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first fundamental form 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 ru = ∂r/∂u and rv = ∂r/∂v is a basis for the tangent space, then the form is given by αru + βrv↦Eα2 + 2Fαβ + Gβ2, where E = ru·ru, F = ru·rv, G = rv·rv. 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.
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