Writing a calculus book that's more rigorous than typical books is hard because if you go too hard, people will say that you've written a real analysis book and the point of calculus is to introduce certain concepts without going full analysis. This book seems to have at least avoided the trap of trying to be too rigorous about the concept of convergence and spending more time on introducing vocabulary to talk about functions and talking about intersections with linear algebra.
>> the author’s wish to present ... mathematics, as intuitively and informally as possible, without compromising logical rigor
The books in the West in general kept getting less rigorous, with time. I don't see Asian or Russian books doing this. The audience getting less receptive to rigor and wishing for more visuals and informal talk. When they get to higher studies and research, would they be able to cope with steep curve of more formalism and rigor?
It correlates student loans with the destruction of academic integrity. The idea is school administrators want to capture as many student loan dollars as possible, and that means maximizing the number of enrolled students. To that end, complexity, rigor and difficulty are all reduced as much as possible. Students are prevented from failing, since if they fail they might drop out, reducing profits. I even remember one article which draws comparisons with russian education.
This one is a hard pass. The book needs tighter editing and more rigorous reviewing.
It tries to serve all at once, but ends up in an awkward middle ground. Not deep enough to function as a real analysis text for Mathematicians, yet full of proofs that Scientists and Engineers do not care about, while failing to deliver the kind of practical rigor, those groups need when using calculus as a tool.
I really appreciate author’s effort and for releasing the PDF to the public. There are however areas I think could be made more helpful.
In the chapter on Trigonometric Functions, it begins with discussion of the derivative of the exponential function is itself, and then moves on to define sine and cosine via second degree derivatives. While interesting, this may not be the most helpful approach for intended readers.
The following chapter on Taylor Series also starts with a long discussion about the value of e. It is interesting, but it can feel a bit distracted from the main topic. The treatment of Taylor polynomials might benefit from a different approach similar to other texts.
I used LLM to rewrite above comment to make it sounds less negative. Then I read those two chapters again. This book adds complexities to simple concepts. There are good math books for non-math majors, but this isn't one of them.
>Calculus is an important part of the intellectual tradition handed down
to us by the Ancient Greeks.
No it isn't? It was discovered by Newton and Leibnitz. If they're talking about Archimedes and integrals, I seem to recall his work on that was only rediscovered through a palimpsest in the last couple of decades and it contributed nothing towards Newton and Leibnitz's work.
So far, I've only had a brief look at this book and what I've seen I like very much.
Many of us—especially those of us who aren't mathematically gifted—learn mathematics in ways that mostly involve procedures, rules and mechanical manipulation rather than through a rigorous step-by-step theoretical framework (well, anyway that's how I leaned the subject).
Somehow I absorbed those foundations more by osmosis than though a full understanding as my early teachers were more concerned with bashing the basics into my head. Sure, later on when confronted with advanced topics I was forced into more rigorous thinking but it wasn't uniform across the whole field.
What I really like about this book is that it confronts people like me who've already learned mathematics to a reasonably advanced level to review those fundamental concepts. The subject of 'What is Calculus?' doesn't start until Chapter 6, p223, and 'Differentiation' at Ch 8, p261. Those first 200 or so pages not only provide a comprehensive and clearly explained overview of those basic fundamentals but they ensure the reader has good understanding of them before the main subject is introduced.
I'd highly recommend this book either as a refresher or as an adjunct to one's current learning.
Without minimizing the quality or your book, I actually like subject matter books that encompass prerequisite knowledge into the text without forcing the reader to read another book in parallel (e.g. Calculus for Machine Learning by Jason Brownlee or No Bullshit Guide to Math & Physics by Ivan Savov). Though not saying that these books are better, they appeal to my learning style a bit more. Learning institutions tend to force students to take too many courses in parallel when they should find a way to join the subjects, whenever possible, without having to break the instruction into multiple semesters, just to sell more books.
Honest question: what kind of rigor and abstraction can help us apply maths to solve problems? Don't get me wrong: I enjoy studying abstract maths and was pretty good at it in school. It's just that when it comes to what to study to make one a more effective problem solver in engineering, I was wonder I can best allocate time. For instance, I find studying probability models more helpful than studying the measure theory when it comes to applied data science or statistics. I also find studying books like Mathematical Methods for Physics and Engineering, which focuses a lot more on intuition and applications than rigor, is more effective for me than going pure math books.
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[ 4.2 ms ] story [ 49.1 ms ] threadThe books in the West in general kept getting less rigorous, with time. I don't see Asian or Russian books doing this. The audience getting less receptive to rigor and wishing for more visuals and informal talk. When they get to higher studies and research, would they be able to cope with steep curve of more formalism and rigor?
https://professorconfess.blogspot.com/
It correlates student loans with the destruction of academic integrity. The idea is school administrators want to capture as many student loan dollars as possible, and that means maximizing the number of enrolled students. To that end, complexity, rigor and difficulty are all reduced as much as possible. Students are prevented from failing, since if they fail they might drop out, reducing profits. I even remember one article which draws comparisons with russian education.
1 - Proof based calculus for math majors
2 - Technique based calculus for hard science majors
3 - Watered down calculus for soft science and business majors (yes, there are a few schools that are exceptions to this)
If he can pull off unifying 1 and 2, good for him!
Now you'll get things in a much easier way, for both programming and math.
It tries to serve all at once, but ends up in an awkward middle ground. Not deep enough to function as a real analysis text for Mathematicians, yet full of proofs that Scientists and Engineers do not care about, while failing to deliver the kind of practical rigor, those groups need when using calculus as a tool.
I really appreciate author’s effort and for releasing the PDF to the public. There are however areas I think could be made more helpful.
In the chapter on Trigonometric Functions, it begins with discussion of the derivative of the exponential function is itself, and then moves on to define sine and cosine via second degree derivatives. While interesting, this may not be the most helpful approach for intended readers.
The following chapter on Taylor Series also starts with a long discussion about the value of e. It is interesting, but it can feel a bit distracted from the main topic. The treatment of Taylor polynomials might benefit from a different approach similar to other texts.
No it isn't? It was discovered by Newton and Leibnitz. If they're talking about Archimedes and integrals, I seem to recall his work on that was only rediscovered through a palimpsest in the last couple of decades and it contributed nothing towards Newton and Leibnitz's work.
Sadly, no. It just seems to start with a gentle version of real analysis, leading into basic Calculus.
Many of us—especially those of us who aren't mathematically gifted—learn mathematics in ways that mostly involve procedures, rules and mechanical manipulation rather than through a rigorous step-by-step theoretical framework (well, anyway that's how I leaned the subject).
Somehow I absorbed those foundations more by osmosis than though a full understanding as my early teachers were more concerned with bashing the basics into my head. Sure, later on when confronted with advanced topics I was forced into more rigorous thinking but it wasn't uniform across the whole field.
What I really like about this book is that it confronts people like me who've already learned mathematics to a reasonably advanced level to review those fundamental concepts. The subject of 'What is Calculus?' doesn't start until Chapter 6, p223, and 'Differentiation' at Ch 8, p261. Those first 200 or so pages not only provide a comprehensive and clearly explained overview of those basic fundamentals but they ensure the reader has good understanding of them before the main subject is introduced.
I'd highly recommend this book either as a refresher or as an adjunct to one's current learning.