This website concerns the systems of non-Newtonian calculus, multiplicative calculus, and nonclassical arithmetic created by Michael Grossman and Robert Katz between 1967 and 1970. (Since 24 June 2021, the website has not been updated regularly.)
This section is intended for reference purposes only. It includes a brief outline for the construction of an arbitrary non-Newtonian calculus, but it is NOT a substitute for the book Non- Newtonian Calculus [15].
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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