What is Divergence in Mathematics ?

DIVERGENCE, ∇

Differentiating a vector function is a simple extension of differentiating scalar quantities.

Suppose r(t) describes the position of a satellite at some time t . Then, for differentiation

with respect to time,

dr(t)

dt = lim

→0

r(t + t)− r(t)

t = v, linear velocity.

Graphically, we again have the slope of a curve, orbit, or trajectory, as shown in Fig. 1.22.

If we resolve r(t) into its Cartesian components, dr/dt always reduces directly to a

vector sum of not more than three (for three-dimensional space) scalar derivatives. In other

coordinate systems (Chapter 2) the situation is more complicated, for the unit vectors are

no longer constant in direction. Differentiation with respect to the space coordinates is

handled in the same way as differentiation with respect to time, as seen in the following

paragraphs.

1.7 Divergence, ∇ 39

FIGURE 1.22 Differentiation of a vector.

In Section 1.6, ∇ was defined as a vector operator. Now, paying attention to both its

vector and its differential properties, we let it operate on a vector. First, as a vector we dot

it into a second vector to obtain

∇ · V =

∂Vx

∂x +

∂Vy

∂y +

∂Vz

∂z

, (1.65a)

known as the divergence of V. This is a scalar, as discussed in Section 1.3.

Example 1.7.1 DIVERGENCE OF COORDINATE VECTOR

Calculate ∇ · r:

∇ · r =

ˆx

∂

∂x + ˆy

∂

∂y + ˆz

∂

∂z

· (ˆxx + ˆyy + ˆzz)

=

∂x

∂x +

∂y

∂y +

∂z

∂z

,

or ∇ · r = 3.

Example 1.7.2 DIVERGENCE OF CENTRAL FORCE FIELD

Generalizing Example 1.7.1,

∇ ·

rf (r)

=

∂

∂x

x f(r)

+

∂

∂y

y f(r)

+

∂

∂z

zf (r)

= 3f (r) +

x2

r

df

dr +

y2

r

df

dr +

z2

r

df

dr

= 3f (r) + r

df

dr

.

40 Chapter 1 Vector Analysis

The manipulation of the partial derivatives leading to the second equation in Example 1.7.2

is discussed in Example 1.6.1. In particular, if f (r) = rn−1,

∇ ·

rrn−1

= ∇ · ˆrrn

= 3rn−1 +(n −1)rn−1

= (n+ 2)rn−1. (1.65b)

This divergence vanishes for n=−2, except at r = 0, an important fact in Section 1.14.

Example 1.7.3 INTEGRATION BY PARTS OF DIVERGENCE

Let us prove the formula

f (r)∇ · A(r)d3r = −

A · ∇f d3r, where A or f or both

vanish at infinity.

To show this, we proceed, as in Example 1.6.3, by integration by parts after writing

the inner product in Cartesian coordinates. Because the integrated terms are evaluated at

infinity, where they vanish, we obtain

f (r)∇ · A(r)d3r =

f

∂Ax

∂x

dx dy dz +

∂Ay

∂y

dy dx dz +

∂Az

∂z

dz dx dy

=−

Ax

∂f

∂x

dx dy dz +Ay

∂f

∂y

dy dx dz +Az

∂f

∂z

dz dx dy

=−

A · ∇f d3r.

A Physical Interpretation

To develop a feeling for the physical significance of the divergence, consider ∇ · (ρv) with

v(x, y, z), the velocity of a compressible fluid, and ρ(x, y, z), its density at point (x, y, z).

If we consider a small volume dx dy dz (Fig. 1.23) at x = y = z = 0, the fluid flowing into

this volume per unit time (positive x-direction) through the face EFGH is (rate of flow

in)EFGH = ρvx |x=0 = dy dz. The components of the flow ρvy and ρvz tangential to this

face contribute nothing to the flow through this face. The rate of flow out (still positive

x-direction) through face ABCD is ρvx |x=dx dy dz. To compare these flows and to find the

net flow out, we expand this last result, like the total variation in Section 1.6.15 This yields

(rate of flow out)ABCD = ρvx|x=dx dy dz

=

ρvx +

∂

∂x

(ρvx)dx

x=0

dy dz.

Here the derivative term is a first correction term, allowing for the possibility of nonuniform

density or velocity or both.16 The zero-order term ρvx |x=0 (corresponding to uniform flow)

15Here we have the increment dx and we show a partial derivative with respect to x since ρvx may also depend on y and z.

