The curvature of a surface. If u and v are local coordinates on a smooth surface, with first and second fundamental forms Edu² + 2Fdudv + Gdv² and Ldu² + 2Mdudv + Ndv², then the Gaussian curvature is K = (LN−M²)/(EG−F²). For a sphere of radius r, then K = 1/r², for a plane or cylinder K = 0, for the hyperbolic plane K =−1; more generally K varies on a surface. If U is a neighbourhood of a point p in the surface and n denotes the Gauss map, then |K(p)| is the limit of area(n(U))/area(U) as the area of U tends to 0. See also Gauss-Bonnet theorem, Theorema Egregium.