Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.8 DIVERGENCE THEOREM AND RELATED THEOREMS

Sincevhas a singularity at the origin it is not differentiable there, i.e.∇·vis not

defined there, but at all other points∇·v= 0, as required for an incompressible

fluid. Therefore, from the divergence theorem, for any closed surfaceS 2 that does

not enclose the origin we have
∮

S 2

v·dS=

∫

V

∇·vdV=0.

Thus we see that the surface integral

∮
Sv·dShas valueQor zero depending
on whether or notSencloses the source. In order that the divergence theorem is

valid forallsurfacesS, irrespective of whether they enclose the source, we write

∇·v=Qδ(r),

whereδ(r) is the three-dimensional Dirac delta function. The properties of this

function are discussed fully in chapter 13, but for the moment we note that it is

defined in such a way that

δ(r−a)=0 forr=a,

∫

V

f(r)δ(r−a)dV=

{

f(a)ifalies inV

0otherwise

for any well-behaved functionf(r). Therefore, for any volumeVcontaining the

source at the origin, we have
∫

V

∇·vdV=Q

∫

V

δ(r)dV=Q,

which is consistent with

∮
Sv·dS=Qfor a closed surface enclosing the source.
Hence, by introducing the Dirac delta function the divergence theorem can be

made valid even for non-differentiable point sources.

The generalisation to several sources and sinks is straightforward. For example,

if a source is located atr=aand a sink atr=bthen the velocity field is

v=

(r−a)Q

4 π|r−a|^3

−

(r−b)Q

4 π|r−b|^3

and its divergence is given by

∇·v=Qδ(r−a)−Qδ(r−b).

Therefore, the integral

∮
Sv·dShas the valueQifSencloses the source,−Qif
Sencloses the sink and 0 ifSencloses neither the source nor sink or encloses

them both. This analysis also applies to other physical systems – for example, in

electrostatics we can regard the sources and sinks as positive and negative point

charges respectively and replacevby the electric fieldE.