First fundamental form

English

Noun

1. Template:Lb 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

   $E={\overrightarrow{x}}_u\cdot {\overrightarrow{x}}_u$

   $F={\overrightarrow{x}}_u\cdot {\overrightarrow{x}}_v$

   $G={\overrightarrow{x}}_v\cdot {\overrightarrow{x}}_v$