= Solution
For each $p\in S$, the <first fundamental form> is defined without coordinates by
$$
I_p(v,w)=v\mathbin{\cdot}w,
\qquad v,w\in T_pS,
$$
the restriction of the Euclidean <inner product> to the <tangent space> of $S$.
If $\phi(u,v)$ is a local parametrization and
$$
\xi=\xi^1\partial_u+\xi^2\partial_v,
\qquad
\eta=\eta^1\partial_u+\eta^2\partial_v,
$$
then
$$
d\phi(\xi)=\xi^1\phi_u+\xi^2\phi_v,
\qquad
d\phi(\eta)=\eta^1\phi_u+\eta^2\phi_v.
$$
Taking their dot product gives
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,
$$
where
$$
\boxed{E=\phi_u\cdot\phi_u,\qquad
F=\phi_u\cdot\phi_v,\qquad
G=\phi_v\cdot\phi_v}.
$$
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