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Non-Newtonian Calculus

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Non-Newtonian calculus, formulated by Robert Katz, Jane Grossman, and Michael Grossman between 1967 and 1972, is a family of non-linear calculi. Classical calculus, primarily developed by Newton and Leibniz in the 17th century, is the “usual” calculus you learn in high school and college. It is linear with additive operators (e.g. derivatives and integrals are both calculated by way of small additions or subtractions). Non-Newtonian calculus is non-linear and doesn’t have additive operators.


After Grossman & Katz’s development of Non-Newtonian calculus, there was a “long period of silence” (Riza & Bugce, 2014) until Bashirov et. al’s 2008 paper Multiplicative calculus and its applications, which appeared in the Journal of Mathematical Analysis and Applications.

Grossman & Katz outlined several branches of Non-Newtonian calculus, including the broad term “multiplicative calculus” and sub-branches of bigeometric calculus and geometric calculus. Other branches, which have largely remained undeveloped, include:


  • Anageometric calculus,
  • Anaharmonic calculus,
  • Anaquadratic calculus,
  • Biharmonic calculus,
  • Biquadratic calculus,
  • Harmonic calculus,
  • Quadratic calculus.

Differences Between Classical and Non-Newtonian Calculus

Non-Newtonian calculus has several major differences from classical calculus. For example, exponential calculus (also called geometric calculus in earlier texts) has multiplicative operators. These operators can only be applied to positive functions, so only positive functions are valid in this particular calculus.

The derivative is defined quite differently as well. In traditional calculus, addition and subtraction are the primary tools to find a derivative. For example, when you find a limit, you’re constantly adding (or subtracting) tiny pieces to approach a particular number. The first non-Newtonian calculus developed, multiplicative calculus, replaces addition and subtraction with division and multiplication. The definition of a derivative changed to focus on a ratio:

non-Newtonian calculus

Non-Newtonian derivative (top) and traditional derivative (bottom).

References

A.E. Bashirov, E.M. Kurpınar, and A. Özyapıcı. Multiplicative calculus and its applications.
Journal of Mathematical Analysis and Applications, 337(1):36–48, 2008
Grossman, M. & Katz, R. (1971). Non-Newtonian Calculus. A Self-contained, Elementary Exposition of the Authors’ Investigations. Lee Press.
Riza, M. & Bugce, E. (2014). Bigeometric Calculus and Runge Kutta Method. Retrieved September 19, 2020 from: https://arxiv.org/pdf/1402.2877.pdf

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Stephanie Glen. "Non-Newtonian Calculus" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/non-newtonian-calculus/
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