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Hyperbolic Partial Differential Equations

Serge Alinhac
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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The first two sentences of the Introduction to Serge Alinhac’s Hyperbolic Partial Differential Equations are both revealing and a promise of things to come: “The aim of the present book is to present hyperbolic partial differential equations at an elementary level. In fact, the required mathematical background is only a third year university course on differential calculus for functions of several variables.” In light of the notoriety PDE enjoys — and has for years: it’s holding strong — and in view of its huge impact on mathematics, pure and applied, the goal Alinhac has set for himself with the book under review is obviously praiseworthy. If undergraduates come to be exposed to PDE early on, as an early upper division offering, as Alinhac intends, then the propaganda of the subject being so very austere will be greatly mitigated and Alinhac’s hope (loc. cit.) that “many readers of this book will eventually do research in the field” will be realized.

But this naturally raises the question, is this book (henceforth just “HypPDE”) truly accessible to upper division undergraduates? I think the answer is a qualified yes, the caveats being, first of all, that a non-trivial amount of maturity and self-motivation be present in the kids’ make-up at this stage, and, second, that the kids, preferably honors-juniors or something, should already be quite comfortable with the business of doing proofs, preferably of theorems in advanced calculus/beginning analysis (as distinct from what is usually done in calculus courses, modulo the exception of a very, very strong honors sequence). However, even with these qualifications in place Alinhac’s optimism is a bit misleading, given that, e.g. already on p. 5 he mentions hypersurfaces, and on p. 7 he presupposes knowledge of the Cauchy-Riemann operator.

Indeed, the young reader of HypPDE should be prepared to do a lot of margin-work and note-taking while traveling through the book’s 150 pages, even if he fits the description Alinhac has in mind in his Introduction. Also, it’s clear that the aforementioned ineffable quality of mathematical maturity is a requirement with a vengeance: the proofs and arguments in HypPDE are often tight and compact. While this makes for elegance and is in itself very pleasing, the novice might well be used to a more discursive style. (I guess that makes HypPDE a good place for as quantum jump in the undergraduate reader’s sophistication.)

HypPDE is split into seven chapters, with the wave equation (properly) prominently featured in the last three. The chapters contain sets of exercises that should clearly be taken very seriously. There are also end notes in all but the first two chapters, directing the reader to further reading on the indicated topics.

HypPDE is a very good book and will reward the considerable effort required of the youthful reader; the more experienced mathematician will also find a lot of good stuff in these pages, all presented well and cogently.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Introduction.- Vector Fields and Integral Curves.- Operators and Systems in the Plane.- Nonlinear First Order Equations.- Conservation Laws in One Dimension Space.- The Wave Equation.- Energy Inequalities for the Wave Equation.- Variable Coefficients Wave Equations and Systems.- Appendices.- Index.