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 [math]\displaystyle{ \Sigma }[/math] be an oriented, closed, two-dimensional surface, and let [math]\displaystyle{ N_\varepsilon(\Sigma) }[/math] denote the set of all points within a distance [math]\displaystyle{ \varepsilon }[/math] of the surface [math]\displaystyle{ \Sigma }[/math]. Then, for [math]\displaystyle{ \varepsilon }[/math] sufficiently small, the volume of [math]\displaystyle{ N_\varepsilon(\Sigma) }[/math] is
- [math]\displaystyle{ V = 2A(\Sigma)\varepsilon + \frac{4\pi}{3} \chi(\Sigma)\varepsilon^3, }[/math]
where [math]\displaystyle{ A(\Sigma) }[/math] is the area of the surface and [math]\displaystyle{ \chi(\Sigma) }[/math] is its Euler characteristic. This expression can be generalized to the case where [math]\displaystyle{ \Sigma }[/math] is a [math]\displaystyle{ q }[/math]-dimensional submanifold of [math]\displaystyle{ n }[/math]-dimensional Euclidean space [math]\displaystyle{ \mathbb{R}^n }[/math].
References
- Weyl, Hermann (1939). "On the volume of tubes". American Journal of Mathematics 61: 461–472.
- 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". https://golem.ph.utexas.edu/category/2010/03/intrinsic_volumes_for_riemanni.html.
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