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 Σ {\displaystyle \Sigma } be an oriented, closed, two-dimensional surface, and let N ε ( Σ ) {\displaystyle N_{\varepsilon }(\Sigma )} denote the set of all points within a distance ε {\displaystyle \varepsilon } of the surface Σ {\displaystyle \Sigma } . Then, for ε {\displaystyle \varepsilon } sufficiently small, the volume of N ε ( Σ ) {\displaystyle N_{\varepsilon }(\Sigma )} is
V = 2 A ( Σ ) ε + 4 π 3 χ ( Σ ) ε 3 , {\displaystyle V=2A(\Sigma )\varepsilon +{\frac {4\pi }{3}}\chi (\Sigma )\varepsilon ^{3},}
where A ( Σ ) {\displaystyle A(\Sigma )} is the area of the surface and χ ( Σ ) {\displaystyle \chi (\Sigma )} is its Euler characteristic. This expression can be generalized to the case where Σ {\displaystyle \Sigma } is a q {\displaystyle q} -dimensional submanifold of n {\displaystyle n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} .
References Weyl, Hermann (1939). "On the volume of tubes". American Journal of Mathematics. 61: 461–472. JSTOR 2371513. Gray, Alfred (2004). "An introduction to Weyl's Tube Formula". Tubes. Progress in Mathematics, volume 221. Springer Science+Business Media. doi:10.1007/978-3-0348-7966-8_1. ISBN 978-3-0348-9639-9. Willerton, Simon (2010-03-12). "Intrinsic Volumes and Weyl's Tube Formula". The n-Category Café. Retrieved 2018-03-10.
