Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 3 / 2H / 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 normalThe first fundamental form coefficients areSince $r_{xx}=r_{yy}=0$ 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: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.