Math Tables:

An Introduction to 
Generalized Calculus
Unit I - Product Calculus
A. Introduction

I. Product Calculus

A. Introduction

We will examine our first non-elementary calculus, the product calculus. The product calculus is defined by a whole new set of building blocks. It gets its name from the product expansion, product, which we will refer to as the discrete product for classification reasons. Unlike the summation, (sum), which is the sum of all terms, the product is the product of all its terms.

One may propose that we replace the (sum) in the discrete iteration block with the product and build a new calculus from there:
Product Calculus Continuous Discrete
Limit ?  
Iteration ? (product) 
discrete product
The other building blocks must be redefined in order to satisfy the conditions mentioned in the preface:

The existence of the other two building blocks is hinted by the following analogy:

summationis to product    as integral is to ?

Furthermore, we may ask:

summation is to d/dx    as product is to ?

In the following sections, we will find out exactly what these building blocks are:
Product Calculus Continuous Discrete
Limit q ^ (1/dx) 
Iteration (continuous product) 
continuous product
discrete product
And from there precipitates a whole new calculus with its own rules and theorems. Yet, it maintains key relationships with elementary calculus.

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