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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 Abraham Robinson's non-standard analysis.
Textbook
In his textbook, Keisler pioneered the pedagogical technique of
an infinite-magnification microscope, so as to represent
graphically, distinct hypperreal numbers infinitely close to each
other. A typical pair of infinitely close hyperreals is represented
in the figure.
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 a useful device in explaining the
derivative.
Examples of a real
statement
To provide a freshman-level explanation of the transfer
principle, Keisler first 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: (x −
y)^{2} = x^{2} −
2xy + y^{2}.
- A trigonometric identity: sin^{2}x +
cos^{2}x = 1.
- A rule for logarithms: If x > 0 and y >
0, then log_{10}(xy) =
log_{10}x +
log_{10}y.
Transfer
principle
- Every real statement that holds for one or more particular real
functions holds for the hyperreal natural extensions of these
functions.
See also
References
- Bishop, Errett
(1977), "Review: H. Jerome Keisler,
Elementary calculus", Bull. Amer. Math. Soc.
83: 205–208, http://projecteuclid.org/euclid.bams/1183538669
- Keisler, H.
Jerome (1976), Elementary Calculus: An
Approach Using Infinitesimals, Prindle Weber &
Schmidt, ISBN
978-0871509116, http://www.math.wisc.edu/~keisler/calc.html
- Keisler, H.
Jerome (1976), Foundations of
Infinitesimal Calculus, Prindle Weber & Schmidt, ISBN
978-0871502155, http://www.math.wisc.edu/~keisler/foundations.html, retrieved 10 jan
2007
A companion to
the textbook Elementary Calculus: An Approach Using
Infinitesimals.
- Davis, Martin
(1977), "Review: J. Donald Monk,
Mathematical logic", Bull. Amer. Math. Soc.
83: 1007–1011, http://projecteuclid.org/euclid.bams/1183539465
- Schubring,
Gert (2005), Conflicts Between Generalization, Rigor, and
Intuition: Number Concepts Underlying the Development of Analysis
in 17th–19th Century France and Germany, Springer,
p. 153, ISBN 0387228365
- Tall, David (1980),
Intuitive infinitesimals
in the calculus (poster), Fourth International Congress on
Mathematics Education, Berkeley, http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1980c-intuitive-infls.pdf
- Artigue, Michèle
(1994), Analysis, Advanced Mathematical Thinking (ed.
David Tall), Springer-Verlag, p. 172, ISBN
0792328124
("The
non-standard analysis revival and its weak impact on
education".)
- Sullivan,
Kathleen (1976), "The Teaching of Elementary
Calculus Using the Nonstandard Analysis Approach", The
American Mathematical Monthly 83:
370–375, http://www.jstor.org/stable/2318657
, Analysis of
experiment to teach freshman calculus from Keisler's book