the first fundamental form
Imagine you are an ant living on a surface, unable to leave it, with a tiny ruler. You want to measure the length of a path you walk, the angle at which two paths cross, and the area of a region — all WITHOUT ever stepping off into the surrounding space. The first fundamental form is exactly the ruler that lets you do this. It is the rule that tells you, in the surface's own (u, v) coordinates, how much real distance corresponds to a small step.
Here is the machinery. Pick a parametrization x(u, v) with tangent vectors x_u and x_v. Form three numbers from their dot products: E = x_u . x_u, F = x_u . x_v, and G = x_v . x_v. These three functions ARE the first fundamental form. The squared length of an infinitesimal step (du, dv) on the surface is then ds^2 = E du^2 + 2F du dv + G dv^2. From E, F, G alone you compute everything metric: the length of a curve (integrate ds along it), the angle between two tangent directions (a formula in E, F, G mirroring the dot-product cosine), and the area of a region (integrate sqrt(E*G - F^2) du dv, since sqrt(E*G - F^2) = |x_u x x_v| is the area of the little parallelogram spanned by x_u and x_v). For a plane in ordinary Cartesian coordinates, E = 1, F = 0, G = 1, and ds^2 = du^2 + dv^2 is just Pythagoras.
The first fundamental form is the heart of intrinsic geometry — the geometry the ant can know from inside, without any view of how the surface sits in space. Its supreme importance is sealed by Gauss's Theorema Egregium: the Gaussian curvature can be computed from E, F, G and their derivatives alone, so even curvature, which seems to be about bending into the third dimension, is secretly intrinsic. A crucial caution: the SAME surface has different E, F, G in different parametrizations (just as a country's shape is recorded by different numbers on different map projections), yet the lengths, angles, and areas they predict are always the same. And two surfaces with the same first fundamental form are bendable into each other without stretching — a flat sheet of paper and a cylinder share the plane's metric, which is why you can roll paper into a tube but never wrap it smoothly around a ball.
A cylinder of radius R, parametrized as x(u, v) = (R cos u, R sin u, v), has x_u = (-R sin u, R cos u, 0) and x_v = (0, 0, 1). So E = x_u . x_u = R^2, F = x_u . x_v = 0, G = x_v . x_v = 1, giving ds^2 = R^2 du^2 + dv^2. Substituting s = R*u (arc length around the circle) turns this into ds^2 = ds_horizontal^2 + dv^2 — exactly the flat plane's metric. That is the precise reason a cylinder is intrinsically flat and unrolls onto paper without stretching.
The cylinder's metric reduces to the plane's: it is intrinsically flat.
The first fundamental form measures distances and angles WITHIN the surface; it says nothing by itself about how the surface bends into space — that is the job of the second fundamental form. A flat plane and a rolled-up cylinder have the SAME first fundamental form yet bend very differently, because bending is extrinsic and the metric is intrinsic.