
567Appendix 1
a1.2.4 divergence oF a vecTor
If
V(, ,)
ˆˆˆ
xxxVeVeVe
123112233
=++
is a differentiable vector eld [i.e., it is dened and differentiable
at each point (x
1
, x
2
, x
3
) in a certain region of space], the divergence of V, written ∇ ∙ V or div V, is
dened by the scalar product
∇⋅ =++
⋅++
V
x
e
x
e
x
eVeVe
∂
∂
∂
∂
∂
∂
1
1
2
2
3
31122
ˆˆˆˆˆ
VVe
V
x
V
x
V
x
33
1
1
2
2
3
3
ˆ
()
=++
∂
∂
∂
∂
∂
∂
.
(A1.34)
It is obvious that the result is a scalar eld. Note the analogy with A ∙ B = A
1
B
1
+ A
2
B
2
+ A
3
B
3
, but
also note that ∇ ∙ V ≠ V ∙ ∇ (bear in mind that ∇ is an operator). V ∙ ∇ is a scalar differential operator:
VV
x
V
x
V
x
⋅∇=++
1
1
2
2
3
3
∂
∂
∂
∂
∂
∂
.
The physical meaning of the divergence is best ...