A BRIEF ACCOUNT
The non-Newtonian calculi were created in the period from 1967 to 1970 by Michael Grossman and Robert Katz. These calculi, of which there are infinitely many, 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.
The first publication about the non-Newtonian calculi was Grossman and Katz’s book Non-Newtonian Calculus [14]. It includes discussions of nine specific non-Newtonian calculi, the general theory of non-Newtonian calculus, and heuristic guides for application.
The non-Newtonian calculus called 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. Furthermore, the geometric derivative is closely related to the well-known logarithmic derivative.
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 nonlinear; in fact, each is 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 discuss the averages (of functions) that arise naturally in the development of non-Newtonian calculus and weighted non-Newtonian calculus. They then use those averages to construct an interesting family of means (of two positive numbers). Included among those means are some well-known ones such as the arithmetic mean, the geometric mean, the harmonic mean, the power means, the logarithmic mean, the identric mean, and the Stolarsky mean. The family of means can be used to yield simple proofs of some familiar inequalities. [13]
An innovative application of non-Newtonian calculus was made by James R. Meginniss of the Claremont Graduate School and Harvey Mudd College. In his article "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis" [15], he used non-Newtonian calculus to create a theory of probability that is adapted to human behavior and decision making. (Proceedings of the American Statistical Association: Business and Economics Statistics, 1980. )
Subsequently, Fernando Córdova-Lepe (Universidad Católica del Maule in Chile) applied the bigeometric derivative to the theory of elasticity in economics. (He refers to the bigeometric derivative as "the multiplicative derivative.") [4,5] Elasticity and its relationship to the bigeometric derivative is also discussed in Non-Newtonian Calculus and Bigeometric Calculus: A System with a Scale-Free Derivative.
Several applications of non-Newtonian calculus were made by Agamirza E. Bashirov and Mustafa Riza of Eastern Mediterranean University in Cyprus, together with Emine Misirli Kurpinar and Ali Ozyapici of Ege University in Turkey. The article "Multiplicative calculus and its applications" [2] was published in 2008 by the Journal of Mathematical Analysis and Applications. The article "Multiplicative finite difference methods" [ 22] was published in 2009 by the Quarterly of Applied Mathematics. And lectures were delivered at the ISAAC Congress in 2007, and at the Congress of the Jangjeon Mathematical Society in 2008 [25]. Their work includes applications to differential equations, calculus of variations, and finite-difference methods.
An application of non-Newtonian calculus to information technology was made in 2008 by S. L. Blyumin of the Lipetsk State Technical University in Russia. [21] Furthermore, the geometric calculus and/or the bigeometric calculus have application to dynamical systems, chaos theory, dimensional spaces, and fractal theory. [1,6,17,19]
It’s natural to speculate about future applications of non-Newtonian calculus, weighted calculus, and meta-calculus. Perhaps scientists, engineers, and mathematicians will use them to define new concepts, to yield new or simpler laws, or to formulate or solve problems.
Note 1. Grossman and Katz knew nothing about non-Newtonian calculus prior to 14 July 1967, when they began their investigation into the matter. 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."
Nevertheless, many years later, information appeared suggesting that some aspects of the geometric calculus and/or the bigeometric calculus might have been known to other people prior to 14 July 1967. [1,6,16,17,19,20] Note 3. Non-Newtonian Calculus is cited by the eminent mathematics-historian Ivor Grattan-Guinness in his book The Rainbow of Mathematics: A History of the Mathematical Sciences [7].
Note 4. The First Nonlinear System of Differential and Integral Calculus is cited in the book The Predator-Prey Model: Do We Live in a Volterra World? by Manfred Peschel and Werner Mende [23]. Note 5. Non-Newtonian calculus is cited in the book The Mathematics of the Energy Crisis by R. Gagliardi and Jerry Pournelle [24]. Note 6. Non-Newtonian calculus is mentioned in the book The New Official Rules by the popular-culture writer Paul Dickson [26]. COMMENTS The [books] on non-Newtonian calculus ... appear to be very useful and innovative.
Professor Kenneth J. Arrow, Nobel-Laureate Stanford University, USA Your ideas [in Non-Newtonian Calculus] seem quite ingenious.
Professor Dirk J. Struik Massachusetts Institute of Technology, USA There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems. This very original piece of mathematics will surely expose a number of missed opportunities in the history of the subject.
Professor Ivor Grattan-Guinness Middlesex University, England The possibilities opened up by the [non-Newtonian] calculi seem to be immense.
