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Simplifying and Refactoring Introductory Calculus (2018)

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Re: Simplifying and Refactoring Introductory Calculus (2018)

#51

Post by srean »

rramadass wrote: post #30

Some good books for introductory Calculus; 1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for... 2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-ha…

Thanks for the list.

I have been on the lookout for a specific book I was suggested as a kid. This was when our high school Physics was traveling a few paces ahead of our mathematics curriculum.

All I remember is that the Indian paperback edition had a blue cover. It was very helpful.

I find it hard to understand the persistent calculus hate that I see on HN. For us it was a very enjoyable experience.

We learned it through two courses that sort of raced each other at a tepid pace -- high school Physics (especially dynamics) and high school mathematics.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#52

Post by simonreiff »

u1hcw9nx wrote: post #44
simonreiff wrote: post #14

Earlier quoted context omitted.

Actually Leibniz invented the modern dy and dx notation and did view the differentials as genuinely nonzero, which is generally speaking a safe assumption. In other words, dy/dx really was a quotient, albeit of really tiny values (at least we assume dx can become arbitrarily small while remaining nonzero). The calculation Leibniz would do looked something like this. First he would consider an equation y = x^2. Then h…

Hyperreal numbers make Leibniz notation literal and algebraically consistent and rigorous, rather than a convenient shorthand for limits. Author suggests using them. Resistance to Hyperreals seems to come from era before there were rigorous definitions for them.

Personally I dislike hyperreals for the reason that we lose the all-powerful Archimidean property (for all real numbers x, there exists a natural number N such that N > x or N = x). But yes, hyperreals are completely consistent. Robinson showed, I want to say in the 1960s, that any statement about the logical consistency of the hyperreals is true if and only if the same is true for the reals. Then hyperreals basically save Leibniz by introducing the standard part function st(), which is rigorously defined.

Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#54

Post by u1hcw9nx »

simonreiff wrote: post #52
u1hcw9nx wrote: post #44

Earlier quoted context omitted.

Hyperreal numbers make Leibniz notation literal and algebraically consistent and rigorous, rather than a convenient shorthand for limits. Author suggests using them. Resistance to Hyperreals seems to come from era before there were rigorous definitions for them.

Personally I dislike hyperreals for the reason that we lose the all-powerful Archimidean property (for all real numbers x, there exists a natural number N such that N > x or N = x). But yes, hyperreals are completely consistent. Robinson showed, I want to say in the 1960s, that any statement about the logical consistency of the hyperreals is true if and only if the same is true for the reals. Then hyperreals basicall…

I think the cost is just a question of tradition what you find harder and the point author makes that it's easier to learn trough non-traditional way.

I see losing Archimedean property as more of a win than loss. It's a one simple thing that turns calculus operations into single algebraic evaluation, where you can avoid dynamic limits. Things become simpler and easier to prove.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#55

Post by mkprc »

imperio59 wrote: post #21
richard_chase wrote: post #16

I think Stewart's Calculus is excellent and it is rightfully the standard textbook. No modifications needed in my opinion.

Which book is that exactly? I'm finding lots of calculus book for "Stewart's Calculus" :(

Calculus by James Stewart, Daniel Clegg, and Saleem Watson

https://www.stewartcalculus.com

Re: Simplifying and Refactoring Introductory Calculus (2018)

#56

Post by ogogmad »

nextaccountic wrote: post #46

It's a bit frustrating that smooth infinitesimal analysis isn't mentioned even once, even as the author lists other systems that add infinitesimals to real numbers. It's a much better system to develop calculus on. It features infinitesimals as normal everyday mathematical objects, rather than the hack that hyperreal numbers are. Of course, I know that something like SIA would never be adopted. The main problem is th…

I think "Smooth Infinitesimal Analysis" is actually a bad idea because I think that the combination of Robinson's Nonstandard Analysis with little/big O notation gives you nilpotent infinitesimals in a more rigorous and elementary way.

