11

Line, surface and volume integrals

In the previous chapter we encountered continuously varying scalar and vector

fields and discussed the action of various differential operators on them. In

addition to these differential operations, the need often arises to consider the

integration of field quantities along lines, over surfaces and throughout volumes.

In general the integrand may be scalar or vector in nature, but the evaluation

of such integrals involves their reduction to one or more scalar integrals, which

are then evaluated. In the case of surface and volume integrals this requires the

evaluation of double and triple integrals (see chapter 6).

11.1 Line integrals

In this section we discusslineorpath integrals, in which some quantity related

to the field is integrated between two given points in space,AandB, along a

prescribed curveCthat joins them. In general, we may encounter line integrals

of the forms ∫

C

φdr,

∫

C

a·dr,

∫

C

a×dr, (11.1)

whereφis a scalar field andais a vector field. The three integrals themselves are

respectively vector, scalar and vector in nature. As we will see below, in physical

applications line integrals of the second type are by far the most common.

The formal definition of a line integral closely follows that of ordinary integrals

and can be considered as the limit of a sum. We may divide the pathCjoining

the pointsAandBintoNsmall line elements ∆rp,p=1,...,N.If(xp,yp,zp)is

any point on the line element ∆rpthen the second type of line integral in (11.1),

for example, is defined as
∫

C

a·dr= lim

N→∞

∑N

p=1

a(xp,yp,zp)·∆rp,

where it is assumed that all|∆rp|→0asN→∞.