A Garden of Integrals is aptly titled. The author, Frank Burk, wrote this book as a doting gardener who tenderly observed his varied collection and loved to show them off. Some of his subjects are rare orchids; others are more commonplace flowers, while still others started as weeds and gradually evolved into something exotic.
The underlying theme of the book is the development of the idea of the integral from ancient to modern times. Each of these gets a chapter of its own: the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Stieltjes, Henstock-Kurtzweil, Wiener, and Feynman. Generally this approach works very well. I can’t think of a comparable book at this level with a scope as broad as this one.
The author begins with an historical introduction, starting with the early developments of Eudoxus and Archimedes and continuing with brief discussions of the inventors of all the integrals discussed later in the book. This chapter is disjointed and disappointing; with a little bit more effort he could have made the connections clearer and the histories more detailed and more interesting. For example, it is amazing that the author does not mention the excitement and the controversy associated with the Feynman integral. He only mentions that physicists and mathematicians have worked on associated convergence questions for the last fifty years. Yet the story of Feynman’s audacious idea is so much more interesting than that!
Once the author begins discussing the individual integrals, he is truly in his element. Most chapters consist of a concise discussion of an integral with a generous number of illustrative examples and exercises. The Lebesgue integral gets more attention than the others, since there are separate chapters on the Lebesgue integral, Lebesgue measure and the Lebesgue-Stieltjes integral. Successive chapters build on one another as we read, for example, how Riemann built on Cauchy’s notion of an integral, how Lebesgue in turn built on Riemann’s ideas, and how the Henstock-Kurtzweil integral extends the Lebesgue integral. With the obvious exception of the Wiener and Feynman integrals, each integral is presented only on the real line. This has the effect of rendering the concepts more concretely, avoiding the complications associated with greater generality, and facilitating comparison of the various kinds of integrals. But of course this also means we miss the perspective we get by working in multiple dimensions.
Who are the intended readers? This is not so clear. A good course in undergraduate analysis is a prerequisite since the author assumes the reader has experience, for example, with uniform continuity, uniform convergence, as well as estimation and bounding of functions. Yet the scope is probably too limited for a graduate course. Perhaps its clearest role is as supplementary reading for a graduate course, or a source of material for advanced undergraduate projects.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.