%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\schrift{11pt}
\input format


\begin{document}

\Blatt{8}{WS 2009/2010, 10.12.2009}
         {Mechanik der Kontinua}
         {Prof. J.~L.~van~Hemmen}

\begin{deflist}{20.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\item[24.]
{\bf Vektordifferentialausdrücke in beliebigen krummlinigen Koordinaten}
\par\smallskip
Gegeben sei das orientierungstreue Koordinatensystem
$x: {\Bbb R}^3 \to {\Bbb R}^3$,
\[
x: \vek{q} = (q_1, q_2, q_3) \mapsto 
\left(\begin{array}{c}
x_1(q_1, q_2, q_3)\\
x_2(q_1, q_2, q_3)\\
x_3(q_1, q_2, q_3)
\end{array}\right)
\]
aus Aufgabe 8. Seien ${\partial x}$ und ${\partial q}$ die
Jacobi-Matrizen der Abbildung $x$ bzw. deren Umkehrabbildung $q$, die
Vektoren
\begin{displaymath}
  \vek{e}_i(\vek{q}) := \frac{\partial x}{\partial q_i}(\vek{q})
\end{displaymath}
und der metrische Tensor $g=(g_{ij})$ mit
\begin{displaymath}
  g_{ij}(\vek{q}) := \vek{e}_i(\vek{q}) \circ \vek{e}_j(\vek{q}) \,.
\end{displaymath}
\begin{enumerate}
\item Zeigen Sie, dass 
  \begin{displaymath}
    \det \partial x = \sqrt{\det g} \,.
  \end{displaymath}
  und dass $g$ invertierbar ist.
\item % MEHR ERLÄUTERNDEN TEXT HINZUFÜGEN!
  Der Vektor $\vek{v}$ sei
  \begin{displaymath}
    \vek{v}(\vek{x}) =  \sum_j v^j(\vek{x})\, \vek{e}_j \,.
  \end{displaymath}
  Bezeichnen wir den Vektor $\vek{v}$ in $q$-Koordinaten als
  \begin{displaymath}
    \vek{v}^q = \left( 
      \begin{array}{c}
        v^1 \\
        v^2 \\
        v^3 
      \end{array} \right) \,.
  \end{displaymath}
  Zeigen Sie, dass
  \begin{displaymath}
    \vek{v}^q = \partial q \, \vek{v}
  \end{displaymath}
  und
  \begin{displaymath}
    v^i = \sum_j g^{-1}_{ij} \vek{e}_j \circ \vek{v} \,.
  \end{displaymath}
\item Zeigen Sie, dass für das Skalarprodukt zweier Vektoren $\vek{u}$
  und $\vek{v}$ gilt
  \begin{displaymath}
    \vek{u} \circ \vek{v} = {\vek{u}^q}^T \, g \, \vek{v}^q \,.
  \end{displaymath}
\item Zeigen Sie, dass für den Gradienten $(\nabla \Phi)^q$ in
  $q$-Koordinaten mit $\nabla^q := (\partial_{q_1}, \partial_{q_2},
  \partial_{q_3} )$ gilt
  \begin{displaymath}
    (\nabla \Phi)^q = g^{-1} \nabla^q \Phi \,.
  \end{displaymath}
\item Zeigen Sie, dass für die Divergenz gilt 
  \begin{displaymath}
    \nabla \circ \vek{v} =
    \frac{1}{\sqrt{\det g}} \sum_i \frac{\partial}{\partial q_i} \left(
      \sqrt{\det g} \, v^i \right) \,.
  \end{displaymath}
  Hinweis: Zeigen Sie zunächst, dass
  \begin{displaymath}
    (\nabla \circ \vek{v}) \det \partial x = 
    \frac{\partial}{\partial q_1} 
    \left[ \vek{v} \circ \left( \vek{e}_2 \times \vek{e}_3 \right) \right] +
    \frac{\partial}{\partial q_2} 
    \left[ \vek{v} \circ \left( \vek{e}_3 \times \vek{e}_1 \right) \right] +
    \frac{\partial}{\partial q_3} 
    \left[ \vek{v} \circ \left( \vek{e}_1 \times \vek{e}_2 \right)
    \right]
  \end{displaymath}
  und dass für eine beliebige Matrix $A \in {\Bbb R}^{3 \times 3}$ und
  beliebige linear unabhängige Vektoren $\vek{e}_1, \vek{e}_2,
  \vek{e}_3 \in {\Bbb R}^3$ gilt
  \begin{displaymath}
    \det(A \vek{e}_1, \vek{e}_2, \vek{e}_3) +
    \det(\vek{e}_1, A \vek{e}_2, \vek{e}_3) +
    \det(\vek{e}_1, \vek{e}_2, A \vek{e}_3) =
    \Sp(A) \, \det(\vek{e}_1, \vek{e}_2, \vek{e}_3) \,.
  \end{displaymath}
\item Zeigen Sie, dass für den Laplace-Operator gilt
  \begin{displaymath}
    \Delta \Phi = \frac{1}{\sqrt{\det g}} \sum_{ik} \frac{\partial}{\partial q_i} \left(
      \sqrt{\det g} \, g^{-1}_{ik} \frac{\partial \Phi}{\partial q_k} \right) \,.
  \end{displaymath}

\item Schreiben Sie den Gradienten, die Divergenz und den
  Laplace-Operator in Zylinderkoordinaten
  \begin{displaymath}
    x(\vek{q}) = x(r,\varphi,h) =
    \left(\begin{array}{c}
        r \cos(\varphi) \\
        r \sin(\varphi) \\
        h
      \end{array}\right)  
  \end{displaymath}

\end{enumerate}

\vspace{4ex}

\textbf{Anmerkung:} Für das Vektorprodukt zweier Vektoren $\vek{u}$ und
$\vek{v}$ gilt
\begin{displaymath}
  \vek{u} \times \vek{v} = \sum_{ijkl} \varepsilon_{ijk} u^i v^j g^{-1}_{kl}
  \sqrt{\det g} \, \vek{e}_l \,,
\end{displaymath}
was man unter Verwendung von $(\vek{u} \times \vek{v})^i = \sum_j
g^{-1}_{ij} \vek{e}_j \circ (\vek{u} \times \vek{v})$ zeigen kann, wir
jedoch nicht benötigen.




\par\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{deflist}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfill 

\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der Übungen Montags 12-14 PH127, Mittwochs 8.30-10 HS2 und
Donnerstags 10-12 HS3. \\
Übungsleitung: Dr. Moritz Franosch, mail@Franosch.org,
http://www.t35.ph.tum.de.


\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
