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 .

References

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  • 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.