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Elementary Calculus: An Infinitesimal approach (the subtitle is sometimes given as An approach using infinitesimals) is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson. The book is available freely online and is currently published by Dover.

Textbook

Keisler's textbook is based on Robinson's construction of the hyperreal numbers. Because this is not a subject widely known Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors which covers the foundational material in more depth.

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. Thus the microscope is used as a device in explaining the derivative.

Reception

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Constructivist Errett Bishop criticized the book for adopting a non-constructive approach. Bishop's review was described as vitriolic by historian Joseph Dauben. In earlier writings, Bishop described what he perceived as a lack of meaning in classical mathematics, a condition he described both as "schizophrenia" and a "debasement of meaning", and predicting its imminent demise (this was 45 years ago). Some commentators feel that his vitriolic opposition was based on his perception of a scarcity of meaning in classical mathematics in general and Keisler's textbook in particular, due to their reliance on the law of excluded middle. Bishop was chosen to review Elementary Calculus by his advisor, Paul Halmos, who characterized non-standard analysis as a "special tool, too special" (see Criticism of non-standard analysis for a conflict of interests involved). G. Stolzenberg contended in a letter published in The Notices that constructivts are capable of the rational minded inquiry necessary to objectively review a textbook that is not constructive. Meanwhile, a recent study notes that Stolzenberg's short letter contains five occurrences of the root "dogma", culminating in a final "spouting of dogma", whereas the root is absent from Keisler's own letter.

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 were not convinced to adopt 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..

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. Hrbacek argues that the definitions of definitions of continuity, derivative, and integration implicitly must be grounded in the ε-δ method in Robinson's theoretical framework. Thus the hope that non-standard calculus could be done without ε-δ methods could not be realized in full.. Both O'Donovan and Hrbacek find infinitesimal techniques to be intuitive to students and a useful teaching tool, and together with Lessmann they put forth their own theory of relative analysis, which allow for infinitesimal techniques but possesses a more general transfer principle that maybe safely used by students.

Transfer principle

Between 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 Non-standard analysis.

In the second edition Keisler introduces the extension principle and the transfer principle in the following form:

Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions.

Keisler then gives a few examples of real statements to which the principle applies:

• Closure law for addition: for any x and y, the sum x + y is defined.
• Commutative law for addition: x + y = y + x.
• A rule for order: if 0 < x < y then 0 < 1/y < 1/x.
• Division by zero is never allowed: x/0 is undefined.
• An algebraic identity: ${\displaystyle (x-y)^{2}=x^{2}-2xy+y^{2}}$.
• A trigonometric identity: ${\displaystyle \sin ^{2}x+\cos ^{2}x=1}$.
• A rule for logarithms: If x > 0 and y > 0, then ${\displaystyle \log _{10}(xy)=\log _{10}x+\log _{10}y}$.

See also

• Criticism of non-standard analysis
• Influence of non-standard analysis
• Non-standard calculus
• Increment theorem

References

• Bishop, Errett (1977), "Review: H. Jerome Keisler, Elementary calculus", Bull. Amer. Math. Soc., 83: 205–208
• Blass, Andreas (1978), "Review: Martin Davis, Applied nonstandard analysis, and K. D. Stroyan and W. A. J. Luxemburg, Introduction to the theory of infinitesimals, and H. Jerome Keisler, Foundations of infinitesimal calculus", Bull. Amer. Math. Soc., 84 (1): 34–41

Blass writes: "I suspect that many mathematicians harbor, somewhere in the back of their minds, the formula ${\displaystyle \int {\sqrt {(dx)^{2}+(dy)^{2}}}}$ for arc length (and quickly factor out dx before writing it down)."

• Davis, Martin (1977), "Review: J. Donald Monk, Mathematical logic", Bull. Amer. Math. Soc., 83: 1007–1011
• Davis, M.; Hausner, M (1978), "Book review. The Joy of Infinitesimals. J. Keisler's Elementary Calculus" (PDF), Mathematical Intelligencer, 1: 168–170.
• Hrbacek, K.; Lessmann, O.; O’Donovan, R. (2010), "Analysis with Ultrasmall Numbers", American Mathematical Monthly, 117 (9): 801–816 {{citation}}: Unknown parameter |month= ignored (help)
• Hrbacek, K. (2007), "Stratified Analysis?", in Van Den Berg, I.; Neves, V. (eds.), The Strength of Nonstandard Analysis, Springer
• Katz, Karin Usadi; Katz, Mikhail G. (2010), "When is .999... less than 1?", The Montana Mathematics Enthusiast, 7 (1): 3–30, arXiv:1007.3018
• Keisler, H. Jerome (1976), Elementary Calculus: An Approach Using Infinitesimals, Prindle Weber & Schmidt, ISBN 978-0871509116
• Keisler, H. Jerome (1976), Foundations of Infinitesimal Calculus, Prindle Weber & Schmidt, ISBN 978-0871502155, retrieved 10 jan 2007 {{citation}}: Check date values in: |accessdate= (help) A companion to the textbook Elementary Calculus: An Approach Using Infinitesimals.
• Keisler, H. Jerome (2011), Elementary Calculus: An Infinitesimal Approach (2nd ed.), New York: Dover Publications, ISBN 978-0-486-48452-5
• Madison, E. W.; Stroyan, K. D. (1977), "Elementary Calculus. by H. Jerome Keisler", The American Mathematical Monthly, 84 (6): 496–500 {{citation}}: Unknown parameter |month= ignored (help)
• O'Donovan, R. (2007), "Pre-University Analysis", in Van Den Berg, I.; Neves, V. (eds.), The Strength of Nonstandard Analysis, Springer
• O'Donovan, R.; Kimber, J. (2006), "Nonstandard analysis at pre-university level: Naive magnitude analysis", in Cultand, N; Di Nasso, M.; Ross (eds.), Nonstandard Methods and Applications in Mathematics, Lecture Notes in Logic, vol. 25 {{citation}}: Unknown parameter |edit-first3= ignored (help)
• Stolzenberg, G. (1978), Notices of the American Mathematical Society, 25 (4): 242 {{citation}}: Missing or empty |title= (help); Unknown parameter |month= ignored (help)
• Sullivan, Kathleen (1976), "The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach", The American Mathematical Monthly, 83 (5), Mathematical Association of America: 370–375, doi:10.2307/2318657, JSTOR 2318657
• Tall, David (1980), Intuitive infinitesimals in the calculus (poster) (PDF), Fourth International Congress on Mathematics Education, Berkeley

Ref-notes

1. ^ Keisler 2011.
2. Keisler 2011, iv.
3. Bishop 1977.
4. Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?", Intellectica, 56 (2), arXiv:1110.5456
5. Stolzenberg 1978.
6. Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?", Intellectica, 56 (2), arXiv:1110.5456
7. Davis & Hausner 1978.
8. Blass 1978.
9. Madison & Stroyan 1977.
10. http://www.math.wisc.edu/oldhome/directories/alumni/1974.htm
11. Sullivan 1976.
12. Tall 1980.
13. Katz & Katz 2010.
14. O'Donovan & Kimber 2006.
15. O'Donovan 2007.
16. ^ Hrbacek 2007.
17. Hrbacek, Lessmann & O’Donovan 2010.
18. O’Donovan 2007. sfn error: no target: CITEREFO’Donovan2007 (help)

v t e Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of nonstandard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle
Related branches	Nonstandard analysis Nonstandard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive nonstandard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental law of homogeneity
Mathematicians	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

• Misplaced Pages neutral point of view disputes from December 2011
• Use dmy dates from May 2011
• Calculus
• Non-standard analysis