For a surface X in ℝ³ with pϵ‎X, the first fundamental form is the restriction of the quadratic form v↦|v|² to the tangent space at p. If r(u,v) is a parametrization of X, so that r_u = ∂r/∂u and r_v = ∂r/∂v is a basis for the tangent space, then the form is given by α‎r_u + β‎r_v↦Eα‎² + 2Fα‎β‎ + Gβ‎², where E = r_u·r_u, F = r_u·r_v, G = r_v·r_v. Any intrinsic (i.e. metric) property of the surface, such as area, length of curves, or Gaussian curvature, can be expressed in terms of E, F, G. See Riemannian manifold, second fundamental form, Theorema Egregium.