Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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.
Free download pdf