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Varieties of Integration (Dolciani Mathematical Expositions), by C. Ray Rosentrater
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Varieties of Integration explores the critical contributions by Riemann, Darboux, Lebesgue, Henstock, Kurzweil, and Stieltjes to the theory of integration and provides a glimpse of more recent variations of the integral such as those involving operator-valued measures. By the first year of graduate school, a young mathematician will have encountered at least three separate definitions of the integral. The associated integrals are typically studied in isolation with little attention paid to the relationships between them or to the historical issues that motivated their definitions. Varieties of Integration redresses this situation by introducing the Riemann, Darboux, Lebesgue, and gauge integrals in a single volume using a common set of examples. This approach allows the reader to see how the definitions influence proof techniques and computational strategies. Then the properties of the integrals are compared in three major areas: the class of integrable functions, the convergence properties of the integral, and the best form of the Fundamental Theorems of Calculus.
- Sales Rank: #189977 in Books
- Published on: 2015-11-16
- Original language: English
- Number of items: 1
- Dimensions: 8.98" h x .91" w x 5.98" l, .0 pounds
- Binding: Hardcover
- 342 pages
Review
The content and the purpose of this book are best described by the text on the back cover: "Early in their studies, young mathematicians are introduced to three different definitions of the integral. Typically, the associated studies take place in isolation from each other and often are unconnected to the historical questions that gave rise to the definitions. Varieties of Integration provides a historically motivated, integrated survey of the most common integral definitions. Making clear pedagogy a priority the integrals are introduced using a common set of text functions and then compared in terms of (1) their common proof and computational strategies, (2) the breadth of their applications, and (3) their flexibility in dealing with limits."
In his preface, the author considers this book a comparative study of four approaches to integration over an interval [a,b]: Riemann, Darboux, Lebesgue, and gauge. He continues by stating that the book can serve as a text for a second course in real analysis. Indeed, this manuscript is written with such users particularly in mind. The prerequisite first course should include the standard topics of supremum, infimum, compactness, the mean value theorem, and sequences of functions.
The book contains numerous examples and counterexamples, which will no doubt help students understand the definitions and concepts in question. Each chapter ends with several pages of exercises, and with a small list of references. This is a handsomely produced book (like most MAA publications), and it appears to be of just the right length and at the right level for the intended audience. --CMS Notices
About the Author
C. Ray Rosentrater is a Professor of Mathematics at Westmont College where he has also served as department chair and Associate Dean for Curriculum. He has been recognized as Westmont s Teacher of the Year in the Natural and Behavioral Sciences and has received the Faculty Research Award. He earned a PhD in mathematics from Indiana University and an MSc in computer science from the University of Toronto. Awarded a Fulbright Fellowship in 1995, he now serves as Westmont s Fulbright Program Advisor. He served multiple terms on the ACMS board including terms as Vice President and President. His other publications include papers in operator theory and articles connecting analysis to computer science and linear algebra to statistics. He co-wrote two chapters in Mathematics through the Eyes of Faith.
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