ratio of VolumeFlux turns out to measure a point characteristic of the vector field called
the divergence. Symbolically, the divergence is a scalar quantity and is defined by
the limiting process
div F = lim∆V→ 0
∆S→ 0
∫∫
R
F·dS
∆V
= lim∆V→ 0
∆S→ 0
Flux
Volume
(8 .6)
Consider the evaluation of this limit in the special case where the closed surface
is a sphere. Consider a sphere of radius > 0 centered at a point P 0 (x 0 , y 0 , z 0 )situated
in a vector field
F=F(x, y, z ) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z )ˆe 2 +F 3 (x, y, z )ˆe 3.
Express the sphere in the parametric form as
x=x 0 +sin θcos φ, 0 ≤φ≤ 2 π
y=y 0 +sin θsin φ, 0 ≤θ≤π
z=z 0 +cos θ
then the position vector to a point on this sphere is given by
r =r (θ, φ ) = (x 0 +sin θcos φ)ˆe 1 + (y 0 +sin θsin φ)ˆe 2 + (z 0 +cos θ)eˆ 3
The coordinate curves on the surface of this sphere are
r (θ 0 , φ ) and r (θ, φ 0 ) θ 0 , φ 0 constants
and one can show the element of surface area on the sphere is given by
dS =|
∂r
∂θ ×
∂r
∂φ |dθ dφ =
√
EG −F^2 dθ dφ =^2 sin θ dθ dφ
Aunit normal to the surface of the sphere is given by
ˆen=
∂r
∂θ ×
∂r
∣ ∂φ
∣∣∂r
∂θ ×
∂r
∂φ
∣∣
∣
= sin θcos φˆe 1 + sin θsin φˆe 2 + cos θˆe 3
The flux integral given by equation (8.6) and integrated over the surface of a sphere
can then be expressed as
φ=
∫∫
R
F·dS=
∫∫
R
F·ˆendS
φ=
∫φ=2π
φ=0
∫θ=π
θ=0
F(x 0 +sin θcos φ, y 0 +sin θsin φ, z 0 +cos θ)ˆen^2 sin θ dθdφ.
