Non-Newtonian calculus

The phrase Non-Newtonian calculus used by Grossman and Katz describes a variety of modifications of the classical calculus of Isaac Newton and Gottfried Leibniz, claimed by the authors to be alternatives.
There are infinitely many such modifications.)
The construction of an arbitrary non-Newtonian calculus involves the real number system and an ordered pair * of arbitrary complete ordered fields.
Let R denote the set of all real numbers, and let A and B denote the respective realms of the two arbitrary complete ordered fields.
Assume that both A and B are subsets of R. (However, we are not assuming that the two arbitrary complete ordered fields are subfields of the real number system.) Consider an arbitrary function f with arguments in A and values in B.
By using the natural operations, natural orderings, and natural topologies for A and B, one can define the following (and other) concepts of the *-calculus: the *-limit of f at an argument a, f is *-continuous at a, f is *-continuous on a closed interval, the *-derivative of f at a, the *-average of a *-continuous function f on a closed interval, and the *-integral of a *-continuous function f on a closed interval.
It turns out that the structure of the *-calculus is similar to that of the classical calculus. For example, there are two Fundamental Theorems of *-calculus, which show that the *-derivative and the *-integral are inversely related; and there is a special class of functions having a constant *-derivative. Furthermore, the classical calculus is one of the infinitely many *-calculi.
A non-Newtonian calculus is defined to be any *-calculus other than the classical calculus.
Relationships to classical calculus
(This section is based on five sources.
History
In August of 1970, Michael Grossman and Robert Katz 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 geometric calculus (July of 1967), and the 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.

In 1972, they completed their book Non-Newtonian Calculus.
* A review of the book Non-Newtonian Calculus in Choice included the following statements about the book: 1) A very interesting book which should be very useful as a basis for a senior undergraduate seminar, but it is not a textbook. 2) It is a worthwhile addition to a college mathematics library. 3) It is clearly written and concise.
* In Mathematical Reviews, Ralph P. Boas, Jr. made the following two assertions: 1) It is not yet clear whether the new calculus provides enough additional insight to justify its use on a large scale. 2) 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.
* A review of Non-Newtonian Calculus in [http://www.mdx.ac.uk/ Middlesex Math Notes (1977)] by Ivor Grattan-Guinness included the following statements: There is enough here 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.
* A [http://books.google.com/books?eiDXOfTOm5FMH58AaC-5TBDA&ctresult&idBkweAQAAIAAJ&dq%22Non-Newtonian+Calculus%22&q=The+greatest+value+of+these+non-Newtonian+calculi+may+prove+to+be+their+ability+to+yield+simpler+physical+laws+than+the+Newtonian+calculus.#search_anchor book review] in the Journal Of The Optical Society Of America by David Pearce MacAdam says "The greatest value of these non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtonian calculus."
* A review in Internationale Mathematische Nachrichten by H. Gollmann says "The possibilities opened up by the calculi seem to be immense."
Citations
* James R. Meginniss (Claremont Graduate School and Harvey Mudd College) used non-Newtonian calculus to create a theory of probability that is adapted to human behavior and decision making.
* Luc Florack and Hans van Assen (both of the Eindhoven University of Technology) used non-Newtonian calculus in an article on biomedical image analysis.
* Ali Uzer (Fatih University in Turkey) used non-Newtonian calculus to develop a multiplicative type of calculus for complex-valued functions of a complex variable.
* Diana Andrada Filip (Babes-Bolyai University of Cluj-Napoca in Romania) and Cyrille Piatecki (Orléans University in France) used non-Newtonian calculus to re-postulate and analyse the neoclassical exogenous growth model in economics.
* S. L. Blyumin (Lipetsk State Technical University in Russia) used non-Newtonian calculus in an article on information technology.
*Emine Misirli and Yusuf Gurefe (both of Ege University in Turkey) used non-Newtonian calculus in an article on the numerical solution of multiplicative differential equations.
*James D. Englehardt (University of Miami) and Ruochen Li (Shenzhen, China) used non-Newtonian calculus in an article on pathogen counts in treated water.
* David Baqaee (Harvard University) used a weighted non-Newtonian calculus
* Fernando Córdova-Lepe (Universidad Católica del Maule in Chile) used the bigeometric derivative in an article on the theory of elasticity in economics.
* The geometric calculus and the bigeometric calculus have application in the study of dimensional spaces. (See Multiplicative calculus and chaos.)
* Ivor Grattan-Guinness (Middlesex University in England) cited Non-Newtonian Calculus
* Manfred Peschel and Werner Mende (both of the German Academy of Sciences Berlin) cited non-Newtonian calculus in a book on the phenomena of growth and structure-building.
* Robert G. Hohlfeld, Thomas W. Drueding, and John F. Ebersole cited non-Newtonian calculus in an article on atmospheric temperature.
* R. Gagliardi and Jerry Pournelle cited Non-Newtonian Calculus
* Ali Ozyapici and Emine Misirli Kurpinar (both of Ege University in Turkey) cited non-Newtonian calculus in a presentation at the 6th ISAAC Congress.
* Ali Ozyapici and Emine Misirli Kurpinar (both of Ege University in Turkey) cited non-Newtonian calculus in a presentation at the 20th International Congress of the Jangjeon Mathematical Society.
* Paul Dickson mentioned non-Newtonian calculus in a book on popular culture.
* Non-Newtonian calculus is mentioned in the journal Science Education International.
* David Malkin cited Non-Newtonian Calculus
* Raymond W. K. Tang and William E. Brigham (both of Stanford University) cited Non-Newtonian Calculus in an article on petroleum engineering.
* Non-Newtonian calculus is mentioned in the journal Ciência e Cultura.
* Non-Newtonian calculus is mentioned in the journal American Statistical Association: 1998 Proceedings of the Section on Bayesian Statistical Science.
* Non-Newtonian Calculus is mentioned in the Australian Journal of Statistics.
* Non-Newtonian calculus is mentioned in the journal Physics in Canada.
* Non-Newtonian Calculus is mentioned in Synthese.
* Non-Newtonian Calculus is mentioned in Mathematical Education.
* Non-Newtonian Calculus is mentioned in the Institute of Mathematical Statistics Bulletin.
* Non-Newtonian Calculus was reviewed in Search.
* Non-Newtonian Calculus was reviewed in the journal Wissenschaftliche Zeitschrift: Mathematisch-Naturwissenschaftliche Reihe.
* Non-Newtonian Calculus was reviewed by M. Dutta in the Indian Journal of History of Science.
* Non-Newtonian Calculus was reviewed by Otakar Zich in Kybernetika.
* Non-Newtonian Calculus was reviewed by Karel Berka in Theory and Decision.
* Non-Newtonian Calculus was reviewed by David Preiss in Aplikace Matematiky.
* Non-Newtonian Calculus was reviewed in Physikalische Blätter.
* Non-Newtonian Calculus was reviewed in "Scientia"; Rivista di Scienza.
* Non-Newtonian Calculus was reviewed in Science Weekly.
* Non-Newtonian Calculus was reviewed in Philosophia mathematica.
* Non-Newtonian Calculus was reviewed in Annals of Science.
* Non-Newtonian Calculus was reviewed in Science Progress.
* Non-Newtonian Calculus was reviewed in Revue du CETHEDEC.
* Non-Newtonian Calculus was reviewed in Allgemeines Statistisches Archiv.
* Non-Newtonian Calculus was reviewed in Il Nuovo Cimento della Societa Italiana di Fisica: A.
* Non-Newtonian Calculus was reviewed in Bollettino della Unione Matematica Italiana.
* Non-Newtonian Calculus was reviewed in Cahiers du Centre d'Etudes de Recherche Opérationnelle.
* Non-Newtonian Calculus was reviewed in the American Mathematical Monthly.
* The First Nonlinear System of Differential And Integral Calculus, a book about the geometric calculus, was reviewed in the American Mathematical Monthly.
* Bigeometric Calculus: A System with a Scale-Free Derivative was reviewed in Mathematics Magazine and listed as "interesting mathematical exposition that occurs outside the mainstream of the mathematics literature".
* Bigeometric Calculus: A System with a Scale-Free Derivative
* The First Systems of Weighted Differential and Integral Calculus
* Meta-Calculus: Differential and Integral
* The article "An introduction to non-Newtonian calculus"
* The article "A new approach to means of two positive numbers" was reviewed in Zentralblatt MATH.
* Each of the following three books was reviewed by K. Strubecker in Zentralblatt MATH.
:1) Non-Newtonian Calculus: Zbl 0493.26001.
* The article "A new approach to means of two positive numbers"
* Each of the following five books was reviewed in the journal ZDM.
:1) Non-Newtonian Calculus: 19861.06873.
* Each of the following six books was reviewed in Internationale Mathematische Nachrichten.
:1) Non-Newtonian Calculus
:1) Non-Newtonian Calculus
:1) Non-Newtonian Calculus
:1) The First Nonlinear System of Differential And Integral Calculus
:1) The First Nonlinear System of Differential and Integral Calculus
:1) The First Nonlinear System of Differential and Integral Calculus
:1) Non-Newtonian Calculus: Volume 33, page 361, 1972.
:2) The First Nonlinear System of Differential and Integral Calculus: Volumes 42-43, page 225, 1980.
* Each of the following six books was reviewed in Industrial Mathematics.
:1) Non-Newtonian Calculus: Volumes 43-45, page 91, 1994 .
:2) The First Nonlinear System of Differential and Integral Calculus: Volumes 28-30, page 143, 1978.
:3) The First Systems of Weighted Differential and Integral Calculus: Volumes 31-33, page 66, 1981.
:4) Meta-Calculus: Differential and Integral: Volumes 31-33, page 83, 1981.
:5) Bigeometric Calculus: A System with a Scale-Free Derivative: Volumes 33-34, page 91, 1983.
:6) Averages: A New Approach: Volumes 33-34, page 91, 1983.
* Each of the following two books was reviewed in Economic Books: Current Selections.
:1) The First Systems of Weighted Differential and Integral Calculus: Volume 9, page 29, 1982.
:2) Meta-Calculus: Differential and Integral: Volume 9, page 29, 1982.
* Non-Newtonian Calculus was reviewed in Mathematical Reviews.
* Each of the following five books was reviewed by Ralph P. Boas, Jr. in Mathematical Reviews.
:1) The First Nonlinear System of Differential and Integral Calculus: Mathematical Reviews, 1981.
:3) Meta-Calculus: Differential and Integral: Mathematical Reviews, 1984.

Main Menu

See Also

Main Menu