Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

In particular, when the surface is closed Ω = 0 ifOis outsideSand Ω = 4πifO

is an interior point.

Surface integrals resulting in vectors occur less frequently. An example is

afforded, however, by the total resultant force experienced by a body immersed in

a stationary fluid in which the hydrostatic pressure is given byp(r). The pressure

is everywhere inwardly directed and the resultant force isF=−

∮
SpdS,taken
over the whole surface.

11.6 Volume integrals

Volume integrals are defined in an obvious way and are generally simpler than

line or surface integrals since the element of volumedVis a scalar quantity. We

may encounter volume integrals of the forms
∫

V

φdV,

∫

V

adV. (11.12)

Clearly, the first form results in a scalar, whereas the second form yields a vector.

Two closely related physical examples, one of each kind, are provided by the total

mass of a fluid contained in a volumeV, given by

∫
Vρ(r)dV, and the total linear
momentum of that same fluid, given by

∫
Vρ(r)v(r)dVwherev(r) is the velocity
field in the fluid. As a slightly more complicated example of a volume integral we

may consider the following.

Find an expression for the angular momentum of a solid body rotating with angular

velocityωabout an axis through the origin.

Consider a small volume elementdVsituated at positionr; its linear momentum isρdV ̇r,
whereρ=ρ(r) is the density distribution, and its angular momentum aboutOisr×ρ ̇rdV.
Thus for the whole body the angular momentumLis

L=

∫

V

(r× ̇r)ρdV.

Putting ̇r=ω×ryields

L=

∫

V

[r×(ω×r)]ρdV=

∫

V

ωr^2 ρdV−

∫

V

(r·ω)rρdV.

The evaluation of the first type of volume integral in (11.12) has already been

considered in our discussion of multiple integrals in chapter 6. The evaluation of

the second type of volume integral follows directly since we can write
∫

V

adV=i

∫

V

axdV+j

∫

V

aydV+k

∫

V

azdV , (11.13)

whereax,ay,azare the Cartesian components ofa. Of course, we could have

writtenain terms of the basis vectors of some other coordinate system (e.g.

spherical polars) but, since such basis vectors are not, in general, constant, they