Discussion
The Gaussian curvature of the hyperbolic paraboloid \(\dfrac{x^{2}}{2} - \dfrac{y^{2}}{3} - z = 0\) is :
Answer: A
The Gaussian curvature is always \(\lt 0\).
For a graph \(z = f(x, y)\),
\[ K = \frac{f_{xx}f_{yy} - f_{xy}^2}{(1 + f_x^2 + f_y^2)^2} \]
Here \(f = x^{2}/2 - y^{2}/3\), so \(f_{xx} = 1\), \(f_{yy} = -2/3\) and \(f_{xy} = 0\). The numerator is \(-2/3\) at every point, and the denominator is positive, so \(K \lt 0\) everywhere.
Every point of a hyperbolic paraboloid is a saddle (hyperbolic) point, so the curvature is never 0 or positive, ruling out the other options.
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