Applied Mathematical Sciences. 134. New York, NY: Springer. xiv, 376 p. DM 144.00; öS 1052.00; sFr. 131.50; £55.50; $ 69.95 (1999).
The book is very specific in the sense that it deals fully on the determinants (of finite dimension) and its applications. It contains a detailed account of all important relations in the analytic theory of determinants from the classical work of Laplace, Cauchy and Jacobi in the eighteenth and nineteenth centuries to almost the end of twentieth century developments, in particular, the investigations of the last few decades, on the presentation of the solutions of nonlinear differential equations in terms of determinants of functions of the independent variables. In this context, as the authors observed, the true title of the book should have been “The analytic theory of determinants with applications to the solutions of certain nonlinear equations of mathematical physics”.
The book begins with a brief note on Grassmann exterior algebra and then proceeds to define a determinant by means of Grassmann identities. The treatment mostly consists of extensive application of column vectors and scaled cofactors. The first five chapters are purely mathematical in nature and contain old and new proofs of several old theorems together with a number of theorems, identities and conjectures which have not hitherto been published. In order to emphasize that the methods are no less interesting and important as the results, quite a few theorems are given two independent proofs. Chapter 3 develops the classical results e.g., the Laplace expansion, the Cauchy and Jacobi identities, variants, etc.. The whole of Chapter 4 is devoted to particular determinants including alternants, Wronskian and Hankelians. The next Chapter is on further developments of determinant theory including the Cusick and Matsuno identities. The last chapter is devoted to verifications of the determinantal solutions of several (mostly nonlinear) equations which arise in three branches of mathematical physics, namely lattice, relativity and soliton theories. They include the Korteweg-de Vries, Toda and Einstein equations.
The appendix consists mainly of non-determinantal 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 mentioned in the text but also many other contributors to the theory of determinants and related topics.
Though derivative of a determinants is quite an old concept, its application to the solution of nonlinear equations is a recent development. This asserts that the theory of determinant has emerged from the confines of classical algebra into the brighter world of analysis. Notwithstanding, the fact that the contents of this book can be described as complicated application of classical algebra and differentiation, the book proclaims that the determinant theory has a standing in its own right, not merely an adjunct to the matrix theory.