= Solution
For a local parametrization $\phi(u,v)$, write the <first fundamental form> and <second fundamental form> as
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,
\qquad
II=e\,du^2+2f\,du\,dv+g\,dv^2.
$$
The <Gaussian curvature> is
$$
\boxed{K=\frac{eg-f^2}{EG-F^2}.}
$$
To prove the <Theorema Egregium>, set $g_{ij}=\langle\phi_i,\phi_j\rangle$ and decompose
$$
\phi_{ij}=\Gamma^k_{ij}\phi_k+h_{ij}N.
$$
Differentiating $g_{ij}$ and solving for the tangential coefficients gives
$$
\Gamma_{ijk}
=\frac12\left(\partial_i g_{jk}+\partial_jg_{ik}-\partial_kg_{ij}\right),
$$
so every <Christoffel symbol> depends only on $E,F,G$ and their first derivatives. Equality of mixed third derivatives yields the <Gauss equation>
$$
R_{1212}=h_{11}h_{22}-h_{12}^2=eg-f^2.
$$
The <Riemann curvature tensor> is formed from first derivatives and quadratic products of the Christoffel symbols, and hence from the first and second derivatives of $E,F,G$. Consequently
$$
\boxed{K=\frac{R_{1212}}{\det(g_{ij})}
=\frac{R_{1212}}{EG-F^2}}
$$
depends only on the first fundamental form.
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