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\bf{Question}

In each case (i), (ii) find the discriminant
$\Delta\subset\mathbf{R}^2$ for the map
$F:\mathbf{R}^3\rightarrow\mathbf{R}^2$. Describe the geometric
form of the set $F^{-1}(s)\subset\mathbf{R}^3$ for $s$ belonging
to each connected region of the complement of $\Delta$ in
$\mathbf{R}^2$, and also for $s$ belonging to $\Delta$.

\begin{description}

\item[(i)]
$F(x_1,x_2,x_3)=(x_1-(x_2^2+x_3^2),2x_1)$

\item[(ii)]
$F(x_1,x_2,x_3)=(x_1^2-x_2^2-x_3^2,x_1^2+x_2^2+x_3^2).$

\end{description}

[Hint: in (ii) let $x_2^2+x_3^2=r^2$.]




\bf{Answer}

\begin{description}
\item{(i)}
$\ds DF(x) = \left ( \begin{array}{ccc} 1 & -2x_2 & -2x_3 \\ 2 & 0
& 0
\end{array} \right )$ which has rank $<2$ if and only if $x_2=x_3=0$,
i.e singular set $\Sigma_1$ is $x_1$-axis.

We have $F(x_1,0,0)=(x_1,2x_1)$ so discriminant $\Delta$ is the
line $y_2=2y_1$.

The set $F^{-1}(a,b)$ is the intersection of the line $x_1 =
\frac{b}{2}$ with the locus $x_2^2+x_3^2 = \frac{b}{2} - a$:
cylinder if $b > 2a$, empty if $b < 2a$.

Thus $F^{-1}(a,b)$ is a circle if $b >2a$, empty if $b<2a$. Also
one point if $b=2a$.

\item{(ii)}
$\ds DF(x) = \left ( \begin{array}{ccc} 2x_1 & -2x_2 & -2x_3 \\
2x_1 & 2x_2 & 2x_3 \end{array} \right )$: rank $<2$ when $x_1=0$
or $x_2=x_3=0$.

We have $F(0,x_2,x_3)=(x_2^2 + x_3^2)(-1,1)$ and $F(x_1,0,0) =
x_1^2(1,1)$. So $\Delta$ is two half-lines.

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The set $F^{-1}(a,b)$ is the intersection of the sphere $x_1^2 +
x_2^2 + x_3^2 =b$ and the hyperboloid $x_1^2-x_2^2-x_3^2=a$.

The 'sphere' is empty if $b<0$, radius $\sqrt{b}$ otherwise. The
hyperboloid has two sheets if $a>0$, 1 sheet if $a<0$ (cone when
$a=0$).

The nearest points of the hyperboloid to the origin are at
distance $\sqrt{| a |}$.

Hence $\ds F^{-1}(a,b): \left. \begin{array}{ll} b < |a| & :
\textrm{empty} \\ b > |a| & : \textrm{two circles} \end{array}
\right \}$

and for $(a,b) \in\Delta$, $\ds F^{-1}(a,b)= \left \{
\begin{array}{cc} \textrm{two points} & (\pm |a|, 0,0) \ \textrm{if} \
a>0\\ \textrm{circle} & \textrm{if }a<0 \end{array} \right)$.
\end{description}


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