First fundamental form
= First fundamental form
{title2=$I$}
{wiki}
The first fundamental form of a surface $S\subseteq\mathbb R^3$ is the restriction of the ambient <inner product> to each <tangent space> $T_pS$. For a parametrized surface $X(u,v)$, it is
$$
I=E\,du^2+2F\,du\,dv+G\,dv^2,
\qquad
E=X_u^2,\quad F=X_u\cdot X_v,\quad G=X_v^2.
$$