Page 250 - Determinants and Their Applications in Mathematical Physics

6

Applications of Determinants in
Mathematical Physics

6.1 Introduction

This chapter is devoted to verifications of the determinantal solutions of
several equations which arise in three branches of mathematical physics,
namely lattice, relativity, and soliton theories. All but one are nonlinear.
Lattice theory can be defined as the study of elements in a two- or
three-dimensional array under the influence of neighboring elements. For
example, it may be required to determine the electromagnetic state of one
loop in an electrical network under the influence of the electromagnetic
field generated by neighboring loops or to study the behavior of one atom
in a crystal under the influence of neighboring atoms.
Einstein’s theory of general relativity has withstood the test of time and
is now called classical gravity. The equations which appear in this chapter
arise in that branch of the theory which deals with stationary axisymmetric
gravitational fields.
A soliton is a solitary wave and soliton theory can be regarded as a
branch of nonlinear wave theory.
The term determinantal solution needs clarification since it can be ar-
gued that any function can be expressed as a determinant and, hence, any
solvable equation has a solution which can be expressed as a determinant.
The term determinantal solution shall mean a solution containing a deter-
minant which has not been evaluated in simple form and may possibly be
the simplest form of the function it represents. A number of determinants
have been evaluated in a simple form in earlier chapters and elsewhere, but