Appendix 1: Construction of an Arbitrary Non-Newtonian Calculus

Construction: An Outline

The construction of an arbitrary non-Newtonian calculus involves an ordered pair * of arbitrary complete ordered fields.

Let A and B denote the respective realms of the two arbitrary complete ordered fields, and let R denote the set of all real numbers.

Assume that both A and B are subsets of R. (However, we are not assuming that the two arbitrary complete ordered fields are subfields of the real number system.) Consider an arbitrary function f with arguments in A and values in B.

By using the natural operations, natural orderings, and natural topologies for A and B, one can define the following (and other) concepts of the *-calculus: the *-limit of f at an argument a, f is *-continuous at a, f is *-continuous on a closed interval, the *-derivative of f at a, the *-average of a *-continuous function f on a closed interval, and the *-integral of a *-continuous function f on a closed interval.

There are infinitely many *-calculi. And infinitely many of them are markedly different from the classical calculus. But the classical calculus is one of the *- calculi, and  each *-calculus is structurally similar to the classical calculus. For example, each *-calculus has two Fundamental Theorems showing that the *- derivative and the *-integral are inversely related; and for each *-calculus, there is a special class of functions having a constant *-derivative. 

A non-Newtonian calculus is defined to be any *-calculus other than the classical calculus.

NOTE. The article "An introduction to non-Newtonian calculus" [12] includes the construction of an arbitrary non‐Newtonian calculus from an ordered pair of arbitrary complete ordered fields.

Relationships to classical calculus

The *-derivative, *-average, and *-integral can be expressed in terms of their classical counterparts (and vice versa). (However, as indicated in the Applications & Reception  section, there are many situations in which a specific non-Newtonian calculus may be more suitable than the classical calculus.

Again, consider an arbitrary function f with arguments in A and values in B. Let α and β be the ordered-field isomorphisms from R onto A and B, respectively. Let α−1 and β−1 be their respective inverses.

Let D denote the classical derivative, and let D* denote the *-derivative. Finally, for each number t such that α(t) is in the domain of f, let F(t) = β−1(f(α(t))).

Theorem 1. For each number a in A, [D*f](a) exists if and only if [DF](α−1(a)) exists, and if they do exist, then [D*f](a) = β([DF](α−1(a))).

Theorem 2. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A. Then F is classically continuous on the closed interval (contained in R) from α−1(r) to α−1(s), and M* = β(M), where M* is the *-average of f from r to s, and M is the classical (i.e., arithmetic) average of F from α−1(r) to α−1(s).

Theorem 3. Assume f is *-continuous on a closed interval (contained in A) from r to s, where r and s are in A. Then S* = β(S), where S* is the *-integral of f from r to s, and S is the classical integral of F from α−1(r) to α−1(s).

Examples

Let I be the identity function on R. Let j be the function on R such that j(x) = 1/x for each nonzero number x, and j(0) = 0. And let k be the function on R such that k(x) = √x for each nonnegative number x, and k(x) = -√(-x) for each negative number x.

Example 1. If α = I = β, then the *-calculus is the classical calculus.

Example 2. If α = I and β = exp, then the *-calculus is the geometric calculus.

Example 3. If α = exp = β, then the *-calculus is the bigeometric calculus.

Example 4. If α = exp and β = I, then the *-calculus is the anageometric calculus.

Example 5. If α = I and β = j, then the *-calculus is the harmonic calculus.

Example 6. If α = j = β, then the *-calculus is the biharmonic calculus.

Example 7. If α = j and β = I, then the *-calculus is the anaharmonic calculus.

Example 8. If α = I and β = k, then the *-calculus is the quadratic calculus.

Example 9. If α = k = β, then the *-calculus is the biquadratic calculus.

Example 10. If α = k and β = I, then the *-calculus is the anaquadratic calculus.

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