Surface Area of an n-dimensional SphereDate: 07/28/97 at 18:32:35 From: Christine Bollinger Subject: Surface area of an n dimensional sphere I was wondering how to calculate the surface area of a sphere in n dimensions. I read and understand your explanation on how to calculate the VOLUME of an n-dimensional sphere and know that the two are related, but I can't figure out how. Thanks in advance, Christine Bollinger Date: 07/31/97 at 11:44:17 From: Doctor Rob Subject: Re: Surface area of an n dimensional sphere Take a look at this Web page with the Formula for the Surface Area of a sphere in Euclidean N-Space: http://daisy.uwaterloo.ca/~alopez-o/math-faq/node42.html The relation is that the measure of the surface of an n-sphere is the derivative of the volume of the n-sphere with respect to the radius. If you know enough calculus to figure that out, that is your answer. If not, this is how that is computed. Let c = pi^(n/2)/((n/2)!), then V = c*r^n. If we increase the radius to r + a, then we get V' = c*(r+a)^n. The difference in volume is V' - V, and the difference in volume per unit increase in radius is (V' - V)/((r + a) - r) = c*((r+a)^n - r^n)/a Use the Binomial Theorem to expand (r+a)^n, and notice that the first term is canceled by the -r^n term above. All the other terms are divisible by a, so we divide the denominator a into them. The result is = c*C(n,1)*r^(n-1) + c*C(n,2)*r^(n-2)*a + ... Now we consider what happens as a gets very small. All the terms from the second one on get very small, too. If we actually put a = 0, we get the formula S = c*n*r^(n-1). The formula for surface area is then just S = V*n/r. I hope this is what you wanted. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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