Page 480 - Electromagnetics

We now come to an essential tool that we employ throughout the book. Using the
divergence theorem we can rewrite (A.64) as

d ∂F
∇· F dV = ∇· dV + (∇· F)v · dS.
dt V (t) V (t) ∂t S(t)
Replacing ∇· F by the scalar field ρ we have

d ∂ρ
ρ dV = dV + ρv · dS. (A.65)
dt V (t) V (t) ∂t S(t)
In this general form of the transport theorem v is an arbitrary velocity. In most appli-
cations v = u describes the motion of a material substance; then

D ∂ρ
ρ dV = dV + ρu · dS, (A.66)
Dt V (t) V (t) ∂t S(t)
which is the Reynolds transport theorem [214]. The D/Dt notation implies that V (t)
retains exactly the same material elements as it moves and deforms to follow the material
substance.
We may rewrite the Reynolds transport theorem in various forms. By the divergence
theorem we have
d ∂ρ
ρ dV = +∇ · (ρv) dV.
dt V (t) V (t) ∂t
Setting v = u, using (B.42), and using (A.59) for the material derivative of ρ, we obtain

D Dρ
ρ dV = + ρ∇· u dV. (A.67)
Dt V (t) V (t) Dt
We may also generate a vector form of the general transport theorem by taking ρ in
(A.65) to be a component of a vector. Assembling all of the components we have
d ∂A
A dV = dV + A(v · ˆ n) dS. (A.68)
dt V (t) V (t) ∂t S(t)

A.3 Dyadic analysis

Dyadic analysis was introduced in the late nineteenth century by Gibbs to generalize
vector analysis to problems in which the components of vectors are related in a linear
manner. It has now been widely supplanted by tensor theory, but maintains a foothold in
engineering where the transformation properties of tensors are not paramount (except,
of course, in considerations such as those involving special relativity). Terms such as
“tensor permittivity” and “dyadic permittivity” are often used interchangeably.

Component form representation. We wish to write one vector field A(r, t) as a
linear function of another vector field B(r, t):
A = f (B).

© 2001 by CRC Press LLC