first fundamental form

Noun

Meaning

first fundamental form (uncountable)

(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




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}}

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