An infinitely small calculus (comeback) for 40 years

In the title of this article, we use the word "restoration", instead of "Resurrection", which is clearly intended, meaning that the infinitely small calculus has restored the glory of the year, I have taken back my crown ".

In the past, leveniz invented the "infinitely small" and used it for the analysis and research of the local state of the function. After the computation process was completed, the non-zero infinitely small was abandoned, making people a little confused ", even under severe criticism, the "infinitely small" face swept the floor and exited the stage of history. The (ε, Delta) theory of limitation was passed in.

Because the concept of Infinitely small is too tempting for people, in the traditional (ε, Delta) Extreme theory, mathematicians have invented an infinitely small "alternative ", the function with the limit zero is called an infinitely small one. However, the infinite number of such manually created functions is essentially not a "Number", it is a "heterogeneous" of a number, and lacks intuition. In 1960, A. Robinson used model theory (model)
Theory) invented the "non-standard analysis" and restored an infinitely small honor. Although the truth is justified, the theory is too difficult and daunting.

In the 1970s S, Alfred, a well-known Polish logistic expert
Tarsky (1901-1983) Gao tu J. keisler (1936 to date) has taught infinite calculus (based on. robinson's ideas), and achieved a very positive teaching effect. Why is this teaching experiment successful? The reason is that J. Keisler invented a teaching tool: an infinitely small microscope and an infinite telescope, allowing students to see what happened in an infinitely small "area.

On the basis of the successful teaching experiment in Chicago in 1973, J. Keisler published an infinitely small calculus textbook in 1976, which serves as a reference book of calculus for junior college students. So far, 40 years have passed. In the first chapter of this textbook, I started my case and told me straight forward that I had provided a comprehensive and effective mathematical defense (rather than a general philosophical theory) for the "ideology" of laveniz ), especially in chapter 1, chapter 2, and section 5th, the article is well written, with an endless aftertaste.

In order to make readers "have a good eye", on October 16, June 18, I asked clerk Xue Lili to start transcription of the two sections and I will be able to meet you soon. Here, I would like to explain: learning an infinitely small calculus must master basic English and have a certain degree of English reading ability. Otherwise, you cannot fully grasp the spirit of the original book and the rigor of mathematics. Some people do not believe that mathematics is beautiful, because some mathematics instructors have wiped out Mathematics in teaching. As long as you read J. Keisler's teaching material of the infinitely small calculus, you will surely agree that the infinitely small calculus will be "restored.

Currently, we are on the runway of calculus "turning", and those who are unwilling to "turning" will inevitably leave behind. The current situation is that the experts guard the road and the layman is watching the excitement. If you are interested in infinitely small calculus, you may want to keep up with the team and do not leave behind. Although we started very slowly, the deeper we went, the more difficult it was to keep up. In the end, the infinite calculus cannot be used. It depends on facts.

Q: Now, the infinitely small method is fighting the (ε, Delta) limit for 'pk ". Which side are you willing to stand on?