Answer
\[
r \cos \theta=x, r \sin \theta=y, \quad r^{2} \sin 2 \theta=z
\]
Work Step by Step
\[
\begin{array}{l}
\qquad r \sin \theta=y , r \cos \theta=x \\
\Rightarrow z=2 x y=2(r \cos \theta)(r \sin \theta)=2 r^{2} \cos \theta \sin \theta=r^{2} \sin 2 \theta \quad(\because \sin 2 \theta=2 \cos \theta \sin \theta)
\end{array}
\]
So, the surface can be represented parametrically in terms of r and
$\theta$ аs
\[
x=r \cos \theta, y=r \sin \theta, \quad z=r^{2} \sin 2 \theta
\]
\[
r \cos \theta=x, r \sin \theta=y, \quad r^{2} \sin 2 \theta=z
\]
