Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.10 EXERCISES

everywhere except on the axisρ=0,wherevhas a singularity. Therefore

∮
Cv·dr
equals zero for any pathCthat does not enclose the vortex line on the axis and

2 πifCdoes enclose the axis. In order for Stokes’ theorem to be valid for all

pathsC, we therefore set

∇×v=2πδ(ρ),

whereδ(ρ) is the Dirac delta function, to be discussed in subsection 13.1.3. Now,

since∇×v= 0 , except on the axisρ= 0, there exists a scalar potentialψsuch

thatv=∇ψ. It may easily be shown thatψ=φ, the polar angle. Therefore, ifC

does not enclose the axis then
∮

C

v·dr=

∮

dφ=0,

and ifCdoes enclose the axis,
∮

C

v·dr=∆φ=2πn,

wherenis the number of times we traverseC. Thusφis a multivalued potential.

Similar analyses are valid for other physical systems – for example, in magneto-

statics we may replace the vortex lines by current-carrying wires and the velocity

fieldvby the magnetic fieldB.

11.10 Exercises

11.1 The vector fieldFis defined by

F=2xzi+2yz^2 j+(x^2 +2y^2 z−1)k.
Calculate∇×Fand deduce thatFcan be writtenF=∇φ.Determine the form
ofφ.
11.2 The vector fieldQis defined by

Q=

[

3 x^2 (y+z)+y^3 +z^3

]

i+

[

3 y^2 (z+x)+z^3 +x^3

]

j+

[

3 z^2 (x+y)+x^3 +y^3

]

k.

Show thatQis a conservative field, construct its potential function and hence

evaluate the integralJ=

∫

Q·dralong any line connecting the pointAat
(1,− 1 ,1) toBat (2, 1 ,2).
11.3 Fis a vector fieldxy^2 i+2j+xk,andLis a path parameterised byx=ct,y=c/t,
z=dfor the range 1≤t≤2. Evaluate (a)

∫

LFdt,(b)

∫

LFdyand (c)

∫

LF·dr.
11.4 By making an appropriate choice for the functionsP(x, y)andQ(x, y) that appear
in Green’s theorem in a plane, show that the integral ofx−yover the upper half
of the unit circle centred on the origin has the value−^23. Show the same result
by direct integration in Cartesian coordinates.
11.5 Determine the point of intersectionP, in the first quadrant, of the two ellipses

x^2

a^2

+

y^2

b^2

=1 and

x^2

b^2

+

y^2

a^2

=1.

Takingb<a, consider the contourLthat bounds the area in the first quadrant

that is common to the two ellipses. Show that the parts ofLthat lie along the

coordinate axes contribute nothing to the line integral aroundLofxdy−ydx.

Using a parameterisation of each ellipse similar to that employed in the example