Professor H. Gollmann Graz, Austria This [Non-Newtonian Calculus] is an exciting little book. ... The greatest value of these non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtonian calculus. Throughout, this book exhibits a clarity of vision characteristic of important mathematical creations. ... The authors have written this book for engineers and scientists, as well as for mathematicians. ... The writing is clear, concise, and very readable. No more than a working knowledge of [classical] calculus is assumed.
Professor David Pearce MacAdam Cape Cod Community College, USA ... It seems plausible that people who need to study functions from this point of view might well be able to formulate problems more clearly by using [bigeometric] calculus instead of [classical] calculus.
Professor Ralph P. Boas, Jr. Northwestern University, USA We think that multiplicative calculus [i.e., the geometric calculus] can especially be useful as a mathematical tool for economics and finance ... .
Professor Agamirza E. Bashirov Eastern Mediterranean University, Cyprus/ Professor Emine Misirli Kurpinar Ege University, Turkey/ Professor Ali Ozyapici Ege University, Turkey Non-Newtonian Calculus, by Michael Grossman and Robert Katz is a fascinating and (potentially) extremely important piece of mathematical theory. That a whole family of differential and integral calculi, parallel to but nonlinear with respect to ordinary Newtonian (or Leibnizian) calculus, should have remained undiscovered (or uninvented) for so long is astonishing -- but true. Every mathematician and worker with mathematics owes it to himself to look into the discoveries of Grossman and Katz.
Professor James R. Meginniss Claremont Graduate School and Harvey Mudd College, USA Note 7. The comments by Professors Grattan-Guinness, Gollmann, and MacAdam are excerpts from their reviews of the book Non-Newtonian Calculus in Middlesex Math Notes, Internationale Mathematische Nachrichten, and Journal of the Optical Society of America, respectively. The comment by Professor Boas is an excerpt from his review of the book Bigeometric Calculus: A System with a Scale-Free Derivative in Mathematical Reviews. REFERENCES
[1] Dorota Aniszewska (2007) "Multiplicative Runge-Kutta methods", Nonlinear Dynamics, Volume 50, Numbers 1-2, 2007. [2] Agamirza E. Bashirov, Emine Misirli Kurpinar, and Ali Ozyapici. "Multiplicative calculus and its applications", Journal of Mathematical Analysis and Applications, 2008. [3] Duff Campbell. "Multiplicative calculus and student projects", Primus, vol 9, issue 4, 1999. [4] Fernando Córdova-Lepe. "From quotient operation toward a proportional calculus", Journal of Mathematics, Game Theory and Algebra, 2004. [5] Fernando Córdova-Lepe. "The multiplicative derivative as a measure of elasticity in economics". [6] Felix R. Gantmacher. The Theory of Matrices, Volumes 1 and 2, Chelsea Publishing Company, 1959.
[7] Ivor Grattan-Guinnness. The Rainbow of Mathematics: A History of the Mathematical Sciences, pages 332 and 774, ISBN 0393320308, W. W. Norton & Company, 2000. [8] Jane Grossman. Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981. [9] Jane Grossman, Michael Grossman, Robert Katz. Averages: A New Approach, ISBN 0977117049, 1983.
[10] Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0977117014, 1980.
[11] Michael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983.
[12] Michael Grossman. The First Nonlinear System of Differential and Integral Calculus, ISBN 0977117006, 1979.
