18
1 Preliminaries
In the same way, the similar relationship can also be obtained in y-direction and
z-direction, thus the net flux flowing out of the elemental volume V is
S
a · dS =
∂a x
∂ x
xyz +
∂a y
∂ y
xyz +
∂a z
∂z
xyz
The both ends of the above expression are divided by the elemental volume V =
xyz and take limit, we obtain
div a = lim
V →0
S a · dS
V
=
∂a x
∂ x
+
∂a y
∂ y
+
∂a z
∂z
Moreover
∇ · a =
∂
∂ x
i +
∂
∂ y
j +
∂
∂z
k
· (a x i + a y j + a z k) =
∂a x
∂ x
+
∂a y
∂ y
+
∂a z
∂z
so that
div a = ∇ · a =
∂a x
∂ x
+
∂a y
∂ y
+
∂a z
∂z
(1.3.37)
Equation (1.3.37) is the expression of divergence in rectangular coordinate system.
When calculating the divergence of a vector field, to apply this expression is usually
more convenient than to directly quote the definition of divergence.
Equation (1.3.37) shows that the divergence of a vector function a is a scalar
function. At an arbitray point in the vector field, the divergence of vector a is the sum
of its component on each coordinate axis to the partial derivative of each coordinate
variable.
The divergence has the following properties:
(1) The divergence of the sum of the two vector functions is equal to the sum of the
divergences of the vector functions
div(a + b) = div a + div b
(1.3.38)
∇ · (a + b) = ∇ · a + ∇ · b
(1.3.39)
(2) The divergence of the product of the scalar function and vector function is equal
to the product of the scalar function and the divergence of vector function plus
the inner product of the vector function and the gradient of scalar function
div(ϕa) = ϕdiv a + a · grad ϕ
(1.3.40)
∇ · (ϕa) = ϕ∇ · a + a · ∇ϕ
(1.3.41)