Brief Description

The non-Newtonian calculi are alternatives to the classical calculus of Newton and Leibniz. They provide a wide variety of mathematical tools, and are used worldwide in science, engineering, and mathematics.

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. Nevertheless, infinitely many non-Newtonian calculi are markedly different from the classical calculus.    
For example, infinitely many non-Newtonian calculi are nonlinear in the sense that each of them has a nonlinear derivative or integral. Among these calculi are the geometric calculus, bigeometric calculus, harmonic calculus, biharmonic calculus, quadratic calculus, and biquadratic calculus. In fact, the geometric calculus and the bigeometric calculus are multiplicative calculi. (Please see the page Multiplicative Calculus.)

Of course, in the classical calculus the linear functions are the functions having a constant derivative. However, in the geometric calculus, the exponential functions are the functions having a constant derivative. And in the bigeometric calculus, the power functions are the functions having a constant derivative. (The geometric derivative and the bigeometric derivative are closely related to the well-known logarithmic derivative and elasticity, respectively.)

The well-known arithmetic average (of functions) is the natural average in the classical calculus, but the well-known geometric average is the natural average in the geometric calculus. And the well-known harmonic average and quadratic average (or root mean square) are closely related to the natural averages in the harmonic and quadratic calculi, respectively.

Furthermore, unlike the classical derivative, the bigeometric derivative is scale invariant (or scale free), i.e., it is invariant under all changes of scale (or unit) in function arguments and in function values.

Non-Newtonian calculus has been applied to a variety of topics in science, engineering, and mathematics.  In particular, the geometric and bigeometric calculi have been used extensively. For more information, please see the pages Applications & Reception and Citations.

NOTE. The six books on non-Newtonian calculus and related matters by Jane Grossman, Michael Grossman, and Robert Katz are indicated below, and are available at some academic libraries, public libraries, and booksellers such as Amazon.com. On the World Wide Web, each of the books can be read and downloaded, free of charge, at HathiTrust, Google Books, and the Digital Public Library of America.  

  • Michael Grossman and Robert Katz.  Non-Newtonian Calculus, ISBN 0912938013, 1972. (Please see item [15] on the References page.)
  • Michael Grossman. The First Nonlinear System of Differential and Integral Calculus, ISBN 0977117006, 1979. (The geometric calculus) [11]
  • Jane Grossman, Michael Grossman, Robert Katz. The First Systems of Weighted Differential and Integral Calculus, ISBN 0977117014, 1980. [9]
  • Jane Grossman. Meta-Calculus: Differential and Integral, ISBN 0977117022, 1981. [7]
  • Michael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN 0977117030, 1983. [10]
  • Jane Grossman, Michael Grossman, and Robert Katz. Averages: A New Approach, ISBN 0977117049, 1983. [8]

NOTE. Non-Newtonian calculus was recommended as a featured topic for the 21st-century college-mathematics-curriculum, in the keynote speech at the 27th International Conference on Technology in Collegiate Mathematics (ICTCM) on 13 March 2015. The keynote speaker was the mathematics-educator Eric Gaze. His speech is entitled "Complexity, Computation, and Quantitative Reasoning: A Mathematics Curriculum for the 21st Century". (The conference was sponsored by Pearson PLC, the largest education company and the largest book publisher in the world; and the Electronic Proceedings of the conference were hosted by Math Archives (archives.math.utk.edu) with partial support provided by the National Science Foundation.) Please see item [224] on the References page.

NOTE. A special-session (mini-symposium) called "Non-Newtonian Calculus" was held at the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE), 4-8 July 2017, at Rota, Cadiz - Spain. The 2017 CMMSE Scientific Committee included over 50 distinguished scholars in a variety of fields, and CMMSE conferences are attended by numerous scholars worldwide. The special-session on non-Newtonian calculus was organized by Fernando Córdova-Lepe and Marco Mora, both from Universidad Católica del Maule in Chile. Included among the papers delivered at the special session on non-Newtonian calculus: "Introductory elements for the development of a multiplicative statistic" by Carol Pavez Rojas, Fernando Córdova-Lepe, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile); "Linearity and its algebra in the bi-geometrical context" (i.e., multiplicative linear-algebra) by Fernando Córdova-Lepe, Rodrigo del Valle, and Karina Vilches Ponce (all from Universidad Católica del Maule in Chile); and "Bigeometric Complex Calculus" by Agamirza E. Bashirov and Sajedeh Norozpour (both from Eastern Mediterranean University in North Cyprus). "Bigeometric Complex Calculus" was also presented at the special-session "Mathematical Modeling and Computational PDE". [324, 383, 384, 385] From a translation (from Spanish) of the conference-announcement:

