\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

\begin{description}
\item[(i)] The following equations are written in terms of
cylinderical co-ordinates.  What curves or surfaces do they
represent?
\begin{description}
\item[(a)] $\ds \phi = \frac{\pi}{4}, \, z =2$
\item[(b)] $\ds R^2 + z^2 = 9$
\item[(c)] $\ds R=z\tan \alpha$ where $\alpha$ is a constant
\item[(d)] $\ds R\sin \phi = 1, \, z=0$
\end{description}

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\item[(ii)] The following equations are written in terms of
spherical co-ordinates.  What curves do they represent?
\begin{description}
\item[(a)] $\ds r \cos \theta = 1$
\item[(b)] $\ds \sin \theta = \frac{\pi}{4}$
\item[(c)] $\ds \theta = \frac{\pi}{2}, \, r = \cos \phi = 0$
\item[(d)] $\ds \theta = \frac{\pi}{4}, \, r = \cos \theta = 1$
\end{description}
\end{description}
\vspace{.25in}

{\bf Answer}
\begin{description}
\item[(i)]
\begin{description}
\item[(a)] $\ds \phi = \frac{\pi}{4}, \, z =2$

\setlength{\unitlength}{.5in}
\begin{picture}(4,3)
\put(1,1){\line(0,1){2}}

\put(1,1){\line(1,0){2}}

\put(1,1){\line(-1,-1){1}}

\multiput(1,1)(.2,-.2){6}{\circle*{0.05}}

\multiput(2,0)(0,0.2){10}{\circle*{0.05}}

\put(1,2.8){\line(1,-1){1.5}}

\put(2.3,0.7){\makebox(0,0){2}}

\put(1,0.7){\makebox(0,0){$\frac{\pi}{4}$}}

\put(3,1.5){\makebox(0,0)[l]{Gives a half line}}

\end{picture}


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\item[(b)] $\ds R^2 + z^2 = 9 \Rightarrow x^2 + y^2 + z^2 = 1$
gives a sphere.

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\item[(c)] $\ds R=z\tan \alpha$ where $\alpha$ is a constant

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\begin{center}
$\begin{array}{c}
\epsfig{file=cg-15-1.eps, width=40mm}
\end{array}
\ \ \ 
\begin{array}{l}
\textrm{Gives a half cone}
\end{array}$
\end{center}

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\item[(d)] $\begin{array}{rcl} \ds R\sin \phi = 1, & & z=0 \\ y=1 & & z = 0
\end{array}$ gives a line.

\end{description}

\item[(ii)]
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\begin{description}
\item[(a)] $\ds r \cos \theta = 1 \Rightarrow z = 1$ gives a plane

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\item[(b)] $\ds \sin \theta = \frac{\pi}{4} \Rightarrow \theta =$
constant.  Gives a double cone.

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\item[(c)] $\ds \theta = \frac{\pi}{2}, \, r = \cos \phi = 0$
Gives the y axis.

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\item[(d)] $\ds \theta = \frac{\pi}{4}, \, r = \cos \theta = 1$

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\begin{center}
\epsfig{file=cg-15-2.eps, width=40mm}
\end{center}

circle centre is at $(0,0,1)$ and  the radius is 1 in the plane
$z=1$
\end{description}

\end{description}
\end{document}
