Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MULTIPLE INTEGRALS

over the ellipsoidal region

x^2

a^2

+

y^2

b^2

+

z^2

c^2

≤ 1.

6.18 Sketch the domain of integration for the integral

I=

∫ 1

0

∫ 1 /y

x=y

y^3

x

exp[y^2 (x^2 +x−^2 )]dx dy

and characterise its boundaries in terms of new variablesu=xyandv=y/x.
Show that the Jacobian for the change from (x, y)to(u, v)isequalto(2v)−^1 ,and
hence evaluateI.
6.19 Sketch the part of the region 0≤x,0≤y≤π/2 that is bounded by the curves
x=0,y=0,sinhxcosy=1andcoshxsiny= 1. By making a suitable change
of variables, evaluate the integral

I=

∫∫

(sinh^2 x+cos^2 y)sinh2xsin 2ydxdy

over the bounded subregion.
6.20 Define a coordinate systemu, vwhose origin coincides with that of the usual
x, ysystem and whoseu-axis coincides with thex-axis, whilst thev-axis makes
an angleαwith it. By considering the integralI=

∫

exp(−r^2 )dA,whereris the

radial distance from the origin, over the area defined by 0≤u<∞,0≤v<∞,

prove that

∫∞

0

∫∞

0

exp(−u^2 −v^2 − 2 uvcosα)du dv=

α

2sinα

.

6.21 As stated in section 5.11, the first law of thermodynamics can be expressed as

dU=TdS−PdV.

By calculating and equating∂^2 U/∂Y ∂Xand∂^2 U/∂X∂Y,whereXandYare an

unspecified pair of variables (drawn fromP,V,TandS), prove that

∂(S,T)

∂(X, Y)

=

∂(V,P)

∂(X, Y)

.

Using the properties of Jacobians, deduce that

∂(S,T)

∂(V,P)

=1.

6.22 The distances of the variable pointP, which has coordinatesx, y, z, from the fixed
points (0, 0 ,1) and (0, 0 ,−1) are denoted byuandvrespectively. New variables
ξ, η, φare defined by

ξ=^12 (u+v),η=^12 (u−v),

andφis the angle between the planey= 0 and the plane containing the three

points. Prove that the Jacobian∂(ξ, η, φ)/∂(x, y, z) has the value (ξ^2 −η^2 )−^1 and

that

∫∫∫

all space

(u−v)^2

uv

exp

(

−

u+v

2

)

dx dy dz=

16 π

3 e

.

6.23 This is a more difficult question about ‘volumes’ in an increasing number of
dimensions.