Generating the Volume of an n-SphereDate: 02/07/2002 at 03:29:54 From: Elara Subject: Multivariable calculus Hi, Could you help me in finding an equation for the integration of a unit-ball in n-dimensions? Date: 02/07/2002 at 10:15:41 From: Doctor Mitteldorf Subject: Re: Multivariable calculus If you know the volume of an n-1-sphere, you can generate the volume of an n-sphere with an integral. Let's set the radius = 1. Imagine stacking up the (n-1)-spheres, each of which has a radius (1-r^2), and integrating from -1 to 1. For example, if pi*r^2 is the answer for n=2, then for n=3 we have [Integral from -1 to +1] pi*(1-r^2) dr This gives the familiar answer V[3]=4pi/3, given that V[2]=pi. For the next step, n=4, we have [Integral from -1 to +1] 4pi/3 * (1-r^2)^(3/2) dr This gives (1/2)pi^2 for the 4-sphere. In general, you can write V[n+1] = V[n] * [Integral from -1 to +1] (1-r^2)^(n/2) dr If you want a general (non-recursive) formula, it's tricky because each of these integrals is a little different. The answer is worked out in terms of the gamma function in Kevin Brown's MathPages: Volume of n-Spheres and the Gamma Function http://www.mathpages.com/home/kmath163.htm - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ Date: 02/08/2002 at 09:18:26 From: Elara Subject: Multivariable calculus Thank you for your help. Could you tell me if we can equate the volume of a unit n-ball in (n-1)dimensions with the area of the same ball in n-dimensions? Yours, Elara Date: 02/08/2002 at 10:13:56 From: Doctor Mitteldorf Subject: Re: Multivariable calculus The n-dimensional volume of an ball of radius r is related to the n-1-dimensional surface area by the following formula: volume = area/(nr) - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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