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| A BRIEF ACCOUNT
The
non-Newtonian calculi provide a wide variety of mathematical tools for use in
science, engineering, and mathematics. They appear to have considerable
potential for use as alternatives to the classical calculus of Newton
and Leibniz.
There are infinitely many non-Newtonian calculi. Like the classical calculus,
each of them possesses (among other things): a
derivative, an integral, a natural average, a special class of functions
having a constant derivative, and two Fundamental Theorems which reveal
that the derivative and integral are 'inversely' related.
The non-Newtonian calculi were created in the period from 1967 to 1970
by Michael Grossman and Robert Katz. In August of 1970, they constructed
a comprehensive family of calculi consisting of the infinitely many
calculi they created in July of 1967 and infinitely many others. Included in the family are the classical calculus, the
so-called "geometric calculus" (July of 1967), and the so-called "bigeometric
calculus" (August of 1970). All of the calculi can be described simultaneously
within the framework of a general theory. They decided to use the
adjective "non-Newtonian" to indicate any of the calculi other than the
classical calculus.
The first publication about the non-Newtonian calculi was Grossman and Katz's book Non-Newtonian Calculus [14]. It contains discussions of nine specific non-Newtonian calculi
(including the geometric calculus and the bigeometric calculus), the
general theory of non-Newtonian calculus, and heuristic guides for
application.
The geometric calculus (or the "exponential calculus") is the topic of Grossman’s book The First Nonlinear System of Differential and Integral Calculus [12].
Just as the arithmetic average is the natural average in the classical
calculus, the geometric average is the natural average in the geometric
calculus. And in the geometric calculus, the
exponential functions play the role that the linear functions play in
the classical calculus.
A non-Newtonian calculus in which the power functions play that role is presented in Grossman’s book Bigeometric Calculus: A System with a Scale-Free Derivative [11].
In the bigeometric calculus and in the geometric calculus, the
derivative, integral, and natural average are multiplicative.
Each non-Newtonian calculus, as well as the classical calculus, can be 'weighted' in a manner explained in the book The First Systems of Weighted Differential and Integral Calculus [10]
by Jane Grossman, Michael Grossman, and Robert Katz. Natural outgrowths
of the systems of weighted calculus are the systems of meta-calculus,
which are described in Jane Grossman's book Meta-Calculus: Differential and Integral [8].
In
their book Averages: A New Approach [9], Grossman, Grossman, and Katz present a detailed discussion about the averages (of functions) that
arise naturally in the development of non-Newtonian calculus and
weighted non-Newtonian calculus, and a discussion about a family of means (of two positive numbers).
Note. Michael Grossman and Robert Katz knew nothing about
non-Newtonian calculus prior to 14 July 1967, when they began their
development of that subject. Indeed, in Non-Newtonian Calculus (1972), they included the following paragraph (page 82):
"However, since we have nowhere seen a discussion of even one
specific non-Newtonian calculus, and since we have not found a notion
that encompasses the *-average, we are inclined to the view that the
non-Newtonian calculi have not been known and recognized heretofore.
But only the mathematical community can decide that."
Note. In 2008, Michael Grossman encountered discussions that led him to
wonder if a multiplicative (perhaps non-Newtonian) integral or
derivative had been developed by Vito Volterra, who lived from 1860 to
1940. [1,6,16,17]
Note. The six books by Grossman, Grossman,
and Katz on non-Newtonian calculus and related matters are available at
some academic libraries, public libraries, and bookstores such as
Amazon.com. On the Internet, each of the books can be read (free of
charge) at Google Books, and each of them can be read and/or
downloaded (free of charge) at HathiTrust.
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(Last edit: 4/01/2012.)
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Non-Newtonian Calculus at Google Sites.
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