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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Editorial Reviews
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This book provides a stimulating panorama of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Stieltjes, Henstock-Kurzweil, Wiener and Feynman. Each argument is well presented and the main properties are displayed. The book is pleasant to read and can serve as a good reference. --B. Bongiorno, Mathematical Reviews
Book Description
Though there is essentially only one derivative, there is a variety of integrals. In this book the basic properties of each are proved, their similarities and differences are pointed out, and the reasons for their existence and their uses are given. There is no other book like it.
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29 of 30 people found the following review helpful:
5.0 out of 5 stars
A beautiful book!,
By Skymariner (Canberra, Australia) - See all my reviews
This review is from: A Garden of Integrals (Dolciani Mathematical Expositions) (Hardcover)
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
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First Sentence:
Figures 1(a) through (d) demonstrate the general idea of rearranging a given area to form another shape. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
whose subintervals, infimum property, tagged partition, functions converging pointwise, measurability criterion, integrability criteria, sigma algebra, measurable functional, outer measure, upper sums, lower sums, contraction semigroup, dense subspace Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Plancherel's Theorem, Fundamental Theorem of Calculus, Monotone Convergence Theorem, Schrödinger's Equation, Cousin's Lemma, Compare Theorems, Fubini's Theorem, American Mathematical Society, Bounded Convergence Theorem, Norbert Wiener, Schrödinger Equation, Adapted Wish, Carole King Krueger, Recovering Functions, The Integrals of Lehesgue New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Figures 1(a) through (d) demonstrate the general idea of rearranging a given area to form another shape. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
whose subintervals, infimum property, tagged partition, functions converging pointwise, measurability criterion, integrability criteria, sigma algebra, measurable functional, outer measure, upper sums, lower sums, contraction semigroup, dense subspace Key Phrases - Capitalized Phrases (CAPs): (learn more)
New York, Plancherel's Theorem, Fundamental Theorem of Calculus, Monotone Convergence Theorem, Schrödinger's Equation, Cousin's Lemma, Compare Theorems, Fubini's Theorem, American Mathematical Society, Bounded Convergence Theorem, Norbert Wiener, Schrödinger Equation, Adapted Wish, Carole King Krueger, Recovering Functions, The Integrals of Lehesgue New!
Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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