What is "first fundamental form"

n. (context differential geometry English) 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

In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space. The first fundamental form is denoted by the Roman numeral I,

I(x, y) = ⟨x, y⟩.

Let X(u, v) be a parametric surface. Then the inner product of two tangent vectors is

$$\begin{align} & {} \quad \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\ & = ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\ & = Eac + F(ad+bc) + Gbd, \end{align}$$

where E, F, and G are the coefficients of the first fundamental form.

The first fundamental form may be represented as a symmetric matrix.

$$\!\mathrm{I}(x,y) = x^T \begin{pmatrix} E & F \\ F & G \end{pmatrix}y$$