There are two approaches to constructing a textbook in a wide-ranging field such as intermediate-level mathematics. One is to start with the standard body of knowledge and make a judicious selection, paying more attention to what to leave out than what to include. The other is to start from a point of view on the subject and include those parts of the subject matter that reinforce this perspective. C. D. Cantrell's book falls firmly into the second category. Though idiosyncratic, it is a book that I am pleased to add to my shelves. But a second or third-year undergraduate using it as their only text will be deprived of much of the material needed to provide a commonality of mathematical language with those trained on more traditional courses.
The publisher's introduction states: "The advent of powerful desktop computers has revolutionised scientific analysis and engineering design in fields as disparate as particle physics and telecommunications. Modern Mathematical Methods for Physicists and Engineers provides an up-to-date mathematical and computational education for students, researchers and practicing engineers."
The emphasis on computation sets Cantrell's book apart from the other three books reviewed here, which concentrate on more traditional analytical mathematics. It introduces us to rounding and propagation of errors in computation in the first chapter, and proceeds to discuss sets and mappings, the evaluation of functions, groups rings and fields, vector spaces, linear mapping and functionals, norms and inner products, convergence, group representations and finally turns to discussing special functions, concentrating almost exclusively on Bessel functions. It would provide an excellent introduction to some of the computational issues in applying numerics to non-linear dynamical problems, yet mention of such fashionable issues is curiously absent.
Tai L. Chow's book is advertised as being "designed for an intermediate undergraduate course in mathematical physics... primarily for physics undergraduates, but could also be used by students in other subjects, such as engineering, astronomy, and mathematics". It contains a traditional summary of mathematics for physics, such as might have been taught in the 1960s. In those days, it was common for the array of subjects to be split between several little books by different authors. But here in one hefty volume we have vector and tensor analysis, differential equations, matrices, Fourier and Laplace transformations, vector spaces, complex-variable theory and the calculus of residues, special functions of mathematical physics, integral equations, the calculus of variations and probability theory, together with 22 pages on numerical methods.
The presentation is clear, at least to one familiar with the subject matter. For the student, there might be a temptation to dip into the material rather than to master one piece of work before moving on. There is an extensive set of problems, and working through these systematically would doubtless reinforce the text of the book considerably.
Writing a maths textbook is similar to musical composition. The order in which ideas are presented, and their interconnections and relation to one another, is of critical importance to the success of the enterprise. Just as people react differently to various musical composers, they have preferences in textbook styles. One normally advises students to choose books with which they have a personal affinity, regardless of course directors' recommendations.
The other two books in this group approach the teaching of mathematics to scientists from the perspective of teaching a language to people who need to use it. "This novel textbook provides a comprehensive guided tour of the mathematical knowledge and techniques," Roel Snieder's publisher says. "All material is presented in the form of problems." The book certainly departs from the normal format. The topics include Taylor series, coordinate systems, vector differential operators, Gauss and Stokes theorems and the Laplacian conservation laws, Fourier analysis and the Dirac delta function, complex-variable theory, Green's functions, normal-mode analysis, Cartesian tensors and perturbation theory.
Within this framework, there are many intriguing and thought-provoking examples. The publisher says that this book is "instructive, applied and fun". I agree with this assessment. For some students - those who can think beyond the immediate problem and make connections with other examples requiring the same techniques - the teaching method will be very effective. Other students, who can memorise algorithmic solutions accurately and apply them in the specific cases presented, will not be so well served.
For the latter students, the fourth book in this group would be a much better bet. It is a rewrite of the well-known book on engineering mathematics by K. A. Stroud. This book has grown in length and scope since its first edition, which has been on our shelves for many years. It provides instruction in programmed learning "frames", with clear educational objectives set out and assessments so that students can decide whether they have attained the objectives. The level is aimed at that of students in transition from school to university. Provided the student has the staying power to work through the massive collection of frames, it will provide an excellent grounding for the material covered in the other three books reviewed here.
Topics covered include arithmetic, number systems, algebra, expressions and equations, graphs, partial fractions, trigonometry, binomial series, differentiation and integration, complex numbers, hyperbolic functions, determinants and matrices, vector algebra, series, differential equations, Laplace transforms, and statistics and probability. Progress occurs in great detail, and is slow. The book is very useful as a reference book with which students may brush up on poorly remembered technique.
Your reviewer looked at these massive texts and then at the slim compressed volumes covering the same ground from which he learned in the 1960s. Students are now presented with such a quantity of material that it is no wonder they have short attention spans. How are you to keep concentrating when a week's study has covered only a few pages in one of these massive books? Breaking this body of knowledge into smaller units might have been a good strategy.
These four books provide four views of applied mathematics, and complement one another. As a collection, their utility exceeds the sum of their parts.
David Jefferies is senior lecturer in engineering, University of Surrey.
Modern Mathematical Methods for Physicists and Engineers. First edition
Author - C. D. Cantrell
ISBN - 0 521 59180 5 and 598 3
Publisher - Cambridge University Press
Price - £70.00 and £24.95
Pages - 763