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Multiplicative Calculus: Definition

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Multiplicative calculus is a special version of non-Newtonian calculus that uses multiplicative operators. It is built on two basic operations: multiplicative integration and multiplicative derivation. The two operations are inversely related to each other.

Essentially, this type of calculus involves a system in which the addition and subtraction in regular calculus are replaced by multiplication. In the ordinary type of calculus, we base the idea of an integral on an additive rate of change; In this special calculus, we base the idea of an integral on a multiplicative rate of change. There are an infinite number of ways to do this.

Applications of Multiplicative Calculus

Different systems of multiplicative calculi have different applications, but the most widely used systems are geometric calculus and bi-geometric calculus. Geometric calculus is used in biometrical image analysis, as well as to study economic growth, bacterial growth, and radioactive decay. We use bi-geometric calculus to study the theory of elasticity in economics, and it can also help us to understand fractals.

One Example: Geometric Calculus

Perhaps the most commonly used multiplicative systems involve geometric calculus. The basis of this is relatively easy to understand, if you understand some basics about derivatives. The regular formula for a derivative is:
Standard Derivative

The derivative for geometric calculus— the geometric derivative, we could call it— is:
geometric derivative (multiplicative derivative)

Incidentally, we can write this in terms of our ‘standard derivative’ as:
multiplicative calculus derivative

Interestingly enough, exponential functions are the functions that have a constant derivative in geometric calculus, just as straight lines have a constant derivative in standard classical calculus.

References

Bashirov, Kurpinar, & Ozyapici. Multiplicative Calc and Its Applications. Journal of Mathematical Analysis and Applications, Volume 337, Issue 1, 1 January 2008, Pages 36-48. Retrieved https://core.ac.uk/download/pdf/81954511.pdf from on April 7, 2019.
Grossman, Michael. Non-Newtonian Calculus: Multiplicative. Retrieved from https://sites.google.com/site/nonnewtoniancalculus/-multiplicative-calculus on April 7, 2019.
Duff Campbell BA and PhD (1999) Multiplicative Calc and Student Projects, PRIMUS, 9:4, 327-332, DOI: 10.1080/10511979908965938. Retrieved from https://www.tandfonline.com/doi/abs/10.1080/10511979908965938 on April 7, 2019.


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Stephanie Glen. "Multiplicative Calculus: Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/multiplicative-calculus-definition/
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