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\Blatt{2}{WS 2005/2006, 31.10.2005}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

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\item[6.]
{\bf Gau{\3}'scher Integralsatz}
\par\smallskip
Beweisen Sie den Gau{\3}'schen Integralsatz f"ur
\begin{enumerate}
\item den Spezialfall des Geschwindigkeitsfelds in einer zylindrischen
  Röhre (Achse ist die x-Achse), die von einer Flüssigkeit mit
  Geschwindigkeit $\mathbf{v}(\mathbf{x}) = \mathbf{v}(x,y,z) =
  (v(x),0,0)$ durchstömt wird.  Das Integrationsvolumen sei der
  Zylinder. 
  
  Warum ist bei einer inkompressiblen Flüssigkeit $\div
  \mathbf{v}(\mathbf{x})=0$?
  
  Warum gilt für die Dichte 
  \begin{displaymath}
    \frac{\partial
      \varrho(\mathbf{x},t)}{\partial t}=-\div [ \varrho(\mathbf{x},t)
    \mathbf{v}(\mathbf{x},t) ] \,?
  \end{displaymath}
\item ein beliebiges Vektorfeld $\mathbf{v}(\mathbf{x})$ und einen
  achsparallelen Quader als Integrationsvolumen.
\end{enumerate}


\item[7.]
{\bf Rotierende Flüssigkeit}
\par\smallskip
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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\item[8.]
{\bf Koordinatentransformationen}
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Ein Diffeomorphismus $x: {\Bbb R}^3 \to {\Bbb R}^3$,
\[
x: \vek{q} = (q_1, q_2, q_3) \mapsto 
\left(\begin{array}{c}
x_1(q_1, q_2, q_3)\\
x_2(q_1, q_2, q_3)\\
x_3(q_1, q_2, q_3)
\end{array}\right)
\]
heißt Koordinatensystem des ${\Bbb R}^3$.
\begin{enumerate}
\item Zeigen Sie, dass für die Jacobi-Matrizen der Abbildung $x$ und
  deren Umkehrabbildung $q$ gilt
  \begin{displaymath}
    \partial x(\vek{q}) = [ \partial q(\vek{x}) ]^{-1}
  \end{displaymath}
\item Zeigen Sie, dass $( \frac{\partial x}{\partial q_1},
  \frac{\partial x}{\partial q_2},\frac{\partial x}{\partial q_3} )$ eine
  Basis des ${\Bbb R}^3$ ist.
\item Berechnen sie das Verhalten der Komponenten $v^j$ des
  Vektorfeldes
\[
\vek{v}(\vek{x}) =  \sum_j v^j(\vek{x})\, \frac{\partial x}{\partial q_j}
\]
unter einer Koordinatentransformation $(q_1,q_2,q_3) \mapsto
(q'_1,q'_2,q'_3)$.
\end{enumerate}



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\footnotesize
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Besprechung der Übungen am Montag den 7.11.2005 um 14-16 Uhr im
Raum PH 121 (Garching) und am Dienstag, den 8.11.2005 um 9-11 Uhr im
Raum PH HS3 (Garching).\\
Übungsleitung: Moritz Franosch, mail@Franosch.org,
http://www.physik.tu-muenchen.de/lehrstuehle/T35/.


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