= Solution
At a regular point, choose the unit normal $\mathbf n=(\sigma_x\times\sigma_y)/|\sigma_x\times\sigma_y|$. The <first fundamental form> is the induced squared length
$$
I=E\,dx^2+2F\,dx\,dy+G\,dy^2,\quad E=\sigma_x\cdot\sigma_x,\ F=\sigma_x\cdot\sigma_y,\ G=\sigma_y\cdot\sigma_y.
$$
The <second fundamental form> is
$$
II=e\,dx^2+2f\,dx\,dy+g\,dy^2,\quad e=\mathbf n\cdot\sigma_{xx},\ f=\mathbf n\cdot\sigma_{xy},\ g=\mathbf n\cdot\sigma_{yy}.
$$
The <Gaussian curvature> is the determinant of the <shape operator>, $K=(eg-f^2)/(EG-F^2)$, independent of the normal orientation.
For the given graph, $\sigma_x=(1,0,y)$ and $\sigma_y=(0,1,x)$, so with $S=1+x^2+y^2$,
$$
I=(1+y^2)\,dx^2+2xy\,dx\,dy+(1+x^2)\,dy^2,\qquad
\mathbf n=\frac{(-y,-x,1)}{\sqrt S},\qquad II=\frac{2\,dx\,dy}{\sqrt S}.
$$
Therefore
$$
\boxed{K(x,y)=-\frac1{(1+x^2+y^2)^2}.}
$$
The surface is regular everywhere. The printed description “hyperboloid” is a naming error: this graph is a <hyperbolic paraboloid>, and the computation uses the actual parametrization.
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