     "Non-Newtonian Calculus (NNC) ... has been increasing its development through the recoding of the multiplicative world (from the point of view of the standard calculation) as an essentially linear domain, and therein lies the nucleus of importance. Many advances and applications in science, engineering and mathematics are appearing more frequently.
     "This mini symposium will be one of the first international meetings of a dispersed scientific community that has worked or is working on this topic and annoting a mark in the history of the NNC. Taking into account the novelty of the subject, all topics related to NNC (theory and applications) are welcome."

NOTE. Non-Newtonian calculus should not be confused with Abraham Robinson's so-called "non-standard calculus".

The aforementioned geometric calculus should not be confused with David Hestenes' so-called "geometric calculus".

A Quotation from Gauss

The following Carl Friedrich Gauss quotation
is from
Carl Friedrich Gauss: Werke, Volume 8, page 298; and from Memorabilia Mathematica or The Philomath's Quotation Book (1914) by Robert Edouard Moritz, quotation #1215.

"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, indeed 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' 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."

Three Reviews of the Book Non-Newtonian Calculus

The following review was written by David Pearce MacAdam, and appeared in the Journal of the Optical Society of America (Volume 63, January of 1973), a publication of  The Optical Society, which is a member of the American Institute of Physics.

"This [Non-Newtonian Calculus] is an exciting little book, for two reasons: first, its content, and second, its presentation. The content consists primarily of a brief nonaxiomatic description of the fundamentals of various calculi ...

"For each calculus, a gradient, a derivative, and an average are defined; a basic theorem (essentially a mean-value theorem) is stated; an integral is defined; and a fundamental theorem of integral calculus is stated. In Chapter 1, the authors review the essentials of Newtonian calculus and establish the format of presentation that they follow in the later chapters on the various non-Newtonian calculi. Classical problems studied by such men as Galileo and Newton provide examples of, and motivation for, the authors' work. The authors' reference to these problems serves to emphasize the relevance of their results to the concerns of modern science. 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. They have made a dramatic break with tradition and omitted all proofs and many of the mathematical details that place so much of contemporary mathematics writing out of the reach of scientists and engineers. Instead, they have included details that help develop an intuitive conception of their calculi and relate their calculi to well-knows classical problems. The authors apparently feel that their results are of sufficient importance to scientists and engineers to justify these departures from the more traditional style of writing in mathematics. The writing is clear, concise, and very readable. No more than a working knowledge of [classical] calculus is assumed. Mathematicians who feel that their results are of importance to scientists and engineers, but who find little interest among those workers for their results, might consider presenting their work as Grossman and Katz have done. This would do much to reopen channels of communication between mathematicians and scientists and do much to advance both disciplines."

The following excerpt is from a review written by Ivor Grattan-Guinness that appeared in Middlesex Math Notes (Volume 3, pages 47 - 50, 1977), a publication of Middlesex University in London, England.

"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."

The following excerpt is from a review written by H. Gollmann that appeared in Internationale Mathematische Nachrichten (Number 105, 1972), a publication of Österreichische Mathematische Gesellschaft in Vienna, Austria.

"The possibilities opened up by the [non-Newtonian] calculi seem to be immense."

Thanks to David Lukas and Kenneth Lukas for their expert advice about computers, the Internet, the World Wide Web, and website construction. In 2003, they suggested the idea of using the World Wide Web to broadcast information about non-Newtonian calculus. Dave, who set up the first non-Newtonian calculus website, is a computer engineer. Ken, a talented musician and photographer, uses the computer to compose music and to create videos and movies. They knew that posting information on the World Wide Web would be an excellent way to inform people about non-Newtonian calculus. That information has turned out to be of considerable interest to many researchers who until they read about it on the World Wide Web knew nothing about non-Newtonian calculus, but subsequently found applications in various fields.

    Name: Michael Grossman
      E-mail address: smithpith@yahoo.com

      Name: Robert Katz
      Address: 12 Green Street; Rockport, MA 01966; USA

Last Edit
: 21 January 2019