Some applaud on writing expository Calculus essays

This post is mainly used as a reply to xAI's tweet, it is excerpted from my self-introduction letter.



I think there are few good textbooks on calculus written in Chinese for beginning students, most of the textbooks put too much emphasis on rigor, while the rigorous presentation is difficult to grasp and obscures the understanding, so many students struggled with the course. I suffered the same difficulty when I started learning calculus on my own, and I failed the self-learning at least three times until I found good books suitable for self-study and developed the method for self-learning. Although I have found my way to go on my calculus learning, many students still struggle with the course, thus I feel a sense of mission to help ease the learning of calculus for our country’s students, so I started to write expository essays on hard topics of calculus and real analysis. I have written 20 essays (nearly all written in Chinese) , each essay was revised at least five times after completing the first draft, some were refined about twenty times, I am striving for perfection in these writings. A calculus learning guide essay written by myself was viewed 69543 times, a video tutorial on explaining the notation dx, dy, differentials, and derivatives was played over 101000 times, an essay on rebuilding the real number theory in an intuitive but still rigorous approach was viewed 12228 times, which is also the essay I had spent the most effort to write yet, it was because I was not satisfied with those presented in the book, so I created my own. Another worthwhile mentioning essay is about proving the L’Hospital’s Rule, which was viewed 22739 times, it was also because I am not satisfied with proofs presented on three commonly used Chinese real analysis textbooks(they all used hard devised special tricks to help the proof, while I think they should give constructible and informative proofs), so after spending a lot of effort, I figured out my satisfactory proofs.

I have posted some of these essays as answers to some related questions in a Chinese QA site – in order to maximize their influence and help as many students as possible, and I got so many votes from my readers (the following is a screenshot). Please check it at https://www.zhihu.com/people/li-ruo-shui-39/answers/by_votes if needed.




I also created a WeChat "public account"（named 高数变简单） to push my new expository essay to the subscribers, up to now, there are 3510 subscribers, below are part of the user analysis statistical charts(translated by Google).


Besides my expository essays, you can also assess my academic potential according to the questions I asked during my Calculus learning process, you can view them on my Math Stack Exchange page, a recent review of them makes me more confident in my academic potential. I think I am good at asking deep questions, so sometimes one can say I am learning Real analysis instead of Calculus, although I am actually learning Calculus.

These days I am considering to improve my essay on constructing the number continuum, it is also the essay I have spent the most effort to compose yet, the first version was published on my blog in 2018, in the following years I've made some revisions to it until now, the net time it has taken should be at least eight months. The two famous construction of the number continuum was historically done by Georg Cantor and Richard Dedekind respectively, but their constructions are independent of geometric magnitude, and the definition of irrational numbers are not a single symbol or a pair of symbols, such as a ratio of two integers, but an infinite collection, so their theories are difficult to understand, as mathematician Hermann Hankel(1839 - 1873) criticized: “Every attempt to treat the irrational numbers formally and without the concept of (geometric) magnitude must lead to the most abstruse and troublesome artificialities, which, even if they can be carried through with complete rigor, as we have every right to doubt, do not have a higher scientific value.” Therefore I decided to try to put forward an intuitive but still rigorous construction of the number continuum - I am trying to construct the number continuum on Euclidean geometry. The question I had sought for help last time originated from the research project. To learn MAT 404 Geometry in the coming semester might be helpful for the research. Another question I’d like to investigate is “why Leibniz’s indefinite integral notation is superior?”, the question originated from another question “why dx is necessary in ∫f(x)dx?” I have considered. As mathematician Paul Halmos said:“The only way to learn mathematics is to do mathematics.” I like learning by researching or doing.

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