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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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up vote 3 down vote accepted

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:

E := 1+v^2; F := 2*u*v; G := 1+u^2;
A := Matrix([[-diff(E,v,v)/2+diff(F,u,v)-diff(G,u,u)/2, diff(E,u)/2, diff(F,u)-diff(E,v)/2], [diff(F,v)-diff(G,u)/2, E, F], [diff(G,v)/2, F, G]]);
B := Matrix([[0, diff(E,v)/2, diff(G,u)/2], [diff(E,v)/2, E, F], [diff(G,u)/2, F, G]]);
K := simplify((Determinant(A)-Determinant(B))/(E*G-F^2)^2);

Output: $$K=\frac{u^2+v^2}{(1+u^2+v^2-3u^2v^2)^2}$$

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Thank you very much for the answer. Would you please let me know the method of calculation when I am not allowed to use a computer? – ShinyaSakai Jan 5 '13 at 8:51
The method is: (1) find the partial derivatives of $E,F,G$ which appear in the Brioschi formula; (2) calculate two determinants of size $3$ which appear in the Brioschi formula; (3) subtract one determinant from the other, and divide the result by $(EG-F^2)^2$. – user53153 Jan 5 '13 at 8:58
Thanks. Let $I=\omega_1^2+\omega_2^2$, if I should write down $\omega_1$ and $\omega_2$ explicitly when the first fundamental is known, what can I do? I tried a couple of ways, but the answer was still out of reach. If there is an easy method, would you please do me the favor to explain it to me? If it's not easy to explain, I would initiate another question. Thanks again. – ShinyaSakai Jan 7 '13 at 2:58
@ShinyaSakai I don't understand what $\omega_1$ and $\omega_2$ are in your question. – user53153 Jan 7 '13 at 3:05
Sorry, I thought these notations were universal. In the calculation of Gaussian curvature $K$ from the first fundamental form $I$, we let $I=\omega_1^2+\omega_2^2$, and $d\omega_1 = \omega_{12} \Lambda \omega_2$, then $d\omega_{12}=-K\omega_1 \Lambda \omega_2$. I am eager to know how the way to find $\omega_1$ and $\omega_2$ in my example. – ShinyaSakai Jan 7 '13 at 5:32

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