In this post, I introduce a formula to calculate the volume of n-dimenstional sphere. We will use the gamma function. The gamma function is defined as following:
It can be easily proven that the gamma function satisfies following identities:
Thus, the gamma function can be regarded as an extension of factorial function. The gamma function can be used to calculate a few kinds of integrals:
Now, we calculate the volume of n-dimensional spheres.
Theorem. The volume of n-dimensional sphere B(a, r) which has the centre a and the radius r is:
Proof. The volume is invariant under translation. So we only have to calculate the case of a = 0. It is trivial for n = 1, so assume that n ≥ 2. We change the variables as
when 0 ≤ ρ ≤ r, 0 ≤ θ ≤ 2π, 0 ≤ φj ≤ π for j = 0, 1, 2, …, n-2. The the determinant of the Jacobian matric is
Now the volume is calculated as following:
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Geschrieben von Sooji Shin