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 Cordova-Lepe (Universidad Catolica 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. 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 2. The six aforementioned book 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 Book Search.
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].
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 4. 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 Cordova-Lepe. "From quotient operation toward a proportional calculus", Journal of Mathematics, Game Theory and Algebra, 2004.
[5] Fernando Cordova-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.
[17] M. Rybaczuk and P. Stoppel. "The fractal growth of fatigue defects in materials", International Journal of Fracture, Volume 103, Number 1, 2000.
[18] Dick Stanley. "A multiplicative calculus", Primus, vol 9, issue 4, 1999.
[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.
ADDITIONAL READING
* Robert Katz. "Axiomatic Analysis", D. C. Heath & Company, 1964.
LINKS
Amazon.com
http://www.amazon.com/exec/obidos/search-handle-url/104-0205036-1759143?url=index%3Dstripbooks%3Arelevance-above&field-keywords=%22Non-Newtonian+Calculus%22&Go.x=9&Go.y=9
Fernando Cordova-Lepe
http://www.tmat.cl/articulocordova.html
Google Book Search
http://books.google.com/books?q=%22Non-Newtonian+Calculus%22&btnG=Search+Books&as_brr=0
Journal of Mathematical Analysis and Applications
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4NDDM7B-G&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=0fd9c9a00232cc8cb6d29c3892133acb
Libraries
http://www.worldcat.org/wcpa/ow/b0a0284407850a80.html
Robert Edouard Moritz
http://www.emis.de/cgi-bin/JFM-item?33.0303.01
http://en.wikipedia.org/wiki/Robert_Edouard_Moritz
Quarterly of Applied Mathematics
http://www.ams.org/distribution/qam/0000-000-00/S0033-569X-09-01158-2/home.html
"The Rainbow of Mathematics"
http://books.google.com/books
q=grossmann&prev=http://www.google.com/search%3Fsourceid%3Dnavclient%26ie%3DUTF-8%26rls%3DGGLD,GGLD:2003-38,GGLD:en%26q%3Divor%2Bgrattan-guinness&id=mC9GcTdHqpcC&hl
Vito Volterra:
http://en.wikipedia.org/wiki/Vito_volterra
Gauss Quote re "new calculi" (in "Memorabilia Mathematica" by Moritz, #1215):
http://books.google.com/books?id=I-wEAAAAYAAJ&pg=PA198&vq=organic+whole&dq=memorabilia+math&as_brr=1&source=gbs_search_r&cad=1_1#PPA198,M1
ACKNOWLEDGEMENT
Thanks to David Lukas and Kenneth Lukas for constructing previous versions of this website, and for their expert advice on website construction.
(Last edit: 7/14/09.)