This website concerns the systems of non-Newtonian calculus, multiplicative calculus, and nonclassical arithmetic created by Michael Grossman and Robert Katz between 1967 and 1970. (Since 24 June 2021, the website has not been updated regularly.)
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The geometric and bigeometric calculi have often been used by scientists, engineers, and mathematicians. In each of these two calculi, the use of multiplication/division to combine/compare numbers is crucial.
Each of these two calculi is a multiplicative calculus in the sense that its derivative and integral are multiplicative operators. It turns out that infinitely many non-Newtonian calculi are multiplicative calculi, and infinitely many non-Newtonian calculi are not multiplicative calculi. Furthermore, each of the known (to Michael Grossman) multiplicative calculi is a non-Newtonian calculus.
Because there are many multiplicative calculi, the expression "the multiplicative calculus" should be avoided, and no one specific calculus should be named "multiplicative calculus". Nevertheless, some authors have used the name "multiplicative calculus" for the geometric calculus, while others have used the same name for the bigeometric calculus. It is hoped that the scientific community will soon reach accord with regard to names for these two calculi. Our suggestion is simply to use the names "geometric calculus" and "bigeometric calculus", respectively. Interestingly, this matter is discussed by Dorota Aniszewska and Marek Rybaczuk in their article "Multiplicative Hénon map" [288]. (From that article: "There are a few versions of non-Newtonian calculi [Non- Newtonian Calculus], for example the geometric calculus and the bigeometric calculus. In papers each of them is referred by authors as multiplicative calculus, which can be confusing.")
Similarly, the expression "the product calculus" should be avoided, and no one specific calculus should be named "product calculus".
Furthermore, some authors have used the expression "Volterra multiplicative calculus" for a mathematical system created by Vito Volterra in 1887 for the purpose of solving linear (classical) differential equations. Since neither the derivative nor the integral in Volterra's system is a multiplicative operator, the Volterra system is not a multiplicative calculus. [143]
In fact, some authors have erroneously referred to the bigeometric calculus as the "Volterra calculus". The Volterra system is not a non-Newtonian calculus, and is markedly different from both the bigeometric calculus and the geometric calculus. (Vito Volterra, 1860 - 1940, was a brilliant and influential Jewish/Italian scientist.) [143]
NOTE. Various presentations and applications of the geometric and bigeometric calculi are indicated in the Applications & Reception and Citations sections of this website.
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 Internet, 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] in the References section.)
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]
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