Multiplicative calculus

In mathematics, a multiplicative calculus is a system with two multiplicative operators, called a "multiplicative derivative" and a "multiplicative integral", which are inversely related in a manner analogous to the inverse relationship between the derivative and integral in the classical calculus of Newton and Leibniz. The multiplicative calculi provide alternatives to the classical calculus, which has an additive derivative and an additive integral.

There are infinitely many non-Newtonian multiplicative calculi, including the geometric calculus and the bigeometric calculus discussed below.[1] These calculi all have a derivative and/or integral that is not a linear operator.

The geometric calculus is useful in image analysis.[2][3][4][5] The bigeometric calculus is useful in some applications of fractals.[6][7][8][9][10][11][12][13][14][15]

Multiplicative derivatives

Geometric calculus

The classical derivative is

f'(x) = \lim_{h \to 0}{f(x+h) - f(x)\over{h}}

The geometric derivative is

f^{*}(x) = \lim_{h \to 0}{ \left({f(x+h)\over{f(x)}}\right)^{1\over{h}} }

(For the geometric derivative, it is assumed that all values of f are positive numbers.)

This simplifies[16] to

f^{*}(x)=e^{f'(x)\over f(x)}

for functions where the statement is meaningful. Notice that the exponent in the preceding expression represents the well-known logarithmic derivative.

In the geometric calculus, the exponential functions are the functions having a constant derivative.[1] Furthermore, just as the arithmetic average (of functions) is the 'natural' average in the classical calculus, the well-known geometric average is the 'natural' average in the geometric calculus.[1]

Bigeometric calculus

A similar definition to the geometric derivative is the bigeometric derivative

f^{*}(x) = \lim_{h \to 0}{ \left({f((1+h)x)\over{f(x)}}\right)^{1\over{h}} } =  \lim_{k \to 1}{ \left({f(kx)\over{f(x)}}\right)^{1\over{\ln(k)}} }

(For the bigeometric derivative, it is assumed that all arguments and all values of f are positive numbers.)

This simplifies[11] to

f^{*}(x)=e^{xf'(x)\over f(x)}.

for functions where the statement is meaningful. Notice that the exponent in the preceding expression represents the well-known elasticity concept, which is widely used in economics.

In the bigeometric calculus, the power functions are the functions having a constant derivative.[1] Furthermore, 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.

Multiplicative integrals

Each multiplicative derivative has an associated multiplicative integral. For example, the geometric derivative and the bigeometric derivative are inversely-related to the geometric integral and the bigeometric integral, respectively.

Of course, each multiplicative integral is a multiplicative operator, but some product integrals are not multiplicative operators. (See Product integral#Basic definitions.)

Discrete calculus

Just as differential equations have a discrete analog in difference equations with the forward difference operator replacing the derivative, so too there is the forward ratio operator f(x + 1)/f(x) and recurrence relations can be formulated using this operator.[17][18][19] See also Indefinite product.

Complex analysis

History

Between 1967 and 1988, Jane Grossman, Michael Grossman, and Robert Katz produced a number of publications on a subject created in 1967 by the latter two, called "non-Newtonian calculus." The geometric calculus[25] and the bigeometric calculus[26] are among the infinitely many non-Newtonian calculi that are multiplicative.[1] (Infinitely many non-Newtonian calculi are not multiplicative.)

In 1972, Michael Grossman and Robert Katz completed their book Non-Newtonian Calculus. It includes discussions of nine specific non-Newtonian calculi, the general theory of non-Newtonian calculus, and heuristic guides for application. Subsequently, with Jane Grossman, they wrote several other books/articles on non-Newtonian calculus, and on related matters such as "weighted calculus",[27] "meta-calculus",[28] and averages/means.[29][30]

On page 82 of Non-Newtonian Calculus, published in 1972, Michael Grossman and Robert Katz wrote:

"However, since we have nowhere seen a discussion of even one specific non-Newtonian calculus, and since we have not found a notion that encompasses the *-average, we are inclined to the view that the non-Newtonian calculi have not been known and recognized heretofore. But only the mathematical community can decide that."

General theory of non-Newtonian calculus

(This section is based on six sources.[1][2][16][31][32][33])

Construction: an outline

The construction of an arbitrary non-Newtonian calculus involves the real number system and an ordered pair * of arbitrary complete ordered fields.

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

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.

Many, if not most, *-calculi are markedly different from the classical calculus, but the structure of each *-calculus is similar to that of 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. Furthermore, the classical calculus is one of the infinitely many *-calculi.

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

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 Reception-section below, there are situations in which a specific non-Newtonian calculus may be more suitable than the classical calculus.[2][6][16][32][33][34])

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 so-called anageometric calculus.

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

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

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

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

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

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

Reception

"What happens to the old calculus when you restrict its application to positive functions and replace the differential ratio [f(x+h)-f(x)]\over h with the multiplicative one \left[{f(x+h)\over f(x)}\right]^{1/h}\text{?} Answer: the usual derivative f'(x) is replaced with f^{\star}(x)=\left[ \exp(\ln[f(x)])\right]'. So you are left with some avatar of the classical calculus to unfold. The authors of this original paper do play this game. Their stated purpose is to promote this new kind of multiplicative calculus." (Note that f^{\star}(x)=\left[ \exp(\ln[f(x)])\right]' should read f^{\star}(x)=\exp[(\ln\circ f)'(x)].[16])
"In this expository article the authors develop the basics of the so called multiplicative calculus, under which the definition of derivatives and integrals is given in terms of the operations of multiplication and division in contrast to addition and subtraction in the usual definitions. Such an approach was suggested in a book of M. Grossman and R. Katz [“Non-Newtonian Calculus”. Pigeon Cove, Mass.: Lee Press (1972; Zbl 0228.26002)]. Transforming multiplication to addition by logarithms, it is easy to see that for instance a multiplicative derivative equals to exp[(lnf)′]. The authors give also some applications where they consider the usage of the language of multiplicative calculus as more useful than the usage of the usual calculus."

See also

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Further reading

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