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 solitarywave andsoliton theory can be regarded as a

branch of nonlinearwavetheory.

The termdeterminantal solutionneeds 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 termdeterminantal solutionshall 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