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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
Classwork 8 (February 27, 2013) \\ 
Cylindrical Coordinates
\end{center}
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\noindent 1. Use cylindrical coordinates to calculate
the area of a circle of radius $R$.
\bigskip

\noindent 2. Calculate 
\begin{eqnarray} \nonumber
  \nabla\times\hat{z}\,\ln(\rho)
\end{eqnarray} 
in cylindrical coordinates.
\bigskip

\noindent 3. Show Oersted's law
\begin{eqnarray} \nonumber
  \oint \vec{H}\cdot d\vec{r}\ =\ I
\end{eqnarray} 
for the magnetic potential 
\begin{eqnarray} \nonumber
  \vec{A} &=& -\hat{z}\,\frac{\mu_0\,I}{2\pi}\,\ln(\rho)\,,~~
  \vec{B}\ =\ \nabla\times\vec{A}\,, \\ \nonumber
  \vec{H} &=& \mu_0^{-1}\vec{B}\,.
\end{eqnarray} 
\bigskip

\noindent 4. Find the acceleration $\vec{a}$ in cylindrical coordinates.
\vfil\newpage

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
Classwork 9 (March 1, 2013) \\ 
Cylindrical Coordinates
\end{center}
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\noindent 5. Completion of the square: We have
\begin{eqnarray} \nonumber
  x^2 + b\,x + c
\end{eqnarray} 
and want this in the form
\begin{eqnarray} \nonumber
  x'^{\,2} + c'\ .
\end{eqnarray} 
What are the values of $x'$ and $c'$?
\bigskip

\noindent 6. In cylindrical coordinates the equation of 
an ellipse is given by
$$ \frac{p}{\rho} = 1 + \epsilon\,\cos(\phi)\,,~~p>0 $$ 
with Cartesian coordinates $x=\rho\,\cos(\phi)$ and $y=\rho\,
\sin(\phi)$. Assume $0<e<1$ for the eccentricity and transform 
the solution into the form
$$ \frac{x'^2}{a^2}+\frac{y'2}{b^2} = 1\ .$$ 
This means, derive the definitions of $x'$, $y'$, major half-axis
$a$ and minor half-axis $b$ in terms of $x$, $y$, $p$ and~$\epsilon$. 
\bigskip

\noindent 7. Use the definition
$$ \left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1\,. $$ 
to calculate the area of an ellipse. Hint: Make a substitution, so that 
it becomes reduced to the area of a circle.
\bigskip

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