A topic in the Open Knowledge Graph — a free, open map of 15,495 topics and the order to learn them in.
For a surface given by a parametrization x(u, v), the first fundamental form records how to measure lengths on the surface using the parameters. Its coefficients are E = x_u · x_u, F = x_u · x_v and G = x_v · x_v, and the squared length of a small step (du, dv) is E du² + 2F du dv + G dv². From E, F and G alone one can compute lengths of curves, angles between curves and areas of regions on the surface. Properties determined by the first fundamental form are called intrinsic: they can be measured by someone living on the surface, without reference to the surrounding space.
Compute E, F and G for a plane in polar coordinates, a cylinder, a sphere and a surface of revolution, and use them to find arc lengths and areas that can be checked against known answers. Notice that a cylinder and a plane have the same coefficients in suitable coordinates, and connect this to rolling a sheet of paper into a tube without stretching. Keep the dot-product definitions in view so the formulas stay grounded.
Suppose a surface S in space is described by a parametrization x(u, v), so that each pair of parameters gives a point. A curve on the surface can be described by functions u(t) and v(t). To find its length, we need its speed. By the chain rule its velocity is x_u u' + x_v v', where x_u and x_v are the partial derivative vectors. The squared speed is
|x_u u' + x_v v'|² = E u'² + 2F u'v' + G v'²,
with E = x_u · x_u, F = x_u · x_v and G = x_v · x_v. The quadratic expression I = E du² + 2F du dv + G dv² is the first fundamental form. It is the dot product of space, restricted to tangent vectors of the surface and written in the surface's own coordinates.
Everything metric about the surface comes from these three functions. Arc length is the integral of √(E u'² + 2F u'v' + G v'²). The angle θ between the coordinate curves satisfies cos θ = F/√(EG), so F = 0 means the coordinate grid is orthogonal. Area comes from |x_u × x_v|, and the identity |a × b|² = |a|²|b|² - (a · b)² gives the area element √(EG - F²) du dv.
Take the sphere of radius R with x(φ, θ) = (R sin φ cos θ, R sin φ sin θ, R cos φ). A short computation gives E = R², F = 0 and G = R² sin²φ. The area element is R² sin φ dφ dθ, and integrating over the sphere gives 4πR². Near the poles, where sin φ is small, a step in θ covers little distance.
Now compare a plane and a cylinder. The plane x(u, v) = (u, v, 0) has E = 1, F = 0, G = 1. The cylinder x(θ, z) = (cos θ, sin θ, z) also has E = 1, F = 0, G = 1. To an observer confined to either surface, measuring with a tape that bends with the surface, small regions look exactly alike. A sheet of paper rolls into a cylinder without stretching for this reason. Such a correspondence that preserves the first fundamental form is called a local isometry.
Properties that depend only on E, F and G are intrinsic. Gauss proved that the Gaussian curvature, which seems to depend on how the surface bends in space, is intrinsic too. So a sphere has no local isometry to the plane, and every flat map of the Earth distorts something. A metric written in local coordinates is also the starting idea of Riemannian geometry.