Determinants and Their Applications in Mathematical Physics
The last treatise on the theory of determinants, by T. Muir, revised and enlarged by W. H. Metzler, was published by Dover Publications Inc. in 1960. It is an unabridged and corrected republication of the edition ori- nally published by Longman, Green and
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The last treatise on the theory of determinants, by T. Muir, revised and enlarged by W.H. Metzler, was published by Dover Publications Inc. in 1960. It is an unabridged and corrected republication of the edition originally published by Longman, Green and Co. in 1933 and contains a preface by Metzler dated 1928. The Table of Contents of this treatise is given in Appendix 13. A small number of other books devoted entirely to determinants have been published in English, but they contain little if anything of importance that was not known to Muir and Metzler. A few have appeared in German and Japanese. In contrast, the shelves of every mathematics library groan under the weight of books on linear algebra, some of which contain short chapters on determinants but usually only on those aspects of the subject which are applicable to the chapters on matrices. There appears to be tacit agreement among authorities on linear algebra that determinant theory is important only as a branch of matrix theory. In sections devoted entirely to the establishment of a determinantal relation, many authors define a determinant by first defining a matrix M and then adding the words: “Let det M be the determinant of the matrix M” as though determinants have no separate existence. This belief has no basis in history. The origins of determinants can be traced back to Leibniz (1646–1716) and their properties were developed by Vandermonde (1735–1796), Laplace (1749–1827), Cauchy (1789–1857) and Jacobi (1804–1851) whereas matrices were not introduced until the year of Cauchy’s death, by Cayley (1821–1895). In this book, most determinants are defined directly.
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Preface
It may well be perfectly legitimate to regard determinant theory as a branch of matrix theory, but it is such a large branch and has such large and independent roots, like a branch of a banyan tree, that it is capable of leading an independent life. Chemistry is a branch of physics, but it is sufficiently extensive and profound to deserve its traditional role as an independent subject. Similarly, the theory of determinants is sufficiently extensive and profound to justify independent study and an independent book. This book contains a number of features which cannot be found in any other book. Prominent among these are the extensive applications of scaled cofactors and column vectors and the inclusion of a large number of relations containing derivatives. Older books give their readers the impression that the theory of determinants is almost entirely algebraic in nature. If the elements in an arbitrary determinant A are functions of a continuous variable x, then A possesses a derivative with respect to x. The formula for this derivative has been known for generations, but its application to the solution of nonlinear differential equations is a recent development. 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. Some theore
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