16Strictly speaking, ρvx is averaged over face EFGH and the expression ρvx +(∂/∂x)(ρvx)dx is similarly averaged over face

ABCD. Using an arbitrarily small differential volume, we find that the averages reduce to the values employed here.

1.7 Divergence, ∇ 41

FIGURE 1.23 Differential rectangular parallelepiped (in first octant).

cancels out:

Net rate of flow out|x =

∂

∂x

(ρvx)dx dy dz.

Equivalently, we can arrive at this result by

lim

x→0

ρvx( x, 0, 0)−ρvx (0, 0, 0)

x ≡

∂[ρvx(x, y, z)]

∂x

0,0,0

.

Now, the x-axis is not entitled to any preferred treatment. The preceding result for the two

faces perpendicular to the x-axis must hold for the two faces perpendicular to the y-axis,

with x replaced by y and the corresponding changes for y and z: y →z, z→x. This is

a cyclic permutation of the coordinates. A further cyclic permutation yields the result for

the remaining two faces of our parallelepiped. Adding the net rate of flow out for all three

pairs of surfaces of our volume element, we have

net flow out

(per unit time) =

∂

∂x

(ρvx )+

∂

∂y

(ρvy )+

∂

∂z

(ρvz)

dx dy dz

= ∇ · (ρv)dx dy dz. (1.66)

Therefore the net flow of our compressible fluid out of the volume element dx dy dz per

unit volume per unit time is ∇ · (ρv). Hence the name divergence. A direct application is

in the continuity equation

∂ρ

∂t +∇ · (ρv) = 0, (1.67a)

which states that a net flow out of the volume results in a decreased density inside the

volume. Note that in Eq. (1.67a), ρ is considered to be a possible function of time as well

as of space: ρ(x, y, z, t). The divergence appears in a wide variety of physical problems,

42 Chapter 1 Vector Analysis

ranging from a probability current density in quantum mechanics to neutron leakage in a

nuclear reactor.

The combination ∇ · (fV), in which f is a scalar function and V is a vector function,

may be written

∇ · (fV) =

∂

∂x

(f Vx )+

∂

∂y

(f Vy )+

∂

∂z

(f Vz)

=

∂f

∂x

Vx + f

∂Vx

∂x +

∂f

∂y

Vy + f

∂Vy

∂y +

∂f

∂z

Vz +f

∂Vz

∂z

= (∇f ) · V+ f∇ · V, (1.67b)

which is just what we would expect for the derivative of a product. Notice that ∇ as a

differential operator differentiates both f and V; as a vector it is dotted into V (in each

term).

If we have the special case of the divergence of a vector vanishing,

∇ · B = 0, (1.68)

the vector B is said to be solenoidal, the term coming from the example in which B is the

magnetic induction and Eq. (1.68) appears as one of Maxwell’s equations. When a vector

is solenoidal, it may be written as the curl of another vector known as the vector potential.

(In Section 1.13 we shall calculate such a vector potential.)

Exercises

1.7.1 For a particle moving in a circular orbit r = ˆxr cos ωt + ˆyr sin ωt,

(a) evaluate r× ˙r, with ˙r = dr

dt = v.

(b) Show that ¨r +ω2r = 0 with ¨r = dv

dt .

The radius r and the angular velocity ω are constant.

ANS. (a) ˆzωr2.

1.7.2 Vector A satisfies the vector transformation law, Eq. (1.15). Show directly that its time

derivative dA/dt also satisfies Eq. (1.15) and is therefore a vector.

1.7.3 Show, by differentiating components, that

(a) d

dt (A · B) = dA

dt · B+ A · dB

dt ,

(b) d

dt (A×B) = dA

dt ×B+ A× dB

dt ,

just like the derivative of the product of two algebraic functions.

1.7.4 In Chapter 2 it will be seen that the unit vectors in non-Cartesian coordinate systems are

usually functions of the coordinate variables, ei = ei(q1, q2, q3) but |ei| = 1. Show that

either ∂ei/∂qj = 0 or ∂ei/∂qj is orthogonal to ei .

Hint. ∂e2

i /∂qj = 0.

1.8 Curl, ∇× 43

1.7.5 Prove ∇ · (a×b) = b · (∇ ×a)−a · (∇ ×b).

Hint. Treat as a triple scalar product.

1.7.6 The electrostatic field of a point charge q is

E =

q

4πε0 · ˆr

r2 .

Calculate the divergence of E. What happens at the origin?