Derivations of Applied Mathematics

Abstract

The book began in 1983 when a high-school classmate challenged me to prove the Pythagorean theorem on the spot. I lost the dare, but looking the proof up later I recorded it on loose leaves, adding to it the derivations of a few other theorems of interest to me. From such a kernel the notes grew over time, until family and friends suggested that the notes might make the material for the book you hold. The book is neither a tutorial on the one hand nor a bald reference on the other. The book is rather a study reference. In this book, you can look up some particular result directly, or you can begin on page one and read— with toil and commensurate profit—straight through to the end of the last chapter. The book as a whole surveys the general mathematical methods common to engineering, architecture, chemistry and physics. As such, the book serves as a marshal or guide. It concisely arrays and ambitiously reprises the mathematics of the scientist and the engineer, deriving the mathematics it reprises, filling gaps in one’s knowledge while extending one’s mathematical reach. Its focus on derivations is what principally distinguishes this book from the few others1 of its class. No result is presented here but that it is justified in a style engineers, scientists and other applied mathematicians will recognize—not indeed the high style of the professional mathematician, which serves other needs; but the long-established style of applications. Plan Following its introduction in chapter 1 the book comes in three parts. The first part begins with a brief review of classical algebra and geometry and develops thence the calculus of a single complex variable, this calculus being the axle as it were about which higher mathematics turns. The second part laboriously constructs the broadly useful mathematics of matrices and vectors, without which so many modern applications (to the fresh incredulity of each generation of college students) remain analytically intractable—the jewel of this second part being the eigenvalue of chapter 14. The third and final part, the most interesting but also the most advanced, introduces the mathematics of the Fourier transform, probability and the wave equation—each of which is enhanced by the use of special functions, the third part’sunifying theme. Thus, the book’s overall plan, though extensive enough to take several hundred pages to execute, is straightforward enough to describe in a single sentence. The plan is to derive as many mathematical results, useful to scientists, engineers and the like, as possible in a coherent train, recording and presenting the derivations together in an orderly manner in a single volume. What constitutes “useful” or “orderly” is a matter of perspective and judgment, of course. My own peculiar, heterogeneous background in military service, building construction, electrical engineering, electromagnetic analysis and software development, my nativity, residence and citizenship in the United States, undoubtedly bias the selection and presentation to some degree. How other authors go about writing their books, I do not know, but I suppose that what is true for me is true for many of them also: we begin by organizing notes for our own use, then observe that the same notes might prove useful to others, and then undertake to revise the notes and to bring them into a form which actually is useful to others. Whether this book succeeds in the last point is for the reader to judge.