Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.6 HINTS AND ANSWERS

(a) LetRbe a real positive number and defineKmby

Km=

∫R

−R

(

R^2 −x^2

)m

dx.

Show, using integration by parts, thatKmsatisfies the recurrence relation

(2m+1)Km=2mR^2 Km− 1.

(b) For integern, defineIn=KnandJn=Kn+1/ 2. EvaluateI 0 andJ 0 directly

and hence prove that

In=

22 n+1(n!)^2 R^2 n+1

(2n+1)!

and Jn=

π(2n+1)!R^2 n+2

22 n+1n!(n+1)!

.

(c) A sequence of functionsVn(R) is defined by

V 0 (R)=1,

Vn(R)=

∫R

−R

Vn− 1

(√

R^2 −x^2

)

dx, n≥ 1.

Prove by induction that

V 2 n(R)=

πnR^2 n

n!

,V 2 n+1(R)=

πn 22 n+1n!R^2 n+1

(2n+1)!

.

(d) For interest,

(i) show thatV 2 n+2(1)<V 2 n(1) andV 2 n+1(1)<V 2 n− 1 (1) for alln≥3;

(ii) hence, by explicitly writing outVk(R)for1≤k≤8 (say), show that the

‘volume’ of the totally symmetric solid of unit radius is a maximum in

five dimensions.

6.6 Hints and answers

6.1 For integration orderz,y,x, the limits are (0,a−x), (−

√

4 ax,

√

4 ax)and(0,a).

For integration ordery,x,z, the limits are (−

√

4 ax,

√

4 ax), (0,a−z)and(0,a).
V=16a^3 /15.
6.3 1 /360.
6.5 (a) Evaluate

∫

2 b[1−(x/a)^2 ]^1 /^2 dxby settingx=acosφ;
(b)dV=π×a[1−(z/c)^2 ]^1 /^2 ×b[1−(z/c)^2 ]^1 /^2 dz.
6.7 Write sin^3 θas (1−cos^2 θ)sinθwhen integrating|Ψ 2 |^2.
6.9 (a)V=2πc×πa^2 andA=2πa× 2 πc. Settingro=c+aandri=c−agives the
stated results. (b) Show that the centre of gravity of either half is 2a/πfrom the
cylinder.
6.11 Transform to cylindrical polar coordinates.
6.13 4 πa^2 ;4πa^3 /3; a sphere.
6.15 The volume element isρdφdρdz. The integrand for the finalz-integration is
given by 2π[(z^2 lnz)−(z^2 /2)];I=− 5 π/9.
6.17 Setξ=x/a,η=y/b,ζ=z/cto map the ellipsoid onto the unit sphere, and then
change from (ξ, η, ζ) coordinates to spherical polar coordinates;I=4πa^3 bc/15.
6.19 Setu=sinhxcosyandv=coshxsiny;Jxy ,uv=(sinh^2 x+cos^2 y)−^1 and the
integrand reduces to 4uvover the region 0≤u≤1, 0≤v≤1;I=1.
6.21 Terms such asT∂^2 S/∂Y ∂Xcancel in pairs. Use equations (6.17) and (6.16).
6.23 (c) Show that the two expressions mutually support the integration formula given
for computing a volume in the next higher dimension.
(d)(ii) 2,π,4π/3,π^2 /2, 8π^2 /15,π^3 /6, 16π^3 /105,π^4 /24.