Points on the continuum

This is an ambitious and substantial book on a classical subject area that has been extensively covered in the past. The authors are well-known mathematicians; Rubinstein pere is the father figure of moving boundary problems and his son has a substantial reputation. Such a book invites comparisons with its predecessors, such as Courant and Hilbert's standard work. What do the present authors have to add?

The authors set out their programme in the preface. They intend to study partial differential equations as arising naturally out of continuum models of physical processes. Moreover, they will give a rigorous treatment of "classical" methods, which they imply have been somewhat unjustly neglected by "modern mathematical education". And since that education often contains little or no physical background, they will begin with a description of commonly occurring models in the natural sciences, and their formulation in terms of partial differential equations.

Finally, they intend their book to be used by graduate students in mathematics, engineering and natural sciences. They begin, then, with a chapter on the ideas of continuum modelling. Several consistent features of the book are immediately clear. The first is, quite simply, that there is an enormous amount of information. After a discussion of the basics of models of continuous media, we find outlines of the usual models for linear elasticity, viscous and inviscid fluids, electrostatics and electrodynamics, chemical kinetics, equilibrium thermodynamics, thermodynamics of irreversible processes, together with a preliminary discussion of conservation laws. All this, in 28 pages. Second, there is no lack of rigour. The tone is set by the definition of a continuum, on page one; the comments on thermodynamics indicate that the authors have an inclination to a "rational mechanics" approach.

It is perhaps no coincidence that the notation is rather dense - altogether, there is more than a whiff of the formidable and comprehensive Soviet school of textbook writing. Finally, there is a degree of idiosyncrasy in the subject matter and the way it is treated. This has positive and negative aspects.

On the downside, one could cite their definition of a rotational fluid flow as one in which the velocity is the curl of a vector potential (thereby making stationary fluids rotational, and forcing all rotational flows to have zero divergence), their use of the notation (Greek) nu(x) for the Heaviside function, or their frankly eccentric preference for a version of the Laplace transform that is p times the usual definition. On the other hand, subjects such as chemical kinetics, electrodiffusion, the Buckley- Leverett equation and the Stefan problem (which, of course, reflect the particular interests of the authors) make a welcome appearance.

There follows a chapter in which the relationship between continuum models and partial differential equations is further developed -- and then we come to the heart of the matter, a detailed analysis, in the classical style, of the mathematical properties of some of these equations. The ordering is quite conventional, beginning with quasilinear and nonlinear first-order equations (there is a nice treatment of the latter but, oddly, Charpit's equations are not named as such, and neither are obvious applications such as geometric optics or the ship wave equation cited). The classification of second-order quasilinear equations is followed by about 50 pages on hyperbolic equations, 100 on second-order elliptic equations (but not the biharmonic equation), and nearly 200 on the parabolic case. We are led through extremely (perhaps overly) detailed treatments of the main classical theorems of the linear classical theory, and there is a more sporadic treatment of selected nonlinear problems.

The final section turns to some more practical matters, with chapters on Fourier series, integral transforms and asymptotic analysis. The material of the first two of these has largely been better presented elsewhere, while in the third the authors more or less admit that they cannot cover the subject adequately. Lastly, a clutch of appendices about routine methods includes some interesting specialised applications to heat and mass transfer.

There is no doubt that this is a work of considerable and thorough erudition. But the rather distinctive slant and style of the book has consequent drawbacks. One is that not many students will find it easy to develop their intuition for the central ideas. This is partly because informal - intuitive - treatments are largely absent, but also the theory of distributions has been, perhaps rather mistrustfully, relegated to a minor role. (The delta function is not even mentioned in the treatment of Green's functions for elliptic equations.) A second is that the enormous literature on functional-analytic approaches to applied mathematics might almost not exist, if we read only this book.

A more balanced treatment would spell out the comparison; the reader could then judge whether the emphasis on classical methods was justified.

On the positive side, in addition to, and hidden away in, the vast body of information are some fascinating topics that are impossible to find elsewhere. What we need is a user's guide to help us to find them.

Sam Howison is a lecturer in applied mathematics, University of Oxford.