Page 483 - Electromagnetics

can be represented in component form using
ˆ
ˆ
ˆ

AB = i 1 a + i 2 a + i 3 a 3

2
1
where

a = A 1 B 1 i 1 + A 1 B 2 i 2 + A 1 B 3 i 3 ,
1 ˆ ˆ ˆ

a = A 2 B 1 i 1 + A 2 B 2 i 2 + A 2 B 3 i 3 ,
2 ˆ ˆ ˆ
a = A 3 B 1 i 1 + A 3 B 2 i 2 + A 3 B 3 i 3 ,

3 ˆ ˆ ˆ
or using
AB = a 1 i 1 + a 2 i 2 + a 3 i 3
ˆ
ˆ
ˆ
where
ˆ
ˆ
ˆ
a 1 = i 1 A 1 B 1 + i 2 A 2 B 1 + i 3 A 3 B 1 ,

ˆ
ˆ
ˆ
a 2 = i 1 A 1 B 2 + i 2 A 2 B 2 + i 3 A 3 B 2 ,

ˆ
ˆ
ˆ
a 3 = i 1 A 1 B 3 + i 2 A 2 B 3 + i 3 A 3 B 3 .

Note that if we write ¯ a = AB then a ij = A i B j .

A simple dyad AB by itself cannot represent a general dyadic ¯ a; only six independent
quantities are available in AB (the three components of A and the three components
of B), while an arbitrary dyadic has nine independent components. However, it can be
shown that any dyadic can be written as a sum of three dyads:
¯ a = AB + CD + EF.
This is called a vector representation of ¯ a.If V is a vector, the distributive laws
¯ a · V = (AB + CD + EF) · V = A(B · V) + C(D · V) + E(F · V),
V · ¯ a = V · (AB + CD + EF) = (V · A)B + (V · C)D + (V · E)F,
apply.
Dyadic algebra and calculus. The cross product of a vector with a dyadic produces
another dyadic. If ¯ a = AB + CD + EF then by definition
¯ a × V = A(B × V) + C(D × V) + E(F × V),
V × ¯ a = (V × A)B + (V × C)D + (V × E)F.
The corresponding component forms are
ˆ
ˆ
ˆ

¯ a × V = i 1 (a × V) + i 2 (a × V) + i 3 (a × V),
1 2 3
V × ¯ a = (V × a 1 )i 1 + (V × a 2 )i 2 + (V × a 3 )i 3 ,
ˆ
ˆ
ˆ
where we have used (A.70) and (A.71), respectively. Interactions between dyads or
dyadics may also be defined. The dot product of two dyads AB and CD is a dyad given
by
(AB) · (CD) = A(B · C)D = (B · C)(AD).
The dot product of two dyadics can be found by applying the distributive property.
© 2001 by CRC Press LLC