Non-Newtonian calculus

The phrase Non-Newtonian calculus used by Grossman and KatzGrossman and Katz. "Non-Newtonian Calculus", ISBN 0912938013, Lee Press, 1972.] describes a variety of alternatives to the classical calculus of Isaac Newton and Gottfried Leibniz.

There are infinitely many non-Newtonian calculi. Like the classical calculus, each of them possesses (among other things) a derivative and an integral, a special class of functions having a constant derivative, and two Fundamental Theorems which reveal that the derivative and integral are inversely related. Nevertheless, the non-Newtonian calculi differ from the classical calculus in various important ways.

For example, in the classical calculus, the derivative and integral are linear operators, i.e., they are additive and homogeneous. This contrasts sharply with the geometric calculus (a non-Newtonian calculus) and the bigeometric calculus (another non-Newtonian calculus), in each of which the derivative and integral are nonlinear operators. In fact, each of those derivatives and integrals is multiplicative and involutional.

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 well-known arithmetic average (of functions) is the 'natural' average in the classical calculus, but the well-known geometric average (of functions) is the 'natural' average in the geometric calculus.

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 values. The bigeometric derivative is closely related to the concept of elasticity used by economists.

Multiplicative calculus

There are many non-Newtonian calculi whose derivative and integral are multiplicative. The geometric calculus and the bigeometric calculus are examples. Accordingly, the expression "the" multiplicative calculus" should be avoided.

General theory: an outline

This section presents information acquired from three sources. [Michael Grossman, "An introduction to non-Newtonian calculus", International Journal of Mathematical Education in Science and Technology 10 #4 (Oct.-Dec., 1979), 525-528.] Meginniss. "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis", Proceedings of the American Statistical Association: Business and Economics Statistics, 1980.] Bashirov, Misirli Kurpinar, and Ozyapici. [http://linkinghub.elsevier.com/retrieve/pii/S0022247X07003824 "Multiplicative calculus and its applications"] , Journal of Mathematical Analysis and Applications, 2008.] Verify source|date=October 2008

Construction of a non-Newtonian calculus

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

In general, however, the *- calculus is markedly different from the classical calculus. For example, when applied to specific functions, the *-derivative and *-integral yield results quite different from those yielded by the classical derivative and classical integral.

A non-Newtonian calculus is defined to be any *-calculus other than the classical calculus.

Relationships to classical calculus

The *-derivative, *-average, and *-integral can be expressed in terms of their classical counterparts (and vice-versa), although as mentioned in the article multiplicative calculusMarek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001 [http://linkinghub.elsevier.com/retrieve/pii/S0960077900002319 The concept of physical and fractal dimension II. The differential calculus in dimensional spaces] , "Chaos, Solitons, & Fractals" Volume 12, Issue 13, October 2001, Pages 2537-2552 ] there are contexts in which this relationship breaks down. Again, consider an arbitrary function f with arguments in A and values in B Let G and H be the ordered-field isomorphisms from R onto A and B, respectively. Let g and h be their respective inverses. Let D denote the classical derivative, and let D* denote the *-derivative. Finally, for each number t such that G(t) is in the domain of f, let F(t) = h(f(G(t))). Theorem 1. For each number a in A, [D*f] (a) exists if and only if [DF] (g(a)) exists, andif they do exist, then [D*f] (a) = H( [DF] (g(a))). Theorem 2. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A.Then F is classically continuous on the closed interval (contained in R) from g(r) to g(s), and M* = H(M), where M* is the *-average of f from r to s, and M is the classical (i.e., arithmetic) average of F from g(r) to g(s). Theorem 3. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A.Then S* = H(S), where S* is the *-integral of f from r to s, and S is the classical integral of F from g(r) to g(s).

Examples

Let I be the identity function on R. And let j be the function on R (the positive reals) such that j(x) = 1/x for each positive number x. Example 1. In the case where G = I = H, the *-calculus is the classical calculus. Example 2.In the case where G = I and H = exp, the *-calculus is the geometric calculus, in which the 'natural' average is the well-known geometric average. Example 3. In the case where G = exp = H, the *-calculus is the bigeometric calculus. Example 4.In the case where G =exp and H = I, the *-calculus is the so-called anageometric calculus.

