As a math professor who has taught calculus many times, I'd say there are many different things one could hope to learn from a calculus course. I don't think the subject distills well to a single point.
One unusual feature of calculus is that it's much easier to understand at a non-rigorous level than at a rigorous level. I wouldn't say this is true of all of math. For example, if you want to understand why the quadratic formula is true, an informal explanation and a rigorous proof would amount to approximately the same thing.
But, when teaching or learning calculus, if you're willing to say that "the derivative is the instantaneous rate of change of a function", treat dy/dx as the fraction which it looks like (the chain rule gets a lot easier to explain!), and so on, you can make a lot of progress.
In my opinion, the issue with most calculus books is that they don't commit to a rigorous or to a non-rigorous approach. They are usually organized around a rigorous approach to the subject, but then watered down a lot -- in anticipation that most of the audience won't care about the rigor.
I believe it's best to choose a lane and stick to it. Whether that's rigorous or non-rigorous depends on your tastes and interests as a learner. This book won't be for everybody, but I'd call that a strength rather than a weakness.
This is quite overstated. There are other approaches to infinitesimals such as synthetic differential geometry (SDG aka. smooth infinitesimal analysis) that are probably more intuitive in some ways and less so in others. SDG infinitesimals lose the ordering of hyperreals in non-standard analysis and force you to use some non-classical logic (intuitively, smooth infinitesimals are "neither equal nor non-equal to 0", wherein classical reasoning would conflate every infinitesimal with 0), but in return you gain nilpotency (d^n = 0 for any infinitesimal d) which is often regarded as a desirable feature in informal reasoning.
The second editions are still the current edition, so no worry that you might be missing out on something if you go with used copies. If you do want new copies (maybe you can't find used copies or they are in bad shape) take a look at international editions.
A new copy of the international edition for India from a seller in India on AbeBooks is around $15 per volume plus around $19 shipping to the US. Same contents as the US edition but paperback instead of hardback, smaller pages, and rougher paper. (International editions also often replace color with grayscale but that's not relevant in this case because Apostol does not use color)).
You can also find US sellers on AbeBooks that has imported an international edition. That will be around $34 but usually with free shipping.
They also have a hologram sticker alongside a printed warning that they are not for sale or export outside of India, Nepal and a couple of other countries.
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?
Mathematics: Its Content, Methods and Meaning by A.D. Aleksandrov, A.N. Kolomogorov, M.A. Lavrent’ev,.. https://www.goodreads.com/book/show/405880.Mathematics
It's still a masterpiece. Originally published in 1962 in 3 volumes. The English translation has all in one.
That wouldn't be who Kolmogorov complexity is named after? ("The Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program that produces the object as output.") ..Oh it is the same mathematician, Andrey Nikolaevich Kolmogorov. I see he was a major figure with many other concepts, equations and theorems named after him.
Published by Dover Publications and hence quite affordable. See ToC at https://store.doverpublications.com/products/9780486409160
I wonder if it's because more people are going to college who would have otherwise gone to a vocational or trade school? If the audience expands to include people who might not have studied calculus had they not chosen to go to college, I feel like textbooks have to change to accommodate that.
I'll quote Poincare:
Math is not about the study of numbers, but the relationships between them.
The abstraction makes it harder to understand how to apply these rules, but if one breaks through this barrier one is able to apply the rules far more broadly.
----
f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h}
If you actually study this, you may realize that there are an infinite number of equations that allow us to describe a secant line. So why this one? Is there something special? (hint: yes)
Let's call that the "forward derivative". Do you notice that through the secant line explanation that the "backward derivative" also works? That is
f'(x) = lim_{h->0} {f(x) - f(x - h)} / {h}
f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h}
Or tell me about the general case of
f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h}
----
This is just a tiny taste of what rigor holds. You are absolutely right to be frustrated and annoyed, but I hope you understand your conclusion is wrong. Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!). The frustration and pain is quite real! But it is absolutely worth it.
