlesungen/slides/7/kurven.tex @@ -0,0 +1,85 @@ +% +% 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} +\[ +\gamma +\colon +I=[a,b] \to \mathbb{R}^n +: +t\mapsto \gamma(t) += +\begin{pmatrix} +x_1(t)\\ +x_2(t)\\ +\vdots\\ +x_n(t) +\end{pmatrix} +\] +\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] {$\gamma(t)$}; + +\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:$x_2$}]; +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Tangenten} +Ableitung +\[ +\frac{d}{dt}\gamma(t) += +\dot{\gamma}(t) += +\begin{pmatrix} +\dot{x}_1(t)\\ +\dot{x}_2(t)\\ +\vdots\\ +\dot{x}_n(t) +\end{pmatrix} +\] +Lineare Approximation: +\[ +\gamma(t+h) += +\gamma(t) ++ +\dot{\gamma}(t) \cdot h ++ +o(h) +\] +Sinnvoll, weil sowohl $\gamma(t)$ und $\dot{\gamma}(t)$ +in $\mathbb{R}^n$ liegen +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |