Page 86 - Engineering Electromagnetics, 8th Edition

68 ENGINEERING ELECTROMAGNETICS

We first consider the dot products of the unit vectors, discarding the six zero terms,
and obtain the result that we recognize as the divergence of D:

∂D x ∂D y ∂D z
∇ · D = + + = div(D)
∂x ∂y ∂z
The use of ∇ · D is much more prevalent than that of div D, although both usages
have their advantages. Writing ∇ · D allows us to obtain simply and quickly the correct
partial derivatives, but only in rectangular coordinates, as we will see. On the other
hand, div D is an excellent reminder of the physical interpretation of divergence.
We shall use the operator notation ∇ · D from now on to indicate the divergence
operation.
The vector operator ∇ is used not only with divergence, but also with several
other very important operations that appear later. One of these is ∇u, where u is any
scalar field, and leads to

∂ ∂ ∂ ∂u ∂u ∂u
∇u = a x + a y + a z u = a x + a y + a z
∂x ∂y ∂z ∂x ∂y ∂z
The ∇ operator does not have a specific form in other coordinate systems. If we
are considering D in cylindrical coordinates, then ∇ · D still indicates the divergence
of D,or
1 ∂ 1 ∂D φ ∂D z
∇ · D = (ρD ρ ) + +
ρ ∂ρ ρ ∂φ ∂z
where this expression has been taken from Section 3.5. We have no form for ∇ itself
to help us obtain this sum of partial derivatives. This means that ∇u,as yet unnamed
but easily written in rectangular coordinates, cannot be expressed by us at this time
in cylindrical coordinates. Such an expression will be obtained when ∇u is defined
in Chapter 4.
We close our discussion of divergence by presenting a theorem that will be needed
several times in later chapters, the divergence theorem. This theorem applies to any
vector field for which the appropriate partial derivatives exist, although it is easiest
for us to develop it for the electric flux density. We have actually obtained it already
and now have little more to do than point it out and name it, for starting from Gauss’s
law, we have

D · dS = Q = ρ ν dv = ∇ · D dv
S vol vol
The first and last expressions constitute the divergence theorem,

∇ · D dv (17)
D · dS =
S vol
which may be stated as follows:
The integral of the normal component of any vector field over a closed surface is equal to
the integral of the divergence of this vector field throughout the volume enclosed by the
closed surface.