Calculating the Volume of a Sphere in N-Dimensional Space
Date: 9/5/95 at 0:14:40 From: Anonymous Subject: geometry How do I calculate the volume of a sphere in 4-dimension or, more generally, in n-dimension space? My regards, Pawel Kowalczyk
Date: 9/5/95 at 13:7:32 From: Doctor Ken Subject: Re: geometry Hello! This comes from the Frequently Asked Questions of the sci.math newsgroup: Archive-Name: sci-math-faq/surfaceSphere Last-modified: December 8, 1994 Version: 6.2 Formula for the Surface Area of a sphere in Euclidean N -Space This is equivalent to the volume of the N -1 solid which comprises the boundary of an N -Sphere. The volume of a ball is the easiest formula to remember: It's r^N (pi^(N/2))/((N/2)!) . The only hard part is taking the factorial of a half-integer. The real definition is that x! = Gamma (x + 1) , but if you want a formula, it's: (1/2 + n)! = sqrt(pi) ((2n + 2)!)/((n + 1)!4^(n + 1)) To get the surface area, you just differentiate to get N (pi^(N/2))/((N/2)!)r^(N - 1) . There is a clever way to obtain this formula using Gaussian integrals. First, we note that the integral over the line of e^(-x^2) is sqrt(pi). Therefore the integral over N -space of e^(-x_1^2 - x_2^2 - ... - x_N^2) is sqrt(pi)^n . Now we change to spherical coordinates. We get the integral from 0 to infinity of Vr^(N - 1)e^(-r^2) , where V is the surface volume of a sphere. Integrate by parts repeatedly to get the desired formula. It is possible to derive the volume of the sphere from "first principles''. -Doctor Ken, The Geometry Forum
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