16.9 Divergence and curl of a vector field 469
(16.71)
is the electrostatic field of charge q
1, andφ 1 = 1 V 2 q
21 = 1 q
124 πε
0ris the corresponding
(scalar) electrostatic potential field.
The meaning of the gradient of a function of positionf(r)is clarified by considering
how the value of the function changes from point to point in space. We consider
a differential (infinitesimal) displacementr 1 → 1 r 1 + 1 dror, in cartesian components,
(x, y, z) 1 → 1 (x 1 + 1 dx, y 1 + 1 dy, z 1 + 1 dz):
dr 1 = 1 (dx, dy, dz) 1 = 1 idx 1 + 1 jdy 1 + 1 kdz (16.72)
The corresponding change in the function,df 1 = 1 f(r 1 + 1 dr) 1 − 1 f(r), is the total differential
(16.73)
This can be written as the scalar product of the gradient∇fand the displacement dr
(Equations (16.68) and (16.72)),
so that
df 1 = 1 ∇f 1
·1 dr (16.74)
The quantity∇fis therefore the generalization to three dimensions of the ordinary
derivativedf 2 dx; the vector operator ∇is the generalization of the differential
(gradient) operatord 2 dx, and is sometimes written asd 2 dr.
16.9 Divergence and curl of a vector field
The divergence
The divergenceof a vector fieldv(r) is defined as the scalar quantity
(16.75)
In hydrodynamics, the vector field is the velocity field of a fluid, and the divergence is
a flux density; the value of div 1 v(r)dVat a point ris a measure of the net flux, or rate
of flow, of fluid out of a volume elementdVat the point.
divvv==
∂
∂
∂
∂
∂
∂
∇∇
·v
v
vxyzxyz
∂
∂
∂
∂
∂
∂
++ =
∂
∂
f
x
f
y
f
z
dx dy dz
f
ijkijk 11
·()
xx
dx
f
y
dy
f
z
- dz
∂
∂
∂
∂
df
f
x
dx
f
y
dy
f
z
= dz
∂
∂
∂
∂
∂
∂
Er==−∇
q
r
1024 πε