Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

integral in (11.1). If a loop of wireCcarrying a currentIis placed in a magnetic

fieldBthen the forcedFon a small lengthdrof the wire is given bydF=Idr×B,

and so the total (vector) force on the loop is

F=I

∮

C

dr×B.

11.1.3 Line integrals with respect to a scalar

In addition to those listed in (11.1), we can form other types of line integral,

which depend on a particular curveCbut for which we integrate with respect

to a scalardu, rather than the vector differentialdr. This distinction is somewhat

arbitrary, however, since we can always rewrite line integrals containing the vector

differentialdras a line integral with respect to some scalar parameter. If the path

Calong which the integral is taken is described parametrically byr(u)then

dr=

dr

du

du,

and the second type of line integral in (11.1), for example, can be written as
∫

C

a·dr=

∫

C

a·

dr

du

du.

A similar procedure can be followed for the other types of line integral in (11.1).

Commonly occurring special cases of line integrals with respect to a scalar are

∫

C

φds,

∫

C

ads,

wheresis the arc length along the curveC. We can always representCparamet-

rically byr(u), and from section 10.3 we have

ds=

√

dr

du

·

dr

du

du.

The line integrals can therefore be expressed entirely in terms of the parameteru

and thence evaluated.

Evaluate the line integralI=

∫

C(x−y)

(^2) ds,whereCis the semicircle of radiusarunning
fromA=(a,0)toB=(−a,0)and for whichy≥ 0.
The semicircular path fromAtoBcan be described in terms of the azimuthal angleφ
(measured from thex-axis) by
r(φ)=acosφi+asinφj,
whereφruns from 0 toπ. Therefore the element of arc length is given, from section 10.3,
by
ds=

√

dr

dφ

·

dr

dφ

dφ=a(cos^2 φ+sin^2 φ)dφ=adφ.