Mathematical Methods for Physics and Engineering : A Comprehensive Guide

11.8 DIVERGENCE THEOREM AND RELATED THEOREMS

11.8 Divergence theorem and related theorems

The divergence theorem relates the total flux of a vector field out of a closed

surfaceSto the integral of the divergence of the vector field over the enclosed

volumeV; it follows almost immediately from our geometrical definition of

divergence (11.15).

Imagine a volumeV, in which a vector fieldais continuous and differentiable,

to be divided up into a large number of small volumesVi. Using (11.15), we have

for each small volume

(∇·a)Vi≈

∮

Si

a·dS,

whereSiis the surface of the small volumeVi. Summing overiwe find that

contributions from surface elements interior toScancel since each surface element

appears in two terms with opposite signs, the outward normals in the two terms

being equal and opposite. Only contributions from surface elements that are also

parts ofSsurvive. If eachViis allowed to tend to zero then we obtain the

divergence theorem,
∫

V

∇·adV=

∮

S

a·dS. (11.18)

We note that the divergence theorem holds for both simply and multiply con-

nected surfaces, provided that they are closed and enclose some non-zero volume

V. The divergence theorem may also be extended to tensor fields (see chapter 26).

The theorem finds most use as a tool in formal manipulations, but sometimes

it is of value in transforming surface integrals of the form

∫
Sa·dSinto volume
integrals or vice versa. For example, settinga=rwe immediately obtain
∫

V

∇·rdV=

∫

V

3 dV=3V=

∮

S

r·dS,

which gives the expression for the volume of a region found in subsection 11.6.1.

The use of the divergence theorem is further illustrated in the following example.

Evaluate the surface integralI=

∫

Sa·dS,wherea=(y−x)i+x

(^2) zj+(z+x (^2) )kandS
is the open surface of the hemispherex^2 +y^2 +z^2 =a^2 ,z≥ 0.
We could evaluate this surface integral directly, but the algebra is somewhat lengthy. We
will therefore evaluate it by use of the divergence theorem. Since the latter only holds
for closed surfaces enclosing a non-zero volumeV, let us first consider the closed surface
S′=S+S 1 ,whereS 1 is the circular area in thexy-plane given byx^2 +y^2 ≤a^2 ,z=0;S′
then encloses a hemispherical volumeV. By the divergence theorem we have
∫
V
∇·adV=

∮

S′

a·dS=

∫

S

a·dS+

∫

S 1

a·dS.

Now∇·a=−1+0+1=0, so we can write
∫

S

a·dS=−

∫

S 1

a·dS.