Weyl's Formula
Weyl's formula states that one can infer the area V of the drum by counting how rapidly the λn grow. We define N(R) to be the number of eigenvalues smaller than R and we get
where d is the dimension. Weyl also conjectured that the next term in the approximation below would give the perimeter of D. In other words, if A denotes the length of the perimeter (or the surface area in higher dimension), then one should have
where is the volume of a d-dimensional unit ball. For smooth boundary, this was proved by Victor Ivrii in 1980. The manifold is also not allowed to have a two parameter family of periodic geodesics such as a sphere would have.
Read more about this topic: Hearing The Shape Of A Drum
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—Friedrich Nietzsche (1844–1900)