Customer Reviews

A Garden of Integrals (Dolciani Mathematical Expositions)

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This is simply a beautiful exposition of the theory of integration. It lies somewhere in the space between a textbook and a monograph, developing the key results in each of a set of increasingly general theories of integration.

After motivating the contents of the book with a historical overview, beginning with a discussion of the computation of areas between curves in antiquity and moving on the development of the integral and differential calculi, Burk goes on to summarise the history of each of the modern integration theories he goes on to treat in the book. The key problems and personalities involved in the development of each theory are discussed, giving the book a nicely clear and focused historical flavour.

This is not to say that the book is not principally a technical exposition: that is precisely what it is! After providing us with the relevant historical particulars, Burk goes on to discuss the integration theories of Cauchy, Riemann, Lebesgue, and Henstock-Kurzweil; each theory subsumed in generality bt those that follow it. The conceptual difficulties that gave rise to each successive generalisation are discussed in quite a bit of detail, and a number of basic results - such as the Fundamental Theorem of Calculus, and the conditions under which a function is integrable - are proved for each theory. A liberal smattering of well chosen excercises aids the theoretical development, and help the reader to develop a good intuition about the nature and limitations of the various theories discussed.

The final two chapters are devoted to Weiner and the Feynman integrals, both of which are integrals of functionals. They are used respectively to model Brownian motion and to sum quantum mechanical probability amplitudes.

There is a genuine clarity and beauty in Burks's exposition, which makes this moderately technical book a pleasure to read. Whilst it is probably not appropriate as a primary text for students of introductory calculus(Burk draws liberally on the armamentarium of abstract analysis), parts of it could be profitably read by first-year undergraduates, and it would make for excellent auxiliary reading for students of analysis. (If you're a math grad who somehow managed to avoid learning about Lebesgue and Generalised Riemann integration as an undergrad, this is an excellent place to start)

For its historicity and the supreme clarity of its exposition, I would recommend this book to anyone who would like to extend the depth and clarity of his/her understanding of integration theory