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\Blatt{8}{WS 2008/2009, 22.1.2009}
         {Theoretische Biophysik I}
         {Prof. J.~L.~van~Hemmen}

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\item[10.]
{\bf Parametertransformationen und Tensoren}
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Die Abbildung $\vek{x}$
\begin{equation}
  (q^\kappa) \mapsto \vek{x}(q^\kappa)\,, 
\end{equation}
wobei $(q^\kappa) \in {\Bbb R}^n$ und $\vek{x}(q^\kappa) \in {\Bbb
  R}^{n+1}$, sei eine Karte einer Hyperfläche des ${\Bbb R}^n$.
\begin{enumerate}
\item Warum ist $(\partial_\kappa \vek{x})$ eine Basis des ${\Bbb R}^n$?
\item Die Koordinaten $a^\kappa$ eines Vektors $\vek{a}$ der
  Hyperfläche seien gegeben durch
  \begin{equation}
    \vek{a} = a^\kappa \partial_\kappa \vek{x} \,.
  \end{equation}
  Warum sind die $a^\kappa$ durch obige Gleichung eindeutig
  festgelegt?
\item Berechnen Sie das Verhalten der $a^\kappa$ unter einer
  Parametertransformation $q^\kappa \mapsto q'^\kappa$.
\item Warum sind die $a^\kappa$ Komponenten eines kontravarianten
  Tensors?
\item Warum sind die
  \begin{equation}
    a_\lambda := a^\mu g_{\mu\lambda}
  \end{equation}
  Komponenten eines kovarianten Tensors?
\item Warum ist das Skalarprodukt
  \begin{equation}
    \vek{a} \vek{b} = a^\kappa \partial_\kappa \vek{x} \, b^\lambda \partial_\lambda \vek{x} 
  \end{equation}
  invariant gegenüber Parametertransformation (\emph{Tensor 0. Stufe}
  oder \emph{Skalar})?
\item Warum ist es sinnvoll
  \begin{equation}
    {\partial^\lambda}_\mu := \frac{\partial q^\lambda}{\partial
      {q'}^\mu}
  \end{equation}
  als gemischt kontra-kovarianten Tensor aufzufassen?
\item Warum ist
  \begin{equation}
    {\partial_\lambda}^\mu = \frac{\partial {q'}^\lambda}{\partial q^\mu}\,,
  \end{equation}
  somit $ {\partial^\lambda}_\mu = ({\partial_\lambda}^\mu)^{-1}$\,?
  Denken Sie daran, dass sich hierbei die Indizes laut Definition mit
  dem metrischen Tensor nach oben und unten schieben lassen.
\end{enumerate}


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\footnotesize
\rule[.5mm]{\textwidth}{0.1mm}\\
Besprechung der Übungen am Donnerstag 29.1.2009 um 8.30 Uhr im
Raum PH 227. \\
Übungsleitung: Dr. Moritz Franosch, mail@Franosch.org,
http://www.t35.ph.tum.de.

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