First fundamental form in the context of Dot product

A parametric surface is a surface in the Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ which is defined by a parametric equation with two parameters ⁠${\displaystyle \mathbf {r} :\mathbb {R} ^{2}\to \mathbb {R} ^{3}}$⁠. Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem, and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by ${\displaystyle \mathrm {I\!I} }$ (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.