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 ${\displaystyle \Sigma }$ be an oriented, closed, two-dimensional surface, and let ${\displaystyle N_{\epsilon }(\Sigma )}$ denote the set of all points within a distance ${\displaystyle \epsilon }$ of the surface ${\displaystyle \Sigma }$. Then, for ${\displaystyle \epsilon }$ sufficiently small, the volume of ${\displaystyle N_{\epsilon }(\Sigma )}$ is

${\displaystyle V=2A(\Sigma )\epsilon +{\frac {4\pi }{3}}\chi (\Sigma )\epsilon ^{3},}$

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

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.