Radio Science,
Volume 22, Number 7, Pages 1283-1288, December 1987
Some essential
formulas
in dyadic analysis and their applications
C. T. Tai
Radiation Laboratory, Department
of Electrical Engineering
and Computer Sciences,
University of Michigan, Ann Arbor
(Received February 2, 1987' revised June 12, 1987' accepted July 7, 1987.)
Some commonly used formulas in dyadic analysis are reviewed and summarized
in this article. In
particular, we have derived the dyadic-dyadic version of two vector-dyadic Green's theorems. The
application of these
theorems
to electromagnetics
is illustrated by two examples.
1.
INTRODUCTION
Dyadic analysis as a branch of applied mathemat-
ics was enunciated by American physicist J. W.
Gibbs after he introduced vector analysis.
In his orig-
inal treatise [Gibbs and Wilson, 1913], many basic
definitions in dyadic analysis were given, but none of
the integral theorems were discussed. The use of
dyadic analysis in electromagnetics appeared in the
book by Morse and Feshbach [1953] and the one by
Collin [1960]. Van Bladel, to whom this article is
dedicated, was the first author to provide a list of
formulas in dyadic analysis in his book Electro-
magnetic Fields [Van Bladel, 1964].
In view of the growing interest of applying dyadic
Green's function technique to electromagnetic prob-
lems in recent years, it seems desirable to supplement
Van Bladel's list and collect them in one place for the
convenience of people interested in this technique. In
order to present a logical development of various
formulas with different degree of complexity, we
decide to start from the very basic definition of
dyadic analysis. Some repetition of elementary ma-
terial is therefore unavoidable. Finally, we introduce
two dyadic-dyadic Green's theorems which have not
appeared before. Two examples are given to show
the application of these
theorems.
2.
DYADIC
ALGEBRA
2,1.
Dyadic functions or dyadics
A vector function or a vector, F, expressed
in a
Cartesian coordinate system
is defined by
Copyright 1987 by the American Geophysical Union.
Paper number 7S0595.
0048-6604/87/007S-0595508.00
3
F = • F,•,
(1)
i=1
where Fi, with i= 1, 2, 3, denote the three scalar
components of the vector and x i denote the three
unit vectors in the direction of •i, which are com-
monly referred to as x, y, z. Although (1) can be used
to define a vector in any orthogonal or nonorthogon-
al system, we use xi in subsequent
sections
to denote
Cartesian
variables.
Now
we consider
three
distinct
vector
functions
denoted by
3
F•= • Foi ,
j=1,2,3
(2)
i=1
Then a dyadic function or a dyadic is defined by
3
j=l
(3)
where F
j, with
j = 1, 2, 3, are designated
as the three
vector components of F. If we substitute the ex-
pression
for Fj defined
by (2) into (3),
then
• can be
written
in the form
3
3
F= •
• F u •, •
(4)
i=1
j=l
where F u are designated
as the nine scalar compo-
nents of F and the doublet •
as the nine unit
dyadics or dyads each being formed by a pair of unit