Weyl's tube formula
Weyl's tube formula gives the volume of an object defined as the set of all points within a small distance of a manifold.
Let be an oriented, closed, two-dimensional surface, and let denote the set of all points within a distance of the surface. Then, for sufficiently small, the volume of is
where is the area of the surface and is its Euler characteristic. This expression can be generalized to the case where is a -dimensional submanifold of -dimensional Euclidean space.
