For the six-dimensional ball of radius (a purely mathematical

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

For the six-dimensional ball of radius R=2R=2 (a purely mathematical, dimensionless construction), use Vn(R)=πn/2Γ(n/2+1)Rn,Sn−1(R)=2πn/2Γ(n/2)Rn−1,V_n(R) = \frac{\pi^{n/2}}{\Gamma(n/2+1)}R^n,\qquad S_{n-1}(R) = \frac{2\pi^{n/2}}{\Gamma(n/2)}R^{n-1}, with n=6n=6, and Γ(3)=2\Gamma(3)=2, Γ(4)=6\Gamma(4)=6. ((a)) Compute the volume V6(2)V_6(2). ((b)) Compute the area S5(2)S_5(2) of the bounding five-dimensional sphere. ((c)) Check the two results against dVn/dR=Sn−1\mathrm{d}V_n/\mathrm{d}R = S_{n-1}.

Browse the problem library