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\Blatt{1}{WS 2003/2004, 24.10.2003}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

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\item[1.]
{\bf Tensorzerlegung}
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Zeigen Sie, dass sich jeder reelle (kovariante) Tensor 2. Ordnung
$t_{ij}$ als Summe eines spurfreien symmetrischen, eines
antisymmetrischen und eines Tensors der Form $c\, \delta_{ij}$
schreiben l"asst, wobei $c\in {\Bbb R}$ und $\delta_{ij}$ das
Kronecker-Delta bezeichnet.


\item[2.]
{\bf Kreuzprodukt}
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Zeigen Sie dass es zu jedem $\mbox{\boldmath{$\omega$}} \in {\Bbb
  R}^3$ eine eindeutige antisymmetrische Matrix $\Omega$ gibt, f"ur
die f"ur alle ${\bf v} \in {\Bbb R}^3$ gilt
\[
\Omega \, {\bf v} =  \mbox{\boldmath{$\omega$}} \times  {\bf v}
\]

\item[3.]
{\bf Drehung}
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Sei $\Omega$ eine antisymmetrische $3 \times 3$ Matrix und
$\varepsilon>0$ klein.

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\item[(i)] 
Interpretieren Sie geometrisch die Abbildung
\[
D(\varepsilon)\, {\bf v} = (\one + \varepsilon\, \Omega) \, {\bf v}\ .
\]
Benutzen Sie dazu die Darstellung von $\Omega$ aus Aufgabe 2 und
rechnen Sie in einem Koordinatensystem, in dem gilt {\boldmath{$\omega$}}$=(0,0,1)^T$.

\item[(ii)] Berechnen Sie die n-te Iteration $D(\varepsilon)^n\, {\bf
    v}$.
\item[(iii)] Berechnen und interpretieren Sie $\lim_{n\to\infty}
  D(\varphi/n)^n$.
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\item[4.]
{\bf Dehnung}
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Argumentieren Sie geometrisch, analog zu 3., dass der Tensor $c\,
\delta_{ij}$ aus Aufgabe 1 einer Dehnung entspricht.

\item[5.]
{\bf Ableitungen eines Vektorfelds}
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Zeigen Sie, dass die Ableitung $\nabla_i u_j = \frac{\partial
  u_j(x_1,x_2,x_3)}{\partial x_i}$ des Vektorfelds $(x_1,x_2,x_3)
\mapsto {\bf u}(x_1,x_2,x_3) \in {\Bbb R}^3$ ein Tensor ist.
Wenden Sie die Zerlegung aus 1. auf die Ableitung des Vektorfelds an.
Dr"ucken Sie das Resultat mittels der aus der Vektoranalysis bekannten
Operatoren $\rot$ und $\div$ aus.

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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der "Ubungen am 3.11.2003 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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