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The surface area are and the volume of n-dimensional ...
Physics 2400 Spring 2017the surface area are and the volume ofn-dimensional spherespring semester 2017 rozman/Courses/P2400_17S/Last modified: May 5, 20171 IntroductionThe volume of n-dimensional sphere of radiusris proportional torn,Vn(r) =v(n)rn,(1)where the proportionality constant,v(n), is the volume of the n-dimensional unit surface area of n-dimensional sphere of radiusris proportional torn (r) =s(n)rn 1,(2)where the proportionality constant,s(n), is the surface area of the n-dimensional n-dimensional sphere is a union of concentric spherical shells:dVn(r) =Sn(r) dr(3)Therefore the surface area and the volume are related as following:Vn(R) =R 0Sn(r) dr=s(n)R 0rn 1dr=s(n)nRn.(4)Page 1 of 3 Physics 2400n-dimensional sphereSpring 2017 The surface area and the volume of the unit sphere are related as following:v(n) =s(n)n.(5)Consider the integralIn= e x21 x22 .. x2ndVn= 0e r2dVn(r),(6)where dVnis the volume element in cartesian coordinatesdVn= (7)anddVn(r) =s(n)rn 1dr(8)is the volume element in spherical the integrand in the first integral in Eq.
May 05, 2017 · The surface area of n-dimensional sphere of radius ris proportional to rn1. S n(r) = s(n)rn1; (2) where the proportionality constant, s(n), is the surface area of the n-dimensional unit sphere. The n-dimensional sphere is a union of concentric spherical shells: dV n(r) = S n(r)dr (3) Therefore the surface area and the volume are related as ...
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