= Solution
For a parametrization $X(u^1,u^2)$, put $g_{ij}=X_i\cdot X_j$. Decompose the <second derivatives> into tangent and <normal components>:
$$
X_{ij}=\Gamma^k_{ij}X_k+b_{ij}\nu.
$$
These tangent coefficients are the <Christoffel symbols>. Differentiate $g_{ij}$ and combine the resulting three identities to obtain
$$
X_{ij}\cdot X_l=\frac12(\partial_i g_{jl}+\partial_jg_{il}-\partial_lg_{ij}),\qquad \boxed{\Gamma^k_{ij}=\frac12g^{kl}(\partial_i g_{jl}+\partial_jg_{il}-\partial_lg_{ij}).}
$$
Here $(g^{kl})$ is the <inverse matrix>. The formula proves that these symbols depend only on the <first fundamental form> and its first derivatives, not on the <second fundamental form>.
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