first fundamental form

first fundamental form

English

Noun

first fundamental form (uncountable)

1. (differential geometry) 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

${\\displaystyle E={\\vec {x}}_{u}\\cdot {\\vec {x}}_{u}}$

${\\displaystyle F={\\vec {x}}_{u}\\cdot {\\vec {x}}_{v}}$

${\\displaystyle G={\\vec {x}}_{v}\\cdot {\\vec {x}}_{v}}$