A Garden of Integrals

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MAA, Aug 30, 2007 - Mathematics - 281 pages
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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 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.
  

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Contents

An Historical Overview
1
The Cauchy Integral
29
The Riemann Integral
45
The RiemannStieltjes Integral
75
Lebesgue Measure
85
The Lebesgue Integral
111
The LebesgueStieltjes Integral
155
The HenstockKurzweil Integral
169
The Wiener Integral
205
The Feynman Integral
235
Index
279
Copyright

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About the author (2007)

Frank Burk was raised in Liberty, Missouri, and graduated from Liberty High School in 1960, a Merit Scholar. In 1963 he graduated from the University of Missouri with a BA in mathematics. He earned his PhD in mathematics from the University of California, Riverside in 1969. Frank joined the faculty at California State University, Chico as a member of the mathematics department in 1968. He taught 37 years before retiring in 2004. He also taught for Butte College. Frank was the author of two books, this one and Lebesgue Measure and Integration (Wiley, 1998). He was a long-time member of the Mathematical Association of America. Frank was a man of many interests. He was a runner his entire adult life and was one of the founders of the Chico Running Club. Frank earned the rank of Eagle Scout in his youth and continued in scouting as scoutmaster of Troop 230 for 18 years. Frank and his wife were foster parents. He was a proud member of the Greater Chico Kiwanis Club. He was a backpacker and hiked the entire John Muir Trail in 2001. Frank enjoyed reading, gardening, and playing bridge. After a losing battle with cancer, Frank passed away Saturday, Mary 17, 2007, at his home in Chico.

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