Nonstandard Calculus: Simple Definition, Overview

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Nonstandard calculus is the application of infinitesimals, using nonstandard analysis, to infinitesimal calculus.

The ε and δ of traditional calculus.

Nonstandard calculus uses infinitesimals for computing limits and derivatives. Following severe criticism, infinitesimals and infinite numbers were effectively banned from calculus at the end of the nineteenth century, favoring the epsilon and delta approach popularized by Karl Weierstrass. Abraham Robinson (1918-1974) revived the idea of infinitesimals in the 1960s, leading to the development of nonstandard calculus.

Although infinitesimals had been around for some time (albeit barred from modern calculus), Robinson gave them a precise definition. He used the hyperreals (*R), an extension of real numbers to include infinitesimal numbers and infinite numbers. These numbers, which can be defined precisely, behave like very large natural numbers.

It’s important to note that the infinitesimal calculus of Newton and Leibniz and nonstandard calculus are not the same thing: nonstandard calculus is a relatively small field that applies nonstandard analysis to infinitesimal calculus.

Disambiguation note: Sometimes, the term “nonstandard calculus” is used to describe courses other than traditional college calculus, which has a different meaning entirely from the branch of calculus described here. For example, in the paper Revitalization of Nonstandard Calculus, Fetta (1996) the “nonstandard” calculus in question is actually business calculus.

Nonstandard calculus is thought by many to be easier to understand, more intuitive, and conceptually simpler than the more common infinitesimal calculus approach. For example:

The definition of the continuity of a function at a point c is simply that x infinitely close to c implies that f(x) is infinitely close to f(c) (Sullivan, 2014).

However, others consider the formal structure of nonstandard calculus too technical for those without a good background in logic. Complicating the issue is that while real numbers are familiar and easy to understand, nonstandard reals are a challenge to comprehend for most students. That said, some courses have been developed that simplify these challenging details. Many are based on Jerome Keisler’s seminal works Elementary Calculus: An Infinitesimal Approach.

References

Fetta, I. (1996). Revitalization of Nonstandard Calculus. Retrieved May 14, 2020 from: https://files.eric.ed.gov/fulltext/ED417936.pdf
Krakoff, G. (2005). Hyperreals and a Brief Introduction to Non-Standard Analysis. Math 336. Retrieved May 14, 2020 from: https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf
Sullivan, K. (2014). The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly, Vol. 83, No. 5 (May, 1976), pp. 370-375.
Tuckey, C. (1993). Nonstandard Methods in the Calculus of Variations (Chapman & Hall/CRC Research Notes in Mathematics Series) 1st Edition. Chapman and Hall/CRC.

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Stephanie Glen. "Nonstandard Calculus: Simple Definition, Overview" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/nonstandard-calculus/
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