An Introduction to Generalized Calculus

1994,1998, David Manura (Rough Draft Version)

Table of Contents

Preface - The building blocks of a calculus.

I. The Product Calculus
    A. Introduction
    B. The Discrete Iteration -
    C. The Continuous Iteration -
    D. The Continuous Limit -
    E. Product Expansion Theorem

II. The Theory of Generalized Calculus
    A. More Anti-Iterations
    B. Iterations from Anti-Iterations
    C. Anti-Iterations from Iterations
    D. Expansion Theorems
    E. Notation & Manipulating Calculuses

Update: This was one of my early adventures into mathematics, and it's been a long time since I've done anything more on it, but over the years others reading this have brought to my attention various similar pursuits, so I'll mention them here for the benefit of those who maintain an interest in it:

Power Means Calculus Product Calculus, Harmonic Mean Calculus, and Quadratic Mean Calculus. H. Vic Dannon. March, 2008
An introduction to Multigral (Product) and Dx-less Calculus. Daryl Williams.
Non-Newtonian Calculus. Michael Grossman, Robert Katz. 1972
Prof. John Verosky of Western Washington University developed something similar to product calculus referred to as a "biological calculus" (discussions in 1999). Michael Doré briefly derived similar results for product calculus. Others have mentioned similar ideas (Casey Coleman and Shane Taylor). For more details, search for "generalized calculus david" on the messageboard http://math2.org/mmb/ . Surely yet others have also considered replacing addition and multiplication operators in Newtonian calulus with different operators, as has been done with algebra. Thanks to Bob Brown for pointing out the Dannon paper and other discussions.
Wikipedia: Product integral | planetmath: non-Newtonian Calculus
questions

Math2.org Math Tables:

An Introduction to Generalized Calculus