= Solution
For a chosen <unit normal> field $N$, the <second fundamental form> is
$$
II_p(v,w)=\langle-dN_p(v),w\rangle.
$$
In a local parametrization $\phi$, its coefficients, up to the equivalent simultaneous sign convention, are
$$
e=N\cdot\phi_{uu},
\qquad
f=N\cdot\phi_{uv},
\qquad
g=N\cdot\phi_{vv}.
$$
The <Gaussian curvature> is the <determinant> of the <shape operator>, hence
$$
K=\det(-dN)=\frac{eg-f^2}{EG-F^2}.
$$
<Theorema Egregium> states that $K$ is determined entirely by the first fundamental form. In particular, it is intrinsic and is preserved by every <local isometry>.
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