= Solution
For a regular parametrization choose the <unit normal> $n=(r_u\times r_v)/|r_u\times r_v|$. The <first fundamental form> is the squared length induced on tangent vectors:
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,\qquad E=r_u\cdot r_u,\quad F=r_u\cdot r_v,\quad G=r_v\cdot r_v.
$$
The <second fundamental form> records their normal bending:
$$
II=L\,du^2+2M\,du\,dv+N\,dv^2,\qquad L=n\cdot r_{uu},\quad M=n\cdot r_{uv},\quad N=n\cdot r_{vv}.
$$
The <Gaussian curvature> is
$$
\boxed{K=\frac{LN-M^2}{EG-F^2}.}
$$
Regularity ensures $EG-F^2>0$.
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