= Solution
For a regular <parametric surface> $\mathbf r(u,v)$, its <tangent vectors> $\mathbf r_u,\mathbf r_v$ are linearly independent. The <first fundamental form> is the induced <inner product> on the <tangent plane>, written
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,\qquad
E=\mathbf r_u\cdot\mathbf r_u,\quad F=\mathbf r_u\cdot\mathbf r_v,\quad G=\mathbf r_v\cdot\mathbf r_v.
$$
It measures squared <lengths> and <angles>; its <determinant> satisfies $EG-F^2=|\mathbf r_u\times\mathbf r_v|^2>0$. Choose the <unit normal>
$$
N=\frac{\mathbf r_u\times\mathbf r_v}{\sqrt{EG-F^2}}.
$$
The <second fundamental form>, with <shape operator> $S=-dN$, is
$$
II=e\,du^2+2f\,du\,dv+g_2\,dv^2,\qquad
e=\mathbf r_{uu}\cdot N,\quad f=\mathbf r_{uv}\cdot N,\quad g_2=\mathbf r_{vv}\cdot N.
$$
The subscript distinguishes the coefficient $g_2$ from a <genus> or a <Riemannian metric>. In these coordinates the <shape operator> has matrix $I^{-1}II$, so the <Gaussian curvature>, the product of its two <principal curvatures>, is
$$
\boxed{\kappa=\det S=\frac{eg_2-f^2}{EG-F^2}.}
$$
Reversing $N$ reverses $II$ and both <principal curvatures>, leaving the <Gaussian curvature> unchanged.
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