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 $$\Sigma$$ be an oriented, closed, two-dimensional surface, and let $$N_\varepsilon(\Sigma)$$ denote the set of all points within a distance $$\varepsilon$$ of the surface $$\Sigma$$. Then, for $$\varepsilon$$ sufficiently small, the volume of $$N_\varepsilon(\Sigma)$$ is
 * $$V = 2A(\Sigma)\varepsilon + \frac{4\pi}{3} \chi(\Sigma)\varepsilon^3,$$

where $$A(\Sigma)$$ is the area of the surface and $$\chi(\Sigma)$$ is its Euler characteristic. This expression can be generalized to the case where $$\Sigma$$ is a $$q$$-dimensional submanifold of $$n$$-dimensional Euclidean space $$\mathbb{R}^n$$.