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" :(
Simplifying and Refactoring Introductory Calculus (2018)
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Re: Simplifying and Refactoring Introductory Calculus (2018)
#22Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx." What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?
> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
-- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus.
Math uses limits because "dx" as a concept is hard to define and relies on faith that such an object can exist. It behaves as zero when convenient yet is non-zero when that breaks math. Limits have a more rigorous footing.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#23,, additionally, moving limits to the end of a first-year course allows students to develop intuitions around the derivative first before seeing the formal proof of their validity’’ Waiting a year to get from intuition to theorems is a perfect way to ruin math. Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking. At the same time there could be more examples…
> Waiting a year to get from intuition to theorems is a perfect way to ruin math.
It's interesting that you chose to make that point in a thread about calculus specifically. It had pretty shaky foundations for most of its history, and even today, there's a pretty significant disconnect between the mechanics of epsilon-delta and the meaning we assign to the result.
> Math is not supposed to be easy/simple, but it supposed to be a great way to understand systems based thinking.
Math is a means to an end. Making the tool easy to use is a desirable property. I've heard "it's not supposed to be easy" applied to many disciplines, from film photography to software engineering, and I think it's mostly gatekeeping.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#24I 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" :(
James Stewart is the author, there are 8 or 9 edition of the same book. Any will do.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#25I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.
If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.
If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.
[0] https://www.bravernewmath.com/ [1] https://intellectualmathematics.com/calculus/ [2] https://people.math.wisc.edu/~hkeisler/keislercalc-06-03-26.... [3] https://people.math.harvard.edu/~knill/teaching/mathe320_201...
Re: Simplifying and Refactoring Introductory Calculus (2018)
#26I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Manfredo P. do Carmo: Differential Forms and Applications. Short and incredibly beautiful.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#27Got to the place where he says "As you can see, this is identical to the d/dx() operation except that the result is not divided by dx." What does it mean with his d() operator to "divide by dx"? All of a sudden it seems like he has changed dy/dx from unfortunate notation that sort of looks like a division into something that actually is dividing two meaningful things, dy and dx? And so what the hell are dy and dx?
> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? -- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus. Math uses limits because "dx" as a concept is hard to defin…
Infinitesimals don't rely on faith any more than any other mathematical idea. The "normal" calculus sequence being based on limits is solely due to the fact that limits were the first discovered method for rigorously formalizing the subject. Infinitesimals do a better job representing how most people intuitively think about calculus though, as evidenced by being the vehicle through which the entire field was discovered in the first place, and so now that they too rest on a formally rigorous foundation we should probably consider rebuilding calculus education around them.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#28I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyp…
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
Re: Simplifying and Refactoring Introductory Calculus (2018)
#29[dead]
Re: Simplifying and Refactoring Introductory Calculus (2018)
#30Some 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-have - https://store.doverpublications.com/products/9780486404530
3) Calculus: The Princess of Mathematics by H.C.Verma et al. A two-vol must-have affordable set. The author is a well-known Indian Physics professor and this is written specifically for students to "understand" calculus i.e. it is not a typical textbook. - https://garudalife.in/calculus-the-princess-of-mathematics-v...
4) How to Think about Analysis by Lara Alcock. Provides conceptual insight like the Tarasov book above. Checkout the author's other books too. - https://global.oup.com/academic/product/how-to-think-about-a...
I believe we need to study Calculus alongside Probability/Statistics nowadays due to their pervasive use in ML/AI/etc. To that end;
a) Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming. It is by Hamming so one of the best. - https://store.doverpublications.com/products/9780486439457?_...
b) Calculus and Statistics by Michael Gemignani. Similar to the above - https://store.doverpublications.com/products/9780486449937