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\Blatt{3}{WS 2003/2004, 10.11.2003}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

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\item[8.]
{\bf Exponentialreihe}
\par\smallskip
Seien $A$ und $B$ $\mathbb{R}^{d \times d}$-Matrizen.
Zeigen Sie:
\begin{enumerate}
\item 
  \begin{displaymath}
    e^A \, e^B = e^{A+B}
  \end{displaymath}
  Hinweis: Berechnen Sie das Cauchy-Produkt der Reihen $e^A$ und
  $e^B$.
\item Eine Drehung um den Winkel $\varphi$ um die Achse $\vek{n} =
  (n_1,n_2,n_3)^T$ ist eine Hintereinanderausführung von Drehungen um
  die Winkel $\varphi \, n_i$ um die jeweiligen Koordinatenachsen.
\end{enumerate}


\item[9.]
{\bf Koordinatentransformationen}
\par\smallskip
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
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der "Ubungen am 17.11.2003 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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