Question

Let us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR and S(R)-S.),respectively (a) Find the relation between V(0) and S 1) (b) Calculate the Gaussian integral 3. (c) Calculate the same integral in spherical coordinates in terms of the gamma function re)-e'd (d) Obtain the closed forms of S,,(1) and V(1) (e) Calculate r5) and S.,0), p.(1) for n-1, 2, 3. (40 points)

Answer 1

Given that For a concentyic sp hevical shells, the condihon is, Trus, the volume asea and Surfoce area ave vela tdl by, 2. (o) Roho of V) and S- (b) The Gaussian integral is, Co no, dn us the volme element in certesion coovdlirntes dv

xeeome integral in spherical co-ordirmte tse the volume elemtnt in sphericul cordirates is, -Xy tit using gamma function. T(z)-I?

An-141 from (b), He Kneus that 6 Non, we know that 2rin

and V) ane, r(%) y(%) e) closed forrms of Sp n-1 e) find) i substitut t=" 12 lo o n わ Similoviy tn

2 r(z) 31 2조31 = 41T A) 2l 「佳) レr(レ) 3.LE-