= Solution
For a regular parametrization $r(u,v)$, the <first fundamental form> is the induced metric
$$
\boxed{ds^2=E\,du^2+2F\,du\,dv+G\,dv^2,\quad E=r_u\cdot r_u,\ F=r_u\cdot r_v,\ G=r_v\cdot r_v.}
$$
It measures the lengths and angles of tangent vectors. A <geodesic> is a curve whose unit tangent has zero covariant derivative along itself. For an arc-length parametrized <surface> curve in Euclidean three-space, this means its acceleration is normal to the <surface>, or $\gamma''\cdot r_u=\gamma''\cdot r_v=0$.
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