# Gaussian curvature of a z=f(x,y) function [closed]

This is not a homework question.

Just wondering if there is a general formula for the gaussian curvature at point (x,y,f(x,y)) in terms of x, y, and f(x,y).

I didn't see any thing like that on the wikipedia article, but maybe it's hidden behind all these letters/symbols which are not actually described.

By the way, I'm studying computer science. For us, a general formula is one that works most of the time. When it doesn't work, we call this an exception, and it requires special handling :). So here of course, the function is assumed everywhere indefinitely continuous and differentiable and bounded inside any box that would fit the RAM - whatever that means - and also in R3, inside a standard classical euclidean space...

Thank you.

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## closed as off-topic by Ricardo Andrade, Denis Serre, Felipe Voloch, Stefan Kohl, Jeremy RouseNov 26 '14 at 1:06

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "MathOverflow is for mathematicians to ask each other questions about their research. See Math.StackExchange to ask general questions in mathematics." – Ricardo Andrade, Denis Serre, Felipe Voloch, Stefan Kohl, Jeremy Rouse
If this question can be reworded to fit the rules in the help center, please edit the question.

If the surface is in Monge form, i.e. written as the graph of a function $z = f(x,y)$ such that $f(0,0) = f_x(0,0) = f_y(0,0) = 0,$ then there is an easy way to calculate the Gauss curvature. You take a rotation about the $z$-axis which kills off the $xy$-term in the quadratic part of $f.$ Then you will be able to write the surface as $z = \frac{1}{2}(\kappa_1x^2 + \kappa_2y^2) + \cdots,$ where the dots denote terms of order three and above. A rotation is a special orthogonal transformation and does not change the Gauss curvature.
The $\kappa_1$ and $\kappa_2$ will be in terms to the coefficients in the Taylor series of the original expression for $f.$
The numbers $\kappa_1$ and $\kappa_2$ are the principal curvatures of the surface at $(0,0,0).$ Moreover, the $x$-axis and the $y$-axis are the principal directions of the surface at $(0,0,0).$ (Recall that there are always two orthogonal principal directions at each non-umbilic point, i.e. $\kappa_1 \neq \kappa_2,$ of the surface.)The Gauss curvature is given by $K = \kappa_1\kappa_2$ and the mean curvature is given by $H = \frac{1}{2}(\kappa_1 + \kappa_2).$