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How can I calculate the Gauss Curvature?

Let $f(x,y)$ be a continuous function defined on $[a,b] \times[c,d] \subset \mathbb{R}^2$. Let $g(u,v) = \frac{\partial f(x,y)}{\partial u} \frac{\partial f(x,y)}{\partial v}$ be a gauss-type curvature on $[a,b] \times [c,d]$. Given that $g(u,v)$ is symmetric, determine the Gauss curvature $K_g(u,v)$ of the surface $f(x,y) = c$.
The book does not give a general formula for the Gauss curvature, but only one for a surface with an upper flat of the form $x^2 + y^2 — a^2 = 1$. How could I generalize this to arbitrary surfaces?

A:

For surfaces with an upper flat I use the formula for a locally defined and positive curvature function:
If $(u,v)$ is a point on the surface (with $u \in (a,b),v \in (c,d)$) with a quadratic differential
$$Q(u,v) := \det \begin{pmatrix} u & v \\ \frac{\partial^2 f}{\partial u^2} & \frac{\partial^2 f}{\partial u\partial v} \\ \frac{\partial^2 f}{\partial v^2} & \frac{\partial^2 f}{\partial v\partial u}\end{pmatrix}$$
then the corresponding Gauss-curvature is given by:
K_g(u,v) = \frac{Q(u,v)}{\det \begin{pmatrix} u & v \\ \frac{\partial^2 f}{\partial u^2} & \frac{\partial^2 f}{\partial u\partial v}

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