first fundamental form

Noun

 * 1)  the Riemannian metric for 2-dimensional manifolds, i.e. given a surface with regular parametrization x(u,v), the first fundamental form is a set of three functions, {E, F, G}, dependent on u and v, which give information about local intrinsic curvature of the surface. These functions are given by
 * $$ E = \vec x_u \cdot \vec x_u $$
 * $$ F = \vec x_u \cdot \vec x_v $$
 * $$ G = \vec x_v \cdot \vec x_v $$