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Brief Description

The non-Newtonian calculi provide a wide variety of mathematical tools for use in science, engineering, and mathematics. They appear to have considerable potential for use as alternatives to the classical calculus of Newton and Leibniz.

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, many non-Newtonian calculi are markedly different from the classical calculus.    
    
For example, infinitely many non-Newtonian calculi have a nonlinear derivative or integral. Among these calculi are the geometric calculus, bigeometric calculus, harmonic calculus, biharmonic calculus, quadratic calculus, and biquadratic calculus. Furthermore, in the geometric calculus and in the bigeometric calculus, the derivative and integral are both multiplicative. (Please see the "Multiplicative Calculus" section of this website.)

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 values.

Non-Newtonian calculus has application in science, engineering, and mathematics. The areas of application include: fractal theory, image analysis (e.g., in bio-medicine), growth/decay (e. g., in economics and biology), rates of return, the theory of elasticity in economics, finance, the economics of climate change, signal processing, atmospheric temperature (e.g., optical measure theory and inverse transfer theory), information technology, pathogen counts in treated water, actuarial science, tumor therapy in medicine, materials science/engineering, demographics, differential equations (including a multiplicative Lorenz system), calculus of variations, finite-difference methods, averages of functions, means of two positive numbers, weighted calculus, meta-calculus, least-squares methods of approximating, multivariable calculus, complex analysis, functional analysis, probability theory, utility theory, Bayesian analysis, stochastics, decision making, dynamical systems, chaos theory, and dimensional spaces. For more information, please see the Applications and Citations sections of this website.

It is natural to speculate about future applications of non-Newtonian calculus, and of related matters such as weighted calculus, meta-calculus, averages, and means. Perhaps scientists, engineers, and mathematicians will use them to define new concepts, to yield new or simpler laws, or to formulate or solve problems.



Acknowledgement                                                                                                                                                          

Thanks to David Lukas and Kenneth Lukas for constructing previous versions of this website, and for their expert advice on website construction.



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  • Name: Michael Grossman
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Last Edit

15 April 2014
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