Page 481 - Electromagnetics

By this we mean that each component of A is a linear combination of the components of
B:

A 1 (r, t) = a 11 B 1 (r, t) + a 12 B 2 (r, t) + a 13 B 3 (r, t),

A 2 (r, t) = a 21 B 1 (r, t) + a 22 B 2 (r, t) + a 23 B 3 (r, t),

A 3 (r, t) = a 31 B 1 (r, t) + a 32 B 2 (r, t) + a 33 B 3 (r, t).

Here the a ij may depend on space and time (or frequency). The prime on the second

ˆ ˆ ˆ
index indicates that A and B may be expressed in distinct coordinate frames (i 1 , i 2 , i 3 )
and (i 1 , i 2 , i 3 ), respectively. We have
ˆ ˆ ˆ

A 1 = a 11 i 1 + a 12 i 2 + a 13 i 3 · i 1 B 1 + i 2 B 2 + i 3 B 3 ,
ˆ
ˆ

ˆ

ˆ
ˆ

ˆ

A 2 = a 21 i 1 + a 22 i 2 + a 23 i 3 · i 1 B 1 + i 2 B 2 + i 3 B 3 ,
ˆ
ˆ

ˆ
ˆ
ˆ
ˆ

A 3 = a 31 i 1 + a 32 i 2 + a 33 i 3 · i 1 B 1 + i 2 B 2 + i 3 B 3 ,
ˆ
ˆ
ˆ
ˆ
ˆ

ˆ

and since B = i 1 B 1 + i 2 B 2 + i 3 B 3 we can write

ˆ
ˆ
ˆ

ˆ
ˆ
ˆ
A = i 1 (a · B) + i 2 (a · B) + i 3 (a · B)

1 2 3
where

a = a 11 i 1 + a 12 i 2 + a 13 i 3 ,
1 ˆ ˆ ˆ
a = a 21 i 1 + a 22 i 2 + a 23 i 3 ,

2 ˆ ˆ ˆ
a = a 31 i 1 + a 32 i 2 + a 33 i 3 .

ˆ
ˆ
ˆ
3
In shorthand notation
A = ¯ a · B (A.69)
where
ˆ
ˆ
ˆ

¯ a = i 1 a + i 2 a + i 3 a . (A.70)

1
3
2
Written out, the quantity ¯ a looks like
¯ a = a 11 (i 1 i 1 ) + a 12 (i 1 i 2 ) + a 13 (i 1 i 3 ) +
ˆ ˆ
ˆ ˆ
ˆ ˆ
+ a 21 (i 2 i 1 ) + a 22 (i 2 i 2 ) + a 23 (i 2 i 3 ) +
ˆ ˆ
ˆ ˆ
ˆ ˆ
+ a 31 (i 3 i 1 ) + a 32 (i 3 i 2 ) + a 33 (i 3 i 3 ).
ˆ ˆ
ˆ ˆ
ˆ ˆ
Terms such as i 1 i 1 are called dyads, while sums of dyads such as ¯ a are called dyadics.
ˆ ˆ
The components a ij of ¯ a may be conveniently placed into an array:

 
a 11 a 12 a 13

[¯ a] =  a 21 a 22 a 23  .

a 31 a 32 a 33

Writing
   
A 1 B 1
[A] =  A 2  , [B] =  B 2  ,
A 3 B 3
© 2001 by CRC Press LLC