= Solution
For a regular <parametric surface> $r(u,v)$, set
$$
E=r_u\cdot r_u,\quad F=r_u\cdot r_v,\quad G=r_v\cdot r_v,\qquad N=\frac{r_u\times r_v}{\|r_u\times r_v\|}.
$$
The <first fundamental form> is the induced <Riemannian metric>
$$
\mathrm I=E\,du^2+2F\,du\,dv+G\,dv^2.
$$
The <second fundamental form>, for this choice of <unit normal vector>, is
$$
\mathrm{II}=e\,du^2+2f_2\,du\,dv+g_2\,dv^2,\quad e=r_{uu}\cdot N,\quad f_2=r_{uv}\cdot N,\quad g_2=r_{vv}\cdot N.
$$
The <Gaussian curvature> is the product of the <principal curvatures>, equivalently the <determinant> of the <shape operator>:
$$
\boxed{K=\frac{eg_2-f_2^2}{EG-F^2}.}
$$
Reversing the normal reverses the <second fundamental form> and both <principal curvatures>, while leaving $K$ unchanged.
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