Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.1 LINE INTEGRALS

Evaluate the line integralI=

∮

Cxdy,whereCis the circle in thexy-plane defined by

x^2 +y^2 =a^2 ,z=0.

Adopting the usual convention mentioned above, the circleCis to be traversed in the
anticlockwise direction. Taking the circle as a whole meansxis not a single-valued
function ofy. We must therefore divide the path into two parts withx=+

√

a^2 −y^2 for
the semicircle lying to the right ofx=0,andx=−

√

a^2 −y^2 for the semicircle lying to
the left ofx= 0. The required line integral is then the sum of the integrals along the two
semicircles. Substituting forx,itisgivenby

I=

∮

C

xdy=

∫a

−a

√

a^2 −y^2 dy+

∫−a

a

(

−

√

a^2 −y^2

)

dy

=4

∫a

0

√

a^2 −y^2 dy=πa^2.

Alternatively, we can represent the entire circle parametrically, in terms of the azimuthal
angleφ,sothatx=acosφandy=asinφwithφrunning from 0 to 2π. The integral can
therefore be evaluated over the whole circle at once. Noting thatdy=acosφdφ,wecan
rewrite the line integral completely in terms of the parameterφand obtain

I=

∮

C

xdy=a^2

∫ 2 π

0

cos^2 φdφ=πa^2 .

11.1.2 Physical examples of line integrals

There are many physical examples of line integrals, but perhaps the most common

is the expression for the total work done by a forceFwhen it moves its point

of application from a pointAto a pointBalong a given curveC. We allow the

magnitude and direction ofFto vary along the curve. Let the force act at a point

rand consider a small displacementdralong the curve; then the small amount

of work done isdW=F·dr, as discussed in subsection 7.6.1 (note thatdWcan

be either positive or negative). Therefore, the total work done in traversing the

pathCis

WC=

∫

C

F·dr.

Naturally, other physical quantities can be expressed in such a way. For example,

the electrostatic potential energy gained by moving a chargeqalong a pathCin

an electric fieldEis−q

∫
CE·dr. We may also note that Ampere’s law concerning`
the magnetic fieldBassociated with a current-carrying wire can be written as
∮

C

B·dr=μ 0 I,

whereIis the current enclosed by a closed pathCtraversed in a right-handed

sense with respect to the current direction.

Magnetostatics also provides a physical example of the third type of line