Let $r(u,v): \mathbb R^2 \rightarrow \mathbb R^3$ be a surface in $\mathbb R^3$. I know how to calculate the Gaussian curvature when both the first and the second fundamental forms are given. Also, it's not quite difficult if only the first fundamental form is known and it is of the form $I=f(u,v)du^2+g(u,v)dv^2$. But what can I do if the coefficient of $du\ dv$ is not $0$ in the first fundamental form? For example, what is the Gaussian curvature if $I = (1+v^2)du^2+2uv\ du\ dv+(1+u^2)dv^2$? Should I replace $u$ and $v$ by linear combinations of them to kill the middle term? Since this involves a lot of calculations, is there any better method?
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While it's true that the computation of $K$ is much easier in orthogonal and especially in isothermal coordinates, finding such coordinates requires solving a PDE. I don't think it would be worth the effort in this case. Your coefficients $E,F,G$ are about as simple as one could have, the partials are immediately found and the determinants are not too bad. Everything is quite doable by hand.
With apologies for using a proprietary system (Maple), here is the general Brioschi formula: