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\Blatt{2}{WS 2003/2004, 3.11.2003}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

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\item[6.]
{\bf Exponentialreihe}
\par\smallskip
Sei $D$ eine $\mathbb{R}^{d \times d}$-Matrix.
Zeigen Sie:
\begin{enumerate}
\item Die Reihe 
  \begin{displaymath}
    e^D := \sum_{k=0}^\infty \frac{D^k}{k!} 
  \end{displaymath}
  konvergiert.
\item $\lim\limits_{n \to \infty} ( I + \frac{1}{n} D )^n = e^D$
\end{enumerate}

\vspace{2cm}

\item[7.]
{\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}







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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der "Ubungen am 10.11.2003 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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