In fact, in light of this, I'm starting to wish logicians would stop promoting SIA/SDG.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#57

Post by ogogmad »

srean wrote: post #49
ogogmad wrote: post #39

Does anyone know how to simplify the actually hard part of calculus: solving integrals? I refer to the million different substitutions and trig/hyperbolic formulas, along with the endless amount of other heuristics. I wonder if there's a way to bypass or simplify most of that.

https://rulebasedintegration.org/

+1 Interesting! I'm wondering if the rules can be generated automatically instead of listed by hand - it could then prove to be a helpful way to "explain" and generalise the discovery process that gave us modern integration methods.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#58

Post by 2b3a51 »

rramadass wrote: post #30

Some good books for introductory Calculus; 1) Calculus: Basic Concepts for High Schools by Lev Tarasov. Soviet-era book written as a dialogue between the author and reader. Absolutely fantastic (also see his other books on Probability etc.) - https://mirtitles.org/2018/09/04/calculus-basic-concepts-for... 2) Calculus: An Intuitive and Physical Approach by Morris Kline. A classic; any book by Morris Kline is a must-ha…

> Calculus: The Princess of Mathematics by H.C.Verma et al.

Garudalife have a store on Abebooks, so available in the UK at under £12 per volume (each circa 340 pages) but inevitably £18 postage. These look interesting and one day we'll have print on demand in each territory!

Re: Simplifying and Refactoring Introductory Calculus (2018)

#59

Post by yorwba »

nextaccountic wrote: post #46

It's a bit frustrating that smooth infinitesimal analysis isn't mentioned even once, even as the author lists other systems that add infinitesimals to real numbers. It's a much better system to develop calculus on. It features infinitesimals as normal everyday mathematical objects, rather than the hack that hyperreal numbers are. Of course, I know that something like SIA would never be adopted. The main problem is th…

The smooth infinitesimal analysis assumption that all functions are infinitely differentiable indeed removes a lot of edge cases, but only in the way that the C assumption that all memory accesses are within bounds removes a lot of edge cases. Just as a C programmer may have difficulty telling a valid C program from a meaningless sequence of symbols, so does SIA make it difficult to tell a valid assumption from a paradoxical statement that would allow defining a non-differentiable function.

If you have a correct calculation in SIA involving infinitesimals squaring to zero, it can easily be translated into the classical setting using functions that have their magnitude bounded above by a quadratic in a neighborhood of zero, but with the advantage that if you try it on a non-differentiable function, you'll merely fail to prove the quadratic upper bound instead of getting nonsensical results.

Such a "quadratic bound" approach (in brief f'(x) exists if there exist a constant C and a neighborhood of zero where for all h in the neighborhood |f(x + h) - f(x) - f'(x)h| ≤ Ch²) could actually be adopted over the typical limit of (f(x + h) - f(x))/h without abandoning classical logic, but considering the difficulty of proving the bound for many functions of interest, it might not make much of a difference over just assuming differentiability either way.

Re: Simplifying and Refactoring Introductory Calculus (2018)

#60

Post by johnnyb_61820 »

Thanks for the shout-out! I teach calculus to homeschool co-op students regularly, and I got tired of the existing books on the market (I used to teach from Saxon Calculus) so I wrote my own, "Calculus from the Ground Up". This paper represents the principles I used for writing the book.

Interestingly, my reformulation of the Leibniz notation for the second derivative was actually the result of writing the book. I was wanting to write a piece on why the second derivative looks the way it does. I had about 10 different calculus books I was looking through trying to find a solid answer and there was none. So, I eventually decided to try and derive the formula myself. I was quite surprised when I was able to derive a formula, but it was different. I tried to figure out for a while how to get from my formula to the standard one, until I eventually realized that the standard formulation was itself problematic.

It's in the "Calculus from the Ground Up" book as "Appendix B", but I don't use it in the main text so as not to confuse students who take further calculus courses. I found a middle ground for the book which neither forces the new notation nor commits the mistakes of the previous one. The book is not heavy in higher-order derivatives anyway, so the usage is minimal.

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