Page 484 - Electromagnetics

If α is a scalar, then the product α¯ a is a dyadic with components equal to α times
the components of ¯ a. Dyadic addition may be accomplished by adding individual dyadic
components as long as the dyadics are expressed in the same coordinate system. Sub-
traction is accomplished by adding the negative of a dyadic, which is defined through
scalar multiplication by −1.
Some useful dyadic identities appear in Appendix B. Many more can be found in Van
Bladel [202].
The various vector derivatives may also be extended to dyadics. Computations are
ˆ
ˆ
ˆ
easiest in rectangular coordinates, since i 1 = ˆ x, i 2 = ˆ y, and i 3 = ˆ z are constant with
position. The dyadic
¯ a = a x ˆ x + a y ˆ y + a z ˆ z
has divergence

∇· ¯ a = (∇· a x )ˆ x + (∇· a y )ˆ y + (∇· a z )ˆ z,
and curl
∇× ¯ a = (∇× a x )ˆ x + (∇× a y )ˆ y + (∇× a z )ˆ z.

Note that the divergence of a dyadic is a vector while the curl of a dyadic is a dyadic.
The gradient of a vector a = a x ˆ x + a y ˆ y + a z ˆ z is

∇a = (∇a x )ˆ x + (∇a y )ˆ y + (∇a z )ˆ z,
a dyadic quantity.
The dyadic derivatives may be expressed in coordinate-free notation by using the vector
representation. The dyadic AB has divergence
∇· (AB) = (∇· A)B + A · (∇B)

and curl
∇× (AB) = (∇× A)B − A × (∇B).
The Laplacian of a dyadic is a dyadic given by

2
∇ ¯ a =∇(∇· ¯ a) −∇ × (∇× ¯ a).
The divergence theorem for dyadics is

∇· ¯ a dV = ˆ n · ¯ a dS.
V S
Some of the other common differential and integral identities for dyadics can be found
in Van Bladel [202] and Tai [192].

Special dyadics. We say that ¯ a is symmetric if
B · ¯ a = ¯ a · B
T
for any vector B. This requires ¯ a = ¯ a, i.e., a ij = a ji . We say that ¯ a is antisymmetric

if
B · ¯ a =−¯ a · B
T
for any B. In this case ¯ a =−¯ a. That is, a ij =−a ji and a ii = 0. A symmetric dyadic

has only six independent components while an antisymmetric dyadic has only three. The

© 2001 by CRC Press LLC