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Consider the surface $\{(x,y,F(x,y))\} $ where $F:\mathbb{R^2} \to \mathbb{R}$ is smooth. How would would evaluate the Gaussian curvature at the general point $(x,y,F(x,y))$? I've tried writing out the first fundamental form and using the Brioschi formula, but the algebra is quickly becoming untenable, is there any easier way?

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up vote 1 down vote accepted

Look here

under the subheading "Alternative formulas". If you want a proof, compute the first and second fundamental forms first.

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