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Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes given as An approach using infinitesimals. The book is available freely online and is currently published by Dover. [1]
As an application to mathematical education, H. Jerome Keisler wrote Elementary Calculus: An Infinitesimal Approach. [10] Covering nonstandard calculus , it develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements.
Keisler's Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods. The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach.
Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers.
Following Abraham Robinson's work resolving what had long been thought to be inherent logical contradictions in the literal interpretation of Leibniz's notation that Leibniz himself had proposed, that is, interpreting "dx" as literally representing an infinitesimally small quantity, Keisler published Elementary Calculus: An Infinitesimal ...
Jerome Keisler wrote a first-year calculus textbook, Elementary calculus: an infinitesimal approach, based on Robinson's approach. From the point of view of modern infinitesimal theory, Δx is an infinitesimal x-increment, Δy is the corresponding y-increment, and the derivative is the standard part of the infinitesimal ratio:
Elementary Calculus: An Infinitesimal Approach; Nonstandard calculus; Infinitesimal; Archimedes' use of infinitesimals; For further developments: see list of real analysis topics, list of complex analysis topics, list of multivariable calculus topics
Bishop reviewed the book Elementary Calculus: An Infinitesimal Approach by Howard Jerome Keisler, which presented elementary calculus using the methods of nonstandard analysis. Bishop was chosen by his advisor Paul Halmos to review the book.
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