Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

in section 11.3, evaluate the two remaining line integrals and hence find the total
area common to the two ellipses.
11.6 By using parameterisations of the formx=acosnθandy=asinnθfor suitable
values ofn, find the area bounded by the curves

x^2 /^5 +y^2 /^5 =a^2 /^5 and x^2 /^3 +y^2 /^3 =a^2 /^3.

11.7 Evaluate the line integral

I=

∮

C

[

y(4x^2 +y^2 )dx+x(2x^2 +3y^2 )dy

]

around the ellipsex^2 /a^2 +y^2 /b^2 =1.
11.8 Criticise the following ‘proof’ thatπ=0.

(a) Apply Green’s theorem in a plane to the functionsP(x, y)=tan−^1 (y/x)and

Q(x, y)=tan−^1 (x/y), taking the regionRto be the unit circle centred on the

origin.

(b) The RHS of the equality so produced is

∫∫

R

y−x

x^2 +y^2

dx dy,

which, either from symmetry considerations or by changing to plane polar

coordinates, can be shown to have zero value.

(c) In the LHS of the equality, setx=cosθandy=sinθ, yieldingP(θ)=θ

andQ(θ)=π/ 2 −θ. The line integral becomes

∫ 2 π

0

[(π

2

−θ

)

cosθ−θsinθ

]

dθ,

which has the value 2π.

(d) Thus 2π= 0 and the stated result follows.

11.9 A single-turn coilCof arbitrary shape is placed in a magnetic fieldBand carries
a currentI. Show that the couple acting upon the coil can be written as

M=I

∫

C

(B·r)dr−I

∫

C

B(r·dr).

For a planar rectangular coil of sides 2aand 2bplaced with its plane vertical
and at an angleφto a uniform horizontal fieldB, show thatMis, as expected,
4 abBIcosφk.
11.10 Find the vector areaSof the part of the curved surface of the hyperboloid of
revolution
x^2
a^2

−

y^2 +z^2

b^2

=1

that lies in the regionz≥0anda≤x≤λa.
11.11 An axially symmetric solid body with its axisABvertical is immersed in an
incompressible fluid of densityρ 0. Use the following method to show that,
whatever the shape of the body, forρ=ρ(z) in cylindrical polars the Archimedean
upthrust is, as expected,ρ 0 gV,whereVis the volume of the body.
Express the vertical component of the resultant force on the body,−

∫

pdS,

wherepis the pressure, in terms of an integral; note thatp=−ρ 0 gzand that for

an annular surface element of widthdl,n·nzdl=−dρ. Integrate by parts and

use the fact thatρ(zA)=ρ(zB)=0.