Fields, Flows and Waves: An Introduction to Continuum Models , D. F. Parker Springer-Verlag, New York, 2003. $34.95 paper (270 pp.). ISBN 1-85233-708-7
David Parker’s book Fields, Flows and Waves: An Introduction to Continuum Models , based on his lectures at the University of Edinburgh in Scotland, is a fine addition to the Springer Undergraduate Mathematics Series. Parker is a professor in the school of mathematics at the university. His book focuses on the use of classical analytical techniques in the continuum description of real-world phenomena. Topics include heat flow, electrostatics, hydrodynamics, elasticity, electromagnetism, chemical diffusion, and biological modeling.
For the subjects considered, the author provides masterly compact accounts of the physical phenomena, develops an appropriate conservation or balance law in one, two, or three dimensions, and solves interesting problems. Parker takes particular care to examine the physical implications of the mathematical solutions, offering many insightful remarks. The book is firmly rooted in the tradition of excellent British texts in mathematical physics, but Parker has supplemented the traditional coverage with many examples of current scientific or technological interest.
Of the book’s ten chapters, the first seven contain about enough material for a mid-level undergraduate course for applied-mathematics, physics, or engineering majors. In those seven chapters, Parker treats in a clear and concise manner the heat equation, Laplace’s equation, Poisson’s equation, the wave equation, hydrodynamics, and elasticity. Standard methods of solution are illustrated. The author assumes a basic knowledge of multivariable calculus and introduces vector operators, but avoids tensor algebra. An adequate selection of student exercises is included, with solutions and a short bibliography given at the end of the book.
The last three chapters comprise more advanced material, some of which is drawn from the author’s research areas. A thorough study of waves is initiated in chapter 8. Taking acoustic waves as a paradigm, Parker analyzes reflection and refraction at plane surfaces. He also treats waveguides and dispersion, and cogently discusses elastic waves, including Love waves. Chapter 9 contains an account of Maxwell’s equations and electromagnetic waves. Reflection, refraction, and waveguides are again considered, and the example of an optical fiber is treated in detail. Chapter 10 is devoted to chemical and biological models. Just as in meteorology or electromagnetism, continuum models become appropriate in those applications whenever the numbers of molecules, cells, insects, or animals inhabiting a region become sufficiently large that spatial averages provide useful information. Beginning with reaction-diffusion equations, Parker leads the reader into a discussion on growth and dispersion of biological populations. He describes in detail the Volterra–Lotka model for competing species, and the chapter ends with the fascinating subject of biological waves.
Fields, Flows and Waves differs from modern textbooks on continuum mechanics: Parker’s book is “problem-driven,” whereas standard texts, which are influenced by the writings of Clifford Truesdell in the mid-20th century, are more formally organized to reveal the theoretical structure of continuum mechanics, starting with kinematics and proceeding to a discussion of balance laws and, finally, constitutive equations. However, many of Parker’s examples could be used to enrich an advanced undergraduate or beginning graduate course on continuum mechanics. Those examples provide convincing evidence of the enduring value of continuum models in engineering, the physical sciences, and biology.
