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\bf{Question}

\quad Sketch or describe the level sets for each of the following
functions $f:\br^3\to\br$\ :
\medskip
\begin{center}
\begin{tabular}{lllll}$f(x_1,x_2,x_3)\ =\hspace{0.5cm}
$&(a)&$x_1^2+2x_2^2+3x_3^2$ &(d)& $x_1^2$\\
&(b)&$x_1^2-2x_2^2+3x_3^2$&(e)&$x_1^2+x_2^2+x_3$\\
&(c)&$x_1^2+x_2^2$&(f)&$x_1^2+x_3$
\end{tabular}
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\bf{Answer}

\begin{description}
\item{(a)}
Ellipsoids

\item{(b)}
$c=0$: cone, axis = y-axis

$c>0$: hyperboloid of 1 sheet

$c<0$: hyperboloid of 2 sheets

\item{(c)}
Cylinders, axis = x-axis

\item{(d)}
Plane pairs $x=\pm \sqrt{c}$ ($c>0$)

$(y,z)$ plane $x=0$ ($c=0$)

empty when $c<0$.

\item{(e)}
Paraboloids with axis = x-axis (put $x^2+y^2=r^2$ and compare (d)

\item{(f)}
Parabolic 'troughs': in (d) above, replace $y$ by $x$ and then
slide the whole picture along the third (=new y-) axis.
\end{description}




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