Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.5 EXERCISES

and similarly forJyzandJxz. On taking the determinant of (6.14), we therefore

obtain

Jxz=JxyJyz

or, in the usual notation,

∂(x 1 ,...,xn)

∂(z 1 ,...,zn)

=

∂(x 1 ,...,xn)

∂(y 1 ,...,yn)

∂(y 1 ,...,yn)

∂(z 1 ,...,zn)

. (6.16)

As a special case, if the setziistakentobeidenticaltothesetxi,andthe

obvious resultJxx= 1 is used, we obtain

JxyJyx=1

or, in the usual notation,

∂(x 1 ,...,xn)

∂(y 1 ,...,yn)

=

[

∂(y 1 ,...,yn)

∂(x 1 ,...,xn)

]− 1

. (6.17)

The similarity between the properties of Jacobians and those of derivatives is

apparent, and to some extent is suggested by the notation. We further note from

(6.15) that since|A|=|AT|,whereATis the transpose ofA, we can interchange the

rows and columns in the determinantal form of the Jacobian without changing

its value.

6.5 Exercises

6.1 Identify the curved wedge bounded by the surfacesy^2 =4ax,x+z=aand
z= 0, and hence calculate its volumeV.
6.2 Evaluate the volume integral ofx^2 +y^2 +z^2 over the rectangular parallelepiped
bounded by the six surfacesx=±a,y=±bandz=±c.
6.3 Find the volume integral ofx^2 yover the tetrahedral volume bounded by the
planesx=0,y=0,z=0,andx+y+z=1.
6.4 Evaluate the surface integral off(x, y) over the rectangle 0≤x≤a,0≤y≤b
for the functions

(a)f(x, y)=

x

x^2 +y^2

, (b)f(x, y)=(b−y+x)−^3 /^2.

6.5 Calculate the volume of an ellipsoid as follows:

(a) Prove that the area of the ellipse

x^2

a^2

+

y^2

b^2

=1

isπab.

(b) Use this result to obtain an expression for the volume of a slice of thickness

dzof the ellipsoid

x^2

a^2

+

y^2

b^2

+

z^2

c^2

=1.

Hence show that the volume of the ellipsoid is 4πabc/3.