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...
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...
I have strong opinions on how Calc should be introduced - visually.
I think we don't cover basics like the distributive rule in school very well, and that it should be a much more nuts n bolts visual / measuring / counting experience.
Ive attempted to outline how I think this stuff should be taught, by making a video tour of the concepts - from Counting, to Distributive rule / algebra, to Quadratics then the Derivative, here :
All of these things are covered in some great books :
W W Sawyer Vision in Elementary Mathematics
Algebra by Gelfand
Calculus by Thomas
We have superb resources now like 3Blue1Brown, KhanAcademy and ArtOfProblemsolving.com / BeastAcademy .. so you _can_ get your kids a superb math education, even as many schools seemingly give up on teaching Algebra and Calculus.
Got 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?
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 he would consider a nonzero difference so something like y + dy = (x + dx)^2 = x^2 + 2x dx + (dx)^2. At this point he would use his starting equation to subtract y from the LHS and x^2 from the RHS, leaving: dy = 2x dx + (dx)^2. Then he would divide by dx leaving dy/dx = 2x + dx and since dx is infintisimal, he would just lop it off. Suffice it to say, just ignoring the nonzero dx on the RHS, or casting it to 0 while conveniently ignoring the division by 0 on the LHS, was rather disturbing to many critics. A lot of work had to be done by Riemann, Cauchy, and Weierstrass over the following century after Newton and Leibniz invented calculus to answer the question you are asking.
I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.
> 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.
Infinitesimals don't rely on faith anymore 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.
They are differentials. https://en.wikipedia.org/wiki/Differential_(mathematics) has some info but is not great as a beginning introduction. dx is an infinitesimal bit of x, and dy is an infinitesimal bit of y. dx here is the same dx as in an integral, which you can think of as the width of one of the infinite infinitesimally thin rectangles whose areas are being added up to find the area under the curve: https://en.wikipedia.org/wiki/Riemann_integral
,, 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 taught on why these building blocks were historically needed and what they are used for solving nowadays.
> 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.
Looks like this came out nearly 8 years ago, so… how’d it work out? Given the way job titles work these days I guess we could have some Senior Engineers here who learned calculus from this paper…
I’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.
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.
My only experience is as a physics TA and teaching X-ray techs, so take this with a grain of salt. I think the author is trying to address a real problem, but he's not working on the right parts.
First, limits are harder than derivatives. Historically, humans figured out the derivative in the late 1600s, but the modern rigorous definition of the limit didn't exist until the 1800s. Slow-walking the definition of the derivative doesn't fix the problem of understanding limits.
The limit of a function f at a point x is defined as the value y, if it exists, such that for all d > 0 there exists an e > 0 such that for all x' in [x - e, x + e] we have |y - f(x')| < d. That's an earful. But for essentially all limits in introductory calculus we evaluate using two rules: the limit of a continuous function f at a point x is f(x), and the squeeze theorem. So my suggestion is to elevate these to the status of axioms. Introducing another number system does not help when students will not do anything nontrivial with it anyway.
The second problem is that "introductory" calculus includes too much material and is consequently pushed too late in the curriculum and seen as a weed-out course. Students spend too much time on "preparation" that doesn't prepare them for calculus. Studying logarithms and trigonometry is orthogonal, so basically all of "precalculus" is not actually pre-calculus. To me a four-year high school curriculum could be written up just fine with two years of algebra and geometry (not separated), one year of calculus and then statistics, which provides an ideal application for the theory of derivatives when you learn regression. But the author has included multivariable calculus and fiddly techniques for taking derivatives of ugly functions into "introductory" calculus. I think this is a step in the wrong direction. Laborious algebra calculations can be moved into an optional methods course for engineering students; we should be ensuring the core ideas are as accessible as possible so that doctors don't write papers about the trapezoid rule anymore:
Eh. I think that the standard calculus approach is mostly fine, but just needs tweaking.
The only major change that I'd like to make is the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity.
It immediately leads to the notion of the derivative. And after that, it's just a lot of building blocks.
> Again, by using differentials instead of derivatives, we have transformed a number of processes that students find unintuitive
into a single process where the intuition is supplied by the student’s knowledge of algebra.
Ah yes. Algebra. The subject all students love dearly. If only we could get students to love and appreciate calculus as much they love algebra!
I have a hard time even imagining an article that is more disconnected from the reality of teaching calculus to tiny humans.
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-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
I have strong opinions on how Calc should be introduced - visually.
I think we don't cover basics like the distributive rule in school very well, and that it should be a much more nuts n bolts visual / measuring / counting experience.
Ive attempted to outline how I think this stuff should be taught, by making a video tour of the concepts - from Counting, to Distributive rule / algebra, to Quadratics then the Derivative, here :
https://www.youtube.com/playlist?list=PLEInJ-Z4qBKYxbK1Mm13g...
All of these things are covered in some great books :
We have superb resources now like 3Blue1Brown, KhanAcademy and ArtOfProblemsolving.com / BeastAcademy .. so you _can_ get your kids a superb math education, even as many schools seemingly give up on teaching Algebra and Calculus.Honestly it's hard to beat how good 3Blue1Brown is at this... https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQ...
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" :(
James Stewart is the author, there are 8 or 9 edition of the same book. Any will do.
