Divergence theorem and related theorems

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
surface S to the integral of the divergence of the vector field over the enclosed
volume V ; it follows almost immediately from our geometrical definition of
divergence (11.15).
Imagine a volume V , in which a vector field a is continuous and differentiable,
to be divided up into a large number of small volumes Vi . Using (11.15), we have
for each small volume
0
(∇ · a)Vi ≈
a · dS,
Si
where Si is the surface of the small volume Vi . Summing over i we find that
contributions from surface elements interior to S cancel 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 of S survive. If each Vi is allowed to tend to zero then we obtain the
divergence theorem,
0
∇ · a dV = a · dS.
(11.18)
V
S
We note that the divergence theorem holds for both simply and multiply connected 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 S a · dS into volume
integrals or vice versa. For example, setting a = r we immediately obtain
0
∇ · r dV =
3 dV = 3V = r · dS,
V
V
S
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 integral I = S a · dS, where a = (y − x) i + x2 z j + (z + x2 ) k and S
is the open surface of the hemisphere x2 + y 2 + z 2 = a2 , 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 volume V , let us first consider the closed surface
S = S + S1 , where S1 is the circular area in the xy-plane given by x2 + y 2 ≤ a2 , z = 0; S then encloses a hemispherical volume V . By the divergence theorem we have
0
∇ · a dV =
a · dS = a · dS +
a · dS.
S
V
S
Now ∇ · a = −1 + 0 + 1 = 0, so we can write
a · dS = −
a · dS.
S
S1
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S1
LINE, SURFACE AND VOLUME INTEGRALS
y
R
dr
dy
C
dx
n̂ ds
x
Figure 11.11 A closed curve C in the xy-plane bounding a region R. Vectors
tangent and normal to the curve at a given point are also shown.
The surface integral over S1 is easily evaluated. Remembering that the normal to the
surface points outward from the volume, a surface element on S1 is simply dS = −k dx dy.
On S1 we also have a = (y − x) i + x2 k, so that
a · dS =
x2 dx dy,
I=−
S1
R
where R is the circular region in the xy-plane given by x2 + y 2 ≤ a2 . Transforming to plane
polar coordinates we have
2π
a
πa4
I=
ρ2 cos2 φ ρ dρ dφ =
cos2 φ dφ
ρ3 dρ =
.
4
0
0
R
It is also interesting to consider the two-dimensional version of the divergence
theorem. As an example, let us consider a two-dimensional planar region R in
the xy-plane bounded by some closed curve C (see figure 11.11). At any point
on the curve the vector dr = dx i + dy j is a tangent to the curve and the vector
n̂ ds = dy i − dx j is a normal pointing out of the region R. If the vector field a is
continuous and differentiable in R then the two-dimensional divergence theorem
in Cartesian coordinates gives
0
0
∂ax
∂ay
+
dx dy = a · n̂ ds = (ax dy − ay dx).
∂x
∂y
R
C
Letting P = −ay and Q = ax , we recover Green’s theorem in a plane, which was
discussed in section 11.3.
11.8.1 Green’s theorems
Consider two scalar functions φ and ψ that are continuous and differentiable in
some volume V bounded by a surface S. Applying the divergence theorem to the
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11.8 DIVERGENCE THEOREM AND RELATED THEOREMS
vector field φ∇ψ we obtain
0
φ∇ψ · dS =
∇ · (φ∇ψ) dV
S
V
2
φ∇ ψ + (∇φ) · (∇ψ) dV .
=
(11.19)
V
Reversing the roles of φ and ψ in (11.19) and subtracting the two equations gives
0
(φ∇ψ − ψ∇φ) · dS = (φ∇2 ψ − ψ∇2 φ) dV .
(11.20)
S
V
Equation (11.19) is usually known as Green’s first theorem and (11.20) as his
second. Green’s second theorem is useful in the development of the Green’s
functions used in the solution of partial differential equations (see chapter 21).
11.8.2 Other related integral theorems
There exist two other integral theorems which are closely related to the divergence
theorem and which are of some use in physical applications. If φ is a scalar field
and b is a vector field and both φ and b satisfy our usual differentiability
conditions in some volume V bounded by a closed surface S then
0
∇φ dV = φ dS,
(11.21)
0S
V
∇ × b dV = dS × b.
