= Solution
Parametrize the surface by $r(x,y)=(x,y,xy)$. Its tangent vectors are $r_x=(1,0,y)$ and $r_y=(0,1,x)$. Choose the upward <unit normal>
$$
N=\frac{(-y,-x,1)}{\sqrt{1+x^2+y^2}}.
$$
The <first fundamental form> coefficients are
$$
E=1+y^2,\qquad F=xy,\qquad G=1+x^2,
\qquad EG-F^2=1+x^2+y^2.
$$
Since $r_{xx}=r_{yy}=0$ and $r_{xy}=(0,0,1)$, the <second fundamental form> coefficients are $e=g=0$ and $f=(1+x^2+y^2)^{-1/2}$. The determinant of the <shape operator> gives the <Gaussian curvature>:
$$
\boxed{K=\frac{eg-f^2}{EG-F^2}=-\frac1{(1+x^2+y^2)^2}.}
$$
Changing the normal's sign changes both second-form factors and leaves this determinant unchanged. Although the question calls the surface a hyperboloid, $z=xy$ is a <hyperbolic paraboloid>; the calculation uses the stated surface equation.
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