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\begin{document}

\Blatt{zu Koordinatensystemen}{WS 2005/2006, 23.1.2006}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

         
         Bei den angegebenen Koordinaten handelt es sich um
         rechtwinklige, \emph{normierte} Koordinaten.


\begin{deflist}{20.}

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\item[1.]{\bf Kartesische Koordinaten $(x,y,z)$}
\par\smallskip
\begin{minipage}{0.49\textwidth}
\begin{center}
  \includegraphics[width=0.8\textwidth]{Kartesische_Koordinaten.eps}
\end{center}
\end{minipage}
\begin{minipage}{0.49\textwidth}
$$
\grad f=
\left(
 \begin{array}{c}
  \partial_{x}f\\
  \partial_{y}f\\
  \partial_{z}f
 \end{array}
\right)
$$
$$
\div\vec{v}=\partial_{x}v_{x}+\partial_{y}v_{y}+\partial_{z}v_{z}
$$
$$
\rot\vec{v}=
 \left(
  \begin{array}{c}
   \partial_{y}v_{z}-\partial_{z}v_{y}\\
   \partial_{z}v_{x}-\partial_{x}v_{z}\\
   \partial_{x}v_{y}-\partial_{y}v_{x}
  \end{array}
 \right)
$$
$$
\Delta f=\partial^2_{x}f+\partial^2_{y}f+\partial^2_{z}f
$$
$$
\Delta\vec{v}=
\left(
 \begin{array}{c}
  \partial^2_{x}v_{x}+\partial^2_{y}v_{x}+\partial^2_{z}v_{x}\\
  \partial^2_{x}v_{y}+\partial^2_{y}v_{y}+\partial^2_{z}v_{y}\\
  \partial^2_{x}v_{z}+\partial^2_{y}v_{z}+\partial^2_{z}v_{z}
 \end{array}
\right)
$$
\end{minipage}


\item[2.]{\bf Zylinderkoordinaten $(r,\phi,z)$}
\par\smallskip
\begin{minipage}{0.49\textwidth}
\begin{center}
  \includegraphics[width=0.8\textwidth]{Zylinderkoordinaten.eps}
\end{center}
\end{minipage}
\begin{minipage}{0.49\textwidth}
$$
\grad f=
\left(
 \begin{array}{c}
  \partial_{r}f\\
  \frac 1{r}\partial_{\phi}f\\
  \partial_{z}f
 \end{array}
\right)
$$
$$
\div\vec{v}=\frac 1{r}\partial_{r}(rv_{r})+\frac 1{r}\partial_{\phi}v_{\phi}+\partial_{z}v_{z}
$$
$$
\rot\vec{v}=
\left(
 \begin{array}{c}
  \frac 1{r}\partial_{\phi}v_{z}-\partial_{z}v_{\phi}\\
  \partial_{z}v_{r}-\partial_{r}v_{z}\\
  \frac 1{r}(\partial_{r}(rv_{\phi})-\partial_{\phi}v_{r})
 \end{array}
\right)
$$
$$
\Delta f= \frac 1{r}\partial_{r}(r\partial_{r}f)+\frac 1{r^2}\partial_{\phi}^2f+\partial^2_{z}f
$$
\end{minipage}


$$
\Delta\vec{v}=
\left(
 \begin{array}{c}
  \partial^2_{r}v_{r}+\frac 1 {r^2}\partial_{\phi}^2v_{r}+\partial_{z}^2v_{r}+\frac 1{r}\partial_{r}v_{r}-\frac 2{r^2}\partial_{\phi}v_{\phi}
  -\frac{v_{r}}{r^2}\\
  \partial^2_{r}v_{\phi}+\frac 1 {r^2}\partial_{\phi}^2v_{\phi}+\partial_{z}^2v_{\phi}+\frac 1{r}\partial_{r}v_{\phi}+\frac 2{r^2}\partial_{\phi}v_{r}
  -\frac{v_{\phi}}{r^2}\\
  \partial^2_{r}v_{z}+\frac 1 {r^2}\partial_{\phi}^2v_{z}+\partial_{z}^2v_{z}+\frac 1{r}\partial_{r}v_{z}
 \end{array}
\right)
$$


