1.Nounthe 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
=
x
→
u
⋅
x
→
u
{\displaystyle E={\vec {x}}_{u}\cdot {\vec {x}}_{u}}
;
F
=
x
→
u
⋅
x
→
v
{\displaystyle F={\vec {x}}_{u}\cdot {\vec {x}}_{v}}
;
G
=
x
→
v
⋅
x
→
v
{\displaystyle G={\vec {x}}_{v}\cdot {\vec {x}}_{v}}
"