Keisler then gives a few examples of real statements to which the principle applies: In the second edition Keisler introduces the extension principle and the transfer principle in the following form:Įvery real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions. Transfer principle īetween the first and second edition of the Elementary Calculus, much of the theoretical material that was in the first chapter was moved to the epilogue at the end of the book, including the theoretical groundwork of nonstandard analysis. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament". Hrbacek writes that the definitions of continuity, derivative, and integral implicitly must be grounded in the ε–δ method in Robinson's theoretical framework, in order to extend definitions to include nonstandard values of the inputs, claiming that the hope that nonstandard calculus could be done without ε–δ methods could not be realized in full. Revolutions are seldom welcomed by the established party, although revolutionaries often are." Blackley remarked in a letter to Prindle, Weber & Schmidt, concerning Elementary Calculus: An Approach Using Infinitesimals, "Such problems as might arise with the book will be political. His initial point of view was positive, but later he found pedagogical difficulties with the approach to nonstandard calculus taken by this text and others. O'Donovan also described his experience teaching calculus using infinitesimals. Recently, Katz & Katz give a positive account of a calculus course based on Keisler's book. Despite the benefits described by Sullivan, the vast majority of mathematicians have not adopted infinitesimal methods in their teaching. Keisler's student K. Sullivan, as part of her PhD thesis, performed a controlled experiment involving 5 schools, which found Elementary Calculus to have advantages over the standard method of teaching calculus. Shortly after, Martin Davis and Hausner published a detailed favorable review, as did Andreas Blass and Keith Stroyan. Bishop's review was harshly critical see Criticism of nonstandard analysis. The book was first reviewed by Errett Bishop, noted for his work in constructive mathematics. Thus the microscope is used as a device in explaining the derivative. The "infinitesimal microscope" is used to view an infinitesimal neighborhood of a standard real. The standard part function "rounds off" a finite hyperreal to the nearest real number. The derivative of ƒ is then the ( standard part of the) slope of that line (see figure). Similarly, an infinite-magnification microscope will transform an infinitesimal arc of a graph of ƒ, into a straight line, up to an infinitesimal error (only visible by applying a higher-magnification "microscope"). When one examines a curve, say the graph of ƒ, under a magnifying glass, its curvature decreases proportionally to the magnification power of the lens. Similarly, an infinite-resolution telescope is used to represent infinite numbers. In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hyperreal numbers infinitely close to each other. The usual definitions in terms of ε–δ techniques are provided at the end of Chapter 5 to enable a transition to a standard sequence. Keisler defines all basic notions of the calculus such as continuity (mathematics), derivative, and integral using infinitesimals. Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors, which covers the foundational material in more depth. Keisler's textbook is based on Robinson's construction of the hyperreal numbers. The book is available freely online and is currently published by Dover. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. Elementary Calculus: An Infinitesimal Approach AuthorĮlementary Calculus: An Infinitesimal approach is a textbook by H.
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