A Garden of Integrals
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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5e-fine partition absolutely continuous abstract Cauchy problem Borel cylinder Borel sets bounded function bounded variation Calculate Cantor function Cantor set Cauchy integrable collection continuous functions Convergence Theorem countable define derivative discontinuities disjoint domain Example Exercise F is continuous Figure finite number fk(x)dx function F Fundamental Theorem given H-K integrable H-K sums Hint integrable on a,b integral of f interval a,b irrational Lebesgue integrable Lebesgue measurable sets Lebesgue outer measure Lebesgue-Stieltjes length less Mathematical measurable functions measure zero metric space monotone increasing nonnegative open intervals open set outer measure particle partition of a,b Proof Property rational numbers real numbers Riemann integrable Riemann sum Riemann-Stieltjes integral SchrOdinger Problem set of real Show sigma algebra simple functions Stieltjes subintervals have length Suppose Theorem of Calculus Wiener integral Wiener measurable functional xk-i