Nature does nothing in vain, and more is in vain when
less will serve; for Nature is pleased for simplicity,
and affects not the pomp of superfluous causes.
Sir Isaac Newton (1642–1727), Principia
This is a nice and almost exhaustive introduction to the theory of functions of several real variables. The volume is a comprehensive introduction to this important field and it is intended for advanced undergraduates or beginning graduate students in mathematics, physics or engineering. Prerequisites include linear algebra and some elementary topology of Euclidean space; most of these are included in the four appendices of the book.
There is a huge amount of mathematics in this volume, presented carefully and with style. The authors do much more than just pay lip service to the main concepts: they do their darndest to make sure the reader understands these concepts. The authors develop the theory in a logical sequence building one result upon the other, enriching the development with numerous explanatory remarks and historical footnotes. There are groups of exercises and problems, many of them with detailed solutions while others have hints or final answers.
The authors begin with the basics of the geometry and topology of Euclidean space. The book omits none of the usual advanced calculus material, such as limits of functions, continuity, differentiation, and integration. Many fundamental concepts, especially related with differentiation and integration, are treated in depth and with mathematical rigor. The classical theorems of differentiation and integration such as the inverse and implicit function theorems, Lagrange’s multiplier rule, Fubini’s theorem, the change of variables formula, Green’s, Stokes’ and Gauss’s theorems are proved in detail, many of them with novel proofs. More advanced topics, such as Morse’s lemma, Sard’s theorem, the Weierstrass approximation theorem, the Fourier transform, vector fields on spheres, Brouwer’s fixed point theorem, Whitney’s embedding theorem, Picard’s theorem, and Hermite polynomials are discussed in starred sections.
A number of well chosen illustrative examples and counter-examples clarify matters and teach the reader how to apply these results to solve problems in mathematics, science, and economics. These features of the book distinguish it and recommend it to any one interested in a cogent but rapid way to get from introductory graduate analysis to some of the hot topics of the day: the connections with ordinary and partial differential equations and the calculus of variations should be sufficient to tip the balance. The goals of Functions of Several Variables go well beyond the usual prosaic objective of presenting beginning graduate students with a certain standard set of tools in the theory of functions of several variables.
The book is well conceived and well written. It is also richer than many of the current crop of real analysis texts. Additionally, the exposition is solid, the book is loaded with exercises, and is dripping with the authors’ expertise. Teachers wanting a solid and interesting treatment that goes right to the point and does not bore good students with verbose explanations will find this book much to their liking.
Vicentiu D. Radulescu (http://inf.ucv.ro/~radulescu) is a Professorial Fellow at the Mathematics Institute of the Romanian Academy. He received both his Ph.D. and Habilitation at the Université Pierre et Marie Curie (Paris 6) under the coordination of Haim Brezis. His current interests are broadly in nonlinear PDEs and their applications. He wrote more than 150 papers and he published books with Oxford University Press, Cambridge University Press, Springer New York, Springer Heidelberg, Kluwer, and Hindawi.