Example 5.In the case where G = I and H = j, the *-calculus is the so-called harmonic calculus, in which the 'natural' average reduces to the well-known harmonic average when applied to positive-valued functions.

Example 6. In the case where G = j = H, the *-calculus is the so-called biharmonic calculus.

Example 7. In the case where G = j and H = I, the *-calculus is the so-called anaharmonic calculus.

Of course, there are infinitely many other examples.

Criticism

There are conflicting views on the value of non-Newtonian calculus as the reviews below show.

* A book review in the Mathematical Gazette by Douglas Quadling describes the work of Grossman and Katz as trivial and of achieving nothing that couldn't already be achieved with the usual calculus.The Mathematical Gazette, Vol. 68, No. 443 (Mar., 1984), pp. 70-71]

* In [http://www.ams.org/mr-database Mathematical Reviews] , Ralph P. Boas, Jr. made the following two assertions: 1) It is not yet clear whether the geometric 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 cleary by using bigeometric calculus instead of classical calculus.

* A book review in the [http://josaa.osa.org/Issue.cfm 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 [http://www.oemg.ac.at/IMN/index.html Internationale Mathematische Nachrichten] by H. Gollmann says "The possibilities opened up by the [non-Newtonian] calculi seem to be immense."
* Long ago comments by Gauss about alternative calculi are given in "Memorabilia Mathematica" [Robert Edouard Moritz. "Memorabilia Mathematica", pages 197 and 198, quotation # 1215, The Macmillan Company, 1914.]

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.Meginniss. "Non-Newtonian calculus applied to probability, utility, and Bayesian analysis", Proceedings of the American Statistical Association: Business and Economics Statistics, 1980.]

Agamirza E. Bashirov (Eastern Mediterranean University in Cyprus), together with Emine Misirli Kurpinar and Ali Ozyapici (Ege University in Turkey), discovered several applications of multiplicative calculus. Their work includes applications to differential equations and calculus of variations.Bashirov, Misirli Kurpinar, and Ozyapici. [http://linkinghub.elsevier.com/retrieve/pii/S0022247X07003824 "Multiplicative calculus and its applications"] , Journal of Mathematical Analysis and Applications, 2008.]

History

In August of 1970, Michael Grossman and Robert Katz constructed a comprehensive family of calculi, which includes the classical calculus, the geometric calculus, the bigeometric calculus, and infinitely-many other calculi that they constructed in July of 1967. All of these 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". It includes discussions of nine specific non-Newtonian calculi, the general theory of non-Newtonian calculus, and heuristic guides for application. Subsequently, with Jane Grossman, they wrote five other books on non-Newtonian calculus and related matters.

See also Multiplicative calculus#History.

Means of two positive numbers

The *-averages can be used to construct a family of means of two positive numbers. [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, pages 205 -208] Verify credibility|date=October 2008 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.

References

Additional reading

* Michael Grossman, [http://www.informaworld.com/smpp/content~content=a746867201~db=all~order=page An introduction to non-Newtonian calculus] , International Journal of Mathematical Education in Science and Technology 10 #4 (Oct.-Dec., 1979), 525-528.

* Michael Grossman and Robert Katz, [http://www.informaworld.com/smpp/content~content=a746858723~db=all~order=page Isomorphic calculi] ,International Journal of Mathematical Education in Science and Technology, Volume 15, Issue 2 March 1984 , pages 253 - 263.

* Michael Grossman and Robert Katz, [http://www.informaworld.com/smpp/content~content=a746867623~db=all~order=page A new approach to means of two positive numbers] , International Journal of Mathematical Education in Science and Technology, Volume 17, Number 2, pages 205 -208.

* Jane Tang, [http://www.informaworld.com/smpp/content~content=a746854636~db=all~order=page On the construction and interpretation of means] , International Journal of Mathematical Education in Science and Technology, Volume 14, Number 1, pages 55 - 57.

* Jane Grossman, Michael Grossman, and Robert Katz. "Averages: A New Approach", ISBN 0977117049, 1983.

* Grattan-Guinness. "The Rainbow of Mathematics: A History of the Mathematical Sciences", pages 332 and 774, ISBN 0393320308, W. W. Norton & Company, 2000.