(11.22)
V
S
Use the divergence theorem to prove (11.21).
In the divergence theorem (11.18) let a = φc, where c is a constant vector. We then have
0
∇ · (φc) dV = φc · dS.
V
S
Expanding out the integrand on the LHS we have
∇ · (φc) = φ∇ · c + c · ∇φ = c · ∇φ,
since c is constant. Also, φc · dS = c · φdS, so we obtain
0
c · (∇φ) dV = c · φ dS.
V
S
Since c is constant we may take it out of both integrals to give
0
∇φ dV = c · φ dS,
c·
V
S
and since c is arbitrary we obtain the stated result (11.21). Equation (11.22) may be proved in a similar way by letting a = b × c in the
divergence theorem, where c is again a constant vector.
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LINE, SURFACE AND VOLUME INTEGRALS
11.8.3 Physical applications of the divergence theorem
The divergence theorem is useful in deriving many of the most important partial
differential equations in physics (see chapter 20). The basic idea is to use the
divergence theorem to convert an integral form, often derived from observation,
into an equivalent differential form (used in theoretical statements).
For a compressible fluid with time-varying position-dependent density ρ(r, t) and velocity
field v(r, t), in which fluid is neither being created nor destroyed, show that
∂ρ
+ ∇ · (ρv) = 0.
∂t
For an arbitrary volume V in the fluid, the conservation of mass tells us that the rate of
increase or decrease of the mass M of fluid in the volume must equal the net rate at which
fluid is entering or leaving the volume, i.e.
0
dM
= − ρv · dS,
dt
S
where S is the surface bounding V . But the mass of fluid in V is simply M = V ρ dV , so
we have
0
d
ρ dV + ρv · dS = 0.
dt V
S
Taking the derivative inside the first integral on the RHS and using the divergence theorem
to rewrite the second integral, we obtain
∂ρ
∂ρ
∇ · (ρv) dV =
dV +
+ ∇ · (ρv) dV = 0.
∂t
V ∂t
V
V
Since the volume V is arbitrary, the integrand (which is assumed continuous) must be
identically zero, so we obtain
∂ρ
+ ∇ · (ρv) = 0.
∂t
This is known as the continuity equation. It can also be applied to other systems, for
example those in which ρ is the density of electric charge or the heat content, etc. For the
flow of an incompressible fluid, ρ = constant and the continuity equation becomes simply
∇ · v = 0. In the previous example, we assumed that there were no sources or sinks in
the volume V , i.e. that there was no part of V in which fluid was being created
or destroyed. We now consider the case where a finite number of point sources
and/or sinks are present in an incompressible fluid. Let us first consider the
simple case where a single source is located at the origin, out of which a quantity
of fluid flows radially at a rate Q (m3 s−1 ). The velocity field is given by
Qr̂
Qr
=
.
4πr 3
4πr 2
Now, for a sphere S1 of radius r centred on the source, the flux across S1 is
0
v · dS = |v|4πr 2 = Q.
v=
S1
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11.8 DIVERGENCE THEOREM AND RELATED THEOREMS
Since v has a singularity at the origin it is not differentiable there, i.e. ∇ · v is not
defined there, but at all other points ∇ · v = 0, as required for an incompressible
fluid. Therefore, from the divergence theorem, for any closed surface S2 that does
not enclose the origin we have
0
v · dS =
∇ · v dV = 0.
S2
V
/
Thus we see that the surface integral S v · dS has value Q or zero depending
on whether or not S encloses the source. In order that the divergence theorem is
valid for all surfaces S, irrespective of whether they enclose the source, we write
∇ · v = Qδ(r),
where δ(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
for r = a,
#
f(r)δ(r − a) dV =
V
f(a)
if a lies in V
0
otherwise
for any well-behaved function f(r). Therefore, for any volume V containing the
source at the origin, we have
∇ · v dV = Q δ(r) dV = Q,
V
/
V
which is consistent with S v · dS = Q for 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 at r = a and a sink at r = b then the velocity field is
v=
(r − a)Q
(r − b)Q
−
4π|r − a|3
4π|r − b|3
and its divergence is given by
∇ · v = Qδ(r − a) − Qδ(r − b).
/
Therefore, the integral S v · dS has the value Q if S encloses the source, −Q if
S encloses the sink and 0 if S encloses 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 replace v by the electric field E.
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