\vfill

\rule[.5mm]{\textwidth}{0.1mm}\\
{\footnotesize Bitte wenden.}

\newpage 


\item[3.]{\bf Kugelkoordinaten $(r, \theta, \phi)$}
\par\smallskip
\begin{center}
  \includegraphics[width=0.4\textwidth]{Kugelkoordinaten.eps}
\end{center}
$$
\grad f=
\left(
 \begin{array}{c}
  \partial_{r}f\\
  \frac 1{r}\partial_{\theta}f\\
  \frac{1}{r\sin\theta}\partial_{\phi}f
 \end{array}
\right)
$$
$$
\div\vec{v}= \frac 1{r^2}\partial_{r}(r^2v_{r})+\frac1{r\sin\theta}\partial_{\theta}(v_{\theta}\sin\theta)
+\frac 1{r\sin\theta}\partial_{\phi}v_{\phi}
$$
$$
\rot\vec{v}=
\left(
 \begin{array}{c}
  \frac{1}{r\sin\theta}(\partial_{\theta}(v_{\phi}\sin\theta)-\partial_{\phi}v_{\theta})\\
  \frac 1{r}\left(\frac 1{\sin\theta}\partial_{\phi}v_{r}-\partial_{r}(rv_{\phi})\right)\\
  \frac 1{r}(\partial_{r}(rv_{\theta})-\partial_{\theta}v_{r})
 \end{array}
\right)
$$
$$
\Delta f=\frac{1}{r^2}\partial_{r}(r^2\partial_{r}f)+\frac{1}{r^2\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta}f)
+\frac 1{r^2\sin^2\theta}\partial^2_{\phi}f
$$

$$
\Delta\vec{v}=
 \left(
  \begin{array}{l}
   \frac 1{r}\partial^2_{r}(rv_{r})+\frac 1{r^2}\partial^2_{\phi}v_{r}+\frac{1}{r^2\sin^2\phi}\partial^2_{\theta}v_{r}+\frac{\cot\phi}{r^2}
   \partial_{\phi}v_{r}-\frac 2{r^2}\partial_{\phi}v_{\phi}-\frac{2}{r^2\sin\phi}\partial_{\theta}v_{\theta}-\frac{2v_{r}}{r^2}-
   \frac{2\cot\phi}{r^2}v_{\phi}\\
   \frac 1{r}\partial^2_{r}(rv_{\phi})+\frac 1{r^2}\partial^2_{\phi}v_{\phi}+\frac{1}{r^2\sin^2\phi}\partial^2_{\theta}v_{\phi}+
   \frac{\cot\phi}{r^2}\partial_{\phi}v_{\phi}-\frac 2{r^2}\frac{\cot\phi}{\sin\phi}\partial_{\theta}v_{\theta}+
   \frac{2}{r^2}\partial_{\phi}v_{r}-\frac{v_{\phi}}{r^2\sin^2\phi}\\
   \frac 1{r}\partial^2_{r}(rv_{\theta})+\frac 1{r^2}\partial^2_{\phi}v_{\theta}+\frac{1}{r^2\sin^2\phi}\partial^2_{\theta}v_{\theta}+
   \frac{\cot\phi}{r^2}\partial_{\phi}v_{\theta}+\frac 2{r^2\sin\phi}\partial_{\theta}v_{r}+
   \frac{2\cot\phi}{r^2\sin\phi}\partial_{\theta}v_{\phi}-\frac{v_{\theta}}{r^2\sin^2\phi}
  \end{array}
 \right)
$$





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\end{deflist}
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\footnotesize
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"Ubungsleitung: Moritz Franosch, mail@Franosch.org, http://www.physik.tu-muenchen.de/lehrstuehle/T35/.


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