Got 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?
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 he would consider a nonzero difference so something like y + dy = (x + dx)^2 = x^2 + 2x dx + (dx)^2. At this point he would use his starting equation to subtract y from the LHS and x^2 from the RHS, leaving: dy = 2x dx + (dx)^2. Then he would divide by dx leaving dy/dx = 2x + dx and since dx is infintisimal, he would just lop it off. Suffice it to say, just ignoring the nonzero dx on the RHS, or casting it to 0 while conveniently ignoring the division by 0 on the LHS, was rather disturbing to many critics. A lot of work had to be done by Riemann, Cauchy, and Weierstrass over the following century after Newton and Leibniz invented calculus to answer the question you are asking.
I think the best way to understand this is that division by dx is always allowed. It is genuinely a nonzero quantity. Later, we think more in a more abstract way in a tiny neighborhood around (x,y), considering what happens to dy as dx becomes arbitrarily small, but it never vanishes entirely. That explains why we can say dy = 2x dx or dy/dx = 2x and both are completely true and reasonable. I think the author's argument is that d() is a bit easier to understand because we aren't dividing by dx but it makes no sense (to me) that way. If you cannot divide by dx, a nonzero number, then why not? And if you can, why doesn't dy/dx involve zero division, which is clearly not well-defined? I think answering those questions makes calculus a lot easier to understand and that they are in a sense the hardest questions. The notation this author uses doesn't really illuminate those points and the fact that the author realizes that he is basically teaching the students to accept zero division for most of the year suggests he is basically saying we should go back to a Leibniz-era approach to calculus. I would rather make rigorous what is meant by dx/dy and what exactly dx and dy are.
This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number
> 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.
Infinitesimals don't rely on faith anymore 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.
They are differentials. https://en.wikipedia.org/wiki/Differential_(mathematics) has some info but is not great as a beginning introduction. dx is an infinitesimal bit of x, and dy is an infinitesimal bit of y. dx here is the same dx as in an integral, which you can think of as the width of one of the infinite infinitesimally thin rectangles whose areas are being added up to find the area under the curve: https://en.wikipedia.org/wiki/Riemann_integral
,, 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 taught on why these building blocks were historically needed and what they are used for solving nowadays.
> 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.
Looks like this came out nearly 8 years ago, so… how’d it work out? Given the way job titles work these days I guess we could have some Senior Engineers here who learned calculus from this paper…
At the very least, Founding Engineers!
(Pointing out the unrelated absurdity of this title being given out to people often not actually present at a company’s founding)
This guy has an interesting publication history, programming books and what looks like evolutionary biology from a creationist perspective.
I’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...
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.
Manfredo P. do Carmo: Differential Forms and Applications. Short and incredibly beautiful.
Try https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...
Generally most of these 'handwavy' notations are rigidly provable, but only under general assumptions, that might not be true in special cases.
where I heard of this from : https://youtu.be/4ZB2PNUYR1Y
My only experience is as a physics TA and teaching X-ray techs, so take this with a grain of salt. I think the author is trying to address a real problem, but he's not working on the right parts.
First, limits are harder than derivatives. Historically, humans figured out the derivative in the late 1600s, but the modern rigorous definition of the limit didn't exist until the 1800s. Slow-walking the definition of the derivative doesn't fix the problem of understanding limits.
The limit of a function f at a point x is defined as the value y, if it exists, such that for all d > 0 there exists an e > 0 such that for all x' in [x - e, x + e] we have |y - f(x')| < d. That's an earful. But for essentially all limits in introductory calculus we evaluate using two rules: the limit of a continuous function f at a point x is f(x), and the squeeze theorem. So my suggestion is to elevate these to the status of axioms. Introducing another number system does not help when students will not do anything nontrivial with it anyway.
The second problem is that "introductory" calculus includes too much material and is consequently pushed too late in the curriculum and seen as a weed-out course. Students spend too much time on "preparation" that doesn't prepare them for calculus. Studying logarithms and trigonometry is orthogonal, so basically all of "precalculus" is not actually pre-calculus. To me a four-year high school curriculum could be written up just fine with two years of algebra and geometry (not separated), one year of calculus and then statistics, which provides an ideal application for the theory of derivatives when you learn regression. But the author has included multivariable calculus and fiddly techniques for taking derivatives of ugly functions into "introductory" calculus. I think this is a step in the wrong direction. Laborious algebra calculations can be moved into an optional methods course for engineering students; we should be ensuring the core ideas are as accessible as possible so that doctors don't write papers about the trapezoid rule anymore:
https://diabetesjournals.org/care/article/17/2/152/17985/A-M...
Eh. I think that the standard calculus approach is mostly fine, but just needs tweaking.
The only major change that I'd like to make is the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity.
It immediately leads to the notion of the derivative. And after that, it's just a lot of building blocks.
> Again, by using differentials instead of derivatives, we have transformed a number of processes that students find unintuitive into a single process where the intuition is supplied by the student’s knowledge of algebra.
Ah yes. Algebra. The subject all students love dearly. If only we could get students to love and appreciate calculus as much they love algebra!
I have a hard time even imagining an article that is more disconnected from the reality of teaching calculus to tiny humans.