Page 8 - Determinants and Their Applications in Mathematical Physics

Preface vii
(n)
minors of A n are denoted by M , etc., retainer minors are denoted by
ij
(n)
N ij , etc., simple cofactors are denoted by A , etc., and scaled cofactors
ij
are denoted by A , etc. The n may be omitted from any passage if all the
ij
n
determinants which appear in it have the same order. The letter D, some-
times with a suffix x, t, etc., is reserved for use as a differential operator.
The letters h, i, j, k, m, p, q, r, and s are usually used as integer param-
eters. The letter l is not used in order to avoid confusion with the unit
integer. Complex numbers appear in some sections and pose the problem
2
of conflicting priorities. The notation ω = −1 has been adopted since the
letters i and j are indispensable as row and column parameters, respec-
tively, in passages where a large number of such parameters are required.
Matrices are seldom required, but where they are indispensable, they ap-
pear in boldface symbols such as A and B with the simple convention
A = det A, B = det B, etc. The boldface symbols R and C, with suffixes,
are reserved for use as row and column vectors, respectively. Determinants,
their elements, their rejecter and retainer minors, their simple and scaled
cofactors, their row and column vectors, and their derivatives have all been
expressed in a notation which we believe is simple and clear and we wish
to see this notation adopted universally.
The Appendix consists mainly of nondeterminantal relations which have
been removed from the main text to allow the analysis to proceed without
interruption.
The Bibliography contains references not only to all the authors men-
tioned in the text but also to many other contributors to the theory of
determinants and related subjects. The authors have been arranged in al-
phabetical order and reference to Mathematical Reviews, Zentralblatt f¨ur
Mathematik, and Physics Abstracts have been included to enable the reader
who has no easy access to journals and books to obtain more details of their
contents than is suggested by their brief titles.
The true title of this book is The Analytic Theory of Determinants with
Applications to the Solutions of Certain Nonlinear Equations of Mathe-
matical Physics, which satisfies the requirements of accuracy but lacks the
virtue of brevity. Chapter 1 begins with a brief note on Grassmann algebra
and then proceeds to define a determinant by means of a Grassmann iden-
tity. Later, the Laplace expansion and a few other relations are established
by Grassmann methods. However, for those readers who find this form of
algebra too abstract for their tastes or training, classical proofs are also
given. Most of the contents of this book can be described as complicated
applications of classical algebra and differentiation.
In a book containing so many symbols, misprints are inevitable, but we
hope they are obvious and will not obstruct our readers’ progress for long.
All reports of errors will be warmly appreciated.
We are indebted to our colleague, Dr. Barry Martin, for general advice
on computers and for invaluable assistance in algebraic computing with the