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\Blatt{7}{WS 2003/2004, 8.12.2003}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[14.]
{\bf Totale Ableitung}
\par\smallskip
Beweisen Sie für die Funktionen $f: (x,y,z,t) \mapsto f(x,y,z,t)$, $g$
und $h$ und $\lambda \in \mathbb{R}$ folgende Eigenschaften der
totalen Ableitung $\mathrm{D}_t$:
\begin{enumerate}
\item Additivität
  \begin{displaymath}
    \mathrm{D}_t ( f+g ) = \mathrm{D}_t f + \mathrm{D}_t g
  \end{displaymath}
\item Homogenität
  \begin{displaymath}
    \mathrm{D}_t ( \lambda f ) = \lambda \, \mathrm{D}_t f
  \end{displaymath}
\item Produktregel
  \begin{displaymath}
     \mathrm{D}_t ( f \cdot g ) = f \, \mathrm{D}_t g + g \, \mathrm{D}_t f
  \end{displaymath}
\item Kettenregel
  \begin{displaymath}
    \mathrm{D}_t ( h \circ g ) = (h' \circ g) \, \mathrm{D}_t g
  \end{displaymath}
\end{enumerate}


\item[15.]
{\bf Rotierende Flüssigkeit}
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Das Geschwindigkeitsfeld einer Flüssigkeit sei 
\begin{displaymath}
  \vek{v}(\vek{x}) = \vek{\omega} \times \vek{x} \,.
\end{displaymath}
\begin{enumerate}
\item Zeigen Sie, dass für Vektoren $\vek{a}, \vek{b},
  \vek{c} \in \mathbb{R}^3$ gilt
  \begin{displaymath}
    \vek{a} \times ( \vek{b} \times \vek{c} ) =
    (\vek{a} \circ \vek{c}) \vek{b} - (\vek{a} \circ
    \vek{b}) \vek{c} \,.
  \end{displaymath}
\item Berechnen Sie
  \begin{displaymath}
    \rot \vek{v}(\vek{x}) = \nabla \times \vek{v}(\vek{x})
    \,,
  \end{displaymath}
  indem Sie die Formel von 1. verwenden.
\item Berechnen Sie $\rot \vek{v}(\vek{x})$ für den Spezialfall
  $\vek{\omega}=(0,0,\omega_3)$ direkt durch Differenzieren. Wie
  lässt sich das Ergebnis auf
  $\vek{\omega}=(\omega_1,\omega_2,\omega_3)$ verallgemeinern?
\item Berechnen Sie $\div \vek{v}(\vek{x})$.
\item Berechnen Sie den Verzerrungstensor
  \begin{displaymath}
    D=(d_{ij})=\frac{1}{2} \left(
      \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial
        x_i} \right) \,.
  \end{displaymath}
\end{enumerate}



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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der "Ubungen am 15.12.2003 um 14.15 Uhr im Raum PH 127
(Garching).\\
"Ubungsleitung: Moritz Franosch, mail@Franosch.org, http://www.physik.tu-muenchen.de/lehrstuehle/T35/.

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