Elementary Calculus: An Infinitesimal Approach
| Author | H. Jerome Keisler |
|---|---|
| Language | English |
| Subject | Mathematics |
| Publisher | Dover |
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.
Keisler's textbook is based on Robinson's construction of the hyperreal numbers. Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors which covers the foundational material in more depth.
Keisler defines all basic notions of the calculus such as continuity, derivative, and integral using infinitesimals. 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.
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. Similarly, an infinite-resolution telescope is used to represent infinite numbers.
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-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"). The derivative of ƒ is then the (standard part of the) slope of that line (see figure).
Thus the microscope is used as a device in explaining the derivative.
The book was first reviewed by Errett Bishop, noted for his work in constructive mathematics. Bishop's review was harshly critical; see Criticism of non-standard analysis. Shortly after, Martin Davis and Hausner published a detailed favorable review, as did Andreas Blass and Keith Stroyan. Keisler's student K. Sullivan, as part of her Ph.D. thesis, performed a controlled experiment involving 5 schools which found Elementary Calculus to have advantages over the standard method of teaching calculus. Despite the benefits described by Sullivan, the vast majority of mathematicians have not adopted infinitesimal methods in their teaching. Recently, Katz & Katz give a positive account of a calculus course based on Keisler's book. O'Donovan also described his experience teaching calculus using infinitesimals. His initial point of view was positive, but later he found pedagogical difficulties with approach to non-standard calculus taken by this text and others.
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