A beautiful book!

books:

	• A Garden of Integrals (Dolciani Mathematical Expositions) Frank E. Burk Mathematical Association of America, 2007 - 281 pages average customer review:based on 1 review view larger image
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A beautiful book!

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 of several integration theories using an exceptionally clear and historical approach.

Burk begins with a historical overview, discussing the key developments in the computation of areas between curves in antiquity, and the development of Calculus, before summarising the history of each of the modern integration theories treated in the book. A chapter is then devoted to each of the increasingly general theories of Cauchy, Riemann, Lebesgue, and Henstock-Kurzweil; and the conceptual difficulties that motivated their formulation elaborated. Each chapter contains a liberal smattering of excercises to motivate and clarify the theories, as they unfold before us, which complement the theoretical exposition and are helpful in developing an intuition for the subject matter.

The final two chapters are devoted to Weiner and the Feynman integrals, both 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 them, 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, I can't think of a better place to start than this.)

I recommend it for anyone who would like extend the depth and clarity of his/her understanding of integration theory.

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The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there is a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. There is plentiful historical information. The audience for the book is advanced undergraduate mathematics majors, graduate students, and faculty members. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate their richness. Professor Burks clear and well-motivated exposition makes this book a joy to read. The book can serve as a reference, as a supplement to courses that include the theory of integration, and a source of exercises in analysis. There is no other book like it.

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