[13] Michael Grossman and Robert Katz. "A new approach to means of two positive numbers", International Journal of Mathematical Education in Science and Technology, Volume 17, Number 2, March 1986, pages 205 - 208. [14] Michael Grossman and Robert Katz. Non-Newtonian Calculus, ISBN 0912938013, 1972. [15] James R. Meginniss. "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis", Proceedings of the American Statistical Association: Business and Economics Statistics, 1980. [16] Robert Edouard Moritz. "Quotientiation, an extension of the differentiation process", Proceedings of the Nebraska Academy of Sciences, 1901. [19] Wikipedia article. "Multiplicative calculus". [20] Wikipedia article. "Product integral". [21] S. L. Blyumin. "Discrete vs. continuous, in information technology: quantum calculus and its alternatives" ( http://www.google.com/search?hl=en&as_qdr=all&q=%D0%91%D0%BB%D1%8E%D0%BC%D0%B8%D0%BD+%D0%A1.%D0%9B.+non-Newtonian+Calculus&lr=lang_ru ), 2008. [22] Mustafa Riza, Ali Ozyapici, and Emine Misirli. "Multiplicative finite difference methods", Quarterly of Applied Mathematics, 2009. [23] Manfred Peschel and Werner Mende. The Predator-Prey Model: Do We Live in a Volterra World?, page 246, ISBN 0387818480, Springer, 1986.[24] R. Gagliardi and Jerry Pournelle. The Mathematics of the Energy Crisis, Intergalactic Publishing Company, 1978. [25] Ali Ozyapici and Emine Misirli Kurpinar. "Notes on Multiplicative Calculus", 20th International Congress of the Jangjeon Mathematical Society, #67, August 2008. [26] Paul Dickson. The New Official Rules, page 113, Addison-Wesley Publishing Company, 1989. [27] Michael Grossman. "An introduction to non-Newtonian calculus", International Journal of Mathematical Education in Science and Technology, Volume 10, Number 4 ,October 1979, pages 525-528. [28] Michael Grossman and Robert Katz, "Isomorphic calculi", International Journal of Mathematical Education in Science and Technology, Volume 15, Number 2, March 1984 , pages 253 - 263. ADDITIONAL READING Robert Katz. Axiomatic Analysis, D. C. Heath and Company, 1964. S. L. Blyumin. "БИНАРНЫЕ АРИФМЕТИЧЕСКИЕ ОПЕРАЦИИ, ФОРМУЛЫ LINKS Amazon.com:
Fernando Córdova-Lepe: http://www.tmat.cl/articulocordova.html
Google Book Search:
Journal of Mathematical Analysis and Applications:
Libraries:
Robert Edouard Moritz:
http://www.emis.de/cgi-bin/JFM-item?33.0303.01 Quarterly of Applied Mathematics:
Vito Volterra: Gauss Quote re "new calculi" (in "Memorabilia Mathematica" by Moritz, #1215): "Discrete vs. continuous, in information technology: quantum calculus and its alternatives": http://www.google.com/search?hl=en&as_qdr=all&q=%D0%91%D0%BB%D1%8E%D0%BC%D0%B8%D0%BD+%D0%A1.%D0%9B.+non-Newtonian+Calculus&lr=lang_ru The Predator-Prey Model: Do We Live in a Volterra World?: http://books.google.com/books?id=9NEQAQAAIAAJ&dq=The+predator-prey+model%3A+do+we+live+in+a+Volterra+world%3F&q=Grossman#search_anchor The Mathematics of the Energy Crisis:
http://books.google.com/books?client=firefox-a&id=YI-yAAAAIAAJ&dq=mathematics+energy+crisis+gagliardi+non-newtonian+calculus&q=Non-Newtonian+Calculus+by+Grossman+and+Katz#search_anchor The New Official Rules: http://books.google.com/books?id=-mz6IZwQofUC&q=%22Non-Newtonian+Calculus%22&dq=%22Non-Newtonian+Calculus%22&lr=&as_brr=0 20th Jangjeon Congress: http://74.125.155.132/scholar?q=cache:NtantQH2rE0J:scholar.google.com/+%22Non-Newtonian+Calculus%22&hl=enHathi Trust: http://babel.hathitrust.org/cgi/m/mdp/pt?view=image;size=100;id=uc1.b4232942;page=root;seq=3 QUOTATIONS "But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes. We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line." - Roger Joseph Boscovich, as quoted in "Boscovich's mathematics", an article by J. F. Scott, in the book Roger Joseph Boscovich (1961) edited by Lancelot Law Whyte; and as quoted in "Transient pressure analysis in composite reservoirs" (1982) by Raymond W. K. Tang and William E. Brigham. "For each successive class of phenomena, a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature." - Henry John Stephen Smith, as quoted in Nature, Volume 8 (1873), page 450. "In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius." - Carl Friedrich Gauss, as quoted in Gauss, Werke, Bd. 8, page 298; and as quoted in Memorabilia Mathematica (or The Philomath's Quotation Book) (1914) by Robert Edouard Moritz, quotation #1215. "A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it would ever be so. By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." - John von Neumann, as quoted in Out of the Mouths of Mathematicians: A Quotation Book for Philomaths (1993) by R. Schmalz.
ACKNOWLEDGEMENT Thanks to David Lukas and Kenneth Lukas for constructing previous versions of this website, and for their expert advice on website construction.
CONTACT Name: Michael Grossman E-mail: smithpith@yahoo.com
(Last edit: 11/06/09.)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
|