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Burk proves the basic properties of various integrals, draws comparisons and analyses their uses.
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 reasons for their existence and their uses are given, with plentiful historical information. The audience for the book is advanced undergraduate mathematics students, graduate students, and faculty members, of which even the most experienced are unlikely to be aware of all of the integrals in the Garden of Integrals. Professor Burk's clear and well-motivated exposition makes this book a joy to read. There is no other book like it.
Table of contents:
Foreword; 1. An historical overview; 2. The Cauchy integral; 3. The Riemann integral; 4. The Riemann-Stieltjes integral; 5. Lebesgue measure; 6. The Lebesgue integral; 7. The Lebesgue-Stieltjes integral; 8. The Henstock-Kurzweil integral; 9. The Wiener integral; 10. The Feynman integral; Index; About the author.
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 reasons for their existence and their uses are given, with plentiful historical information. The audience for the book is advanced undergraduate mathematics students, graduate students, and faculty members, of which even the most experienced are unlikely to be aware of all of the integrals in the Garden of Integrals. Professor Burk's clear and well-motivated exposition makes this book a joy to read. There is no other book like it.
Table of contents:
Foreword; 1. An historical overview; 2. The Cauchy integral; 3. The Riemann integral; 4. The Riemann-Stieltjes integral; 5. Lebesgue measure; 6. The Lebesgue integral; 7. The Lebesgue-Stieltjes integral; 8. The Henstock-Kurzweil integral; 9. The Wiener integral; 10. The Feynman integral; Index; About the author.