[0] Linking an EpsilonDelta video that covers this exact example in more detail https://www.youtube.com/watch?v=oIhdrMh3UJw
> had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM
No, you wouldn't. You would also learn things out of order. You would be exposed to things without understanding why you are learning them. People who argue this usually learn things the intuitive way (whether from rigorous material or not - what goes on in their mind isn't rigorous), and then they go back and reassess the rigor in the light of that. Then they pretend that they learned from the rigorous exposition. No, they didn't.
It is totally fine to iterate. Learn non-rigorously. Go back to it and iterate on rigor later. As it becomes necessary, and if it ever becomes necessary for your field.
> Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall
Particularly if you are learning from "rigorous" material. But then you go watch some YouTube videos to make up for the absence of didactics in your textbook.
I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes? They are maximally rigorous, therefore maximally great to learn from, right?
> I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes?
So I'm not sure your argument of "out of order" is accurate. The order is what we make. There's no clear optimal way to teach math. Your argument hinges on that. You might argue that the current status quo is working, so why disrupt it, and I'll point around asking if you really think it's so effective when many demonstrate a lack of understanding all around us. That so many struggle with calculus is evidence itself. We need not even acknowledge that there are many children who learn this (and let's certainly not admit that it's far more common for them to learn it in unconventional ways).
> let them sort it out without classes?
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.
Given the choice between a class room of first years who believe (in themselves and) an appearance of calculus knowledge or a room of scared undergrads that recoil from any calculus-inspired argument they 'have never learnt it properly', I'll take the former. I can work with that much more easily. Sure, some things might break - but what's the worst that can happen?
We'll sort out the rigour later while we patch the bruises of overextending some analogies.
> Most people don't need to know calculus
Most people don't need to know how to read. Most people don't need to know how to add. Most people don't need to know how to use a computer. The foolishness of these statements are all subjective and based on what one believes one "needs". Yet, I have no doubt all of these things can improve peoples lives.
I'd argue the same with calculus. While I don't compute derivatives and integrals every day[0], I certainly use calculus every day. That likely sounds weird, but it is only because one thinks that math and computation is the same. When I drive I use calculus as I'm thinking about my rates of change, not only my velocity. Understanding different easing functions[1] I am able to create a smoother ride, be safer, drive faster, and save fuel. All at the same time!
The magic of the rigor is often lost, but the magic is abstraction. That's what we've done here with the car example. I don't need to compute numbers to "do math", I only need to have an abstract formulation. To understand that multiple variables are involved and there are relationships between them, and understanding that there are concepts like a rate of change, the rate of change of the rate of change, and even the rate of change of the rate of change of the rate of change! (the jerk!)[2].
That's still math. It may not be as rigorous, but a rigorous foundation gives you a greater ability to be less rigorous at times and take advantage of the lessons.
So yes, most people "don't need calculus" but learning it can give them a lot of power in how to think. This is true for much of mathematics. You may argue that this is not how it is taught, but with that I'll agree. The inefficiency of how it is taught is orthogonal to the utility of its lessons.
[0] Is a physicist not doing math just when they do symbol manipulation? I can tell you with great confidence, and experience, that much of their job is doing math without the use of numbers. It is about deriving formulations. Relationship!
If you're willing to stretch the definition of what "using" maths is then it can apply to everything and that devalues the concept as a whole. I'm not on the toilet, I'm doing calculus!
The difference may be in two different cavemen. One throws his spear on intuition alone. The other is thinking about the speed he throws, how the animal moves, the wind, and so on. There is a formulation, though not as robust as you'd see in a physics book.
> the definition of what "using" maths is then it can apply to everything
Math is a language, or more accurately a class of languages. If you're formulating your toilet activities, then it might be math. But as you might gather, there's nuance here.
I quoted Poincaré in another comment but I'll repeat here as I think it may help reduce confusion (though may add more)
Math is not the study of numbers, but the relationships between them.
Yes, technique is one thing, but being really good at throwing spears doesn't make you really good at math, is my argument. And most people will encounter maths in a formal setting while lacking the broader perspective that everything is technically "math".
Yet, we need to see the argument from the common person's view, if we're talking about calculus and learning in the traditional sense. The view you stated is quite esoteric and doesn't generalize well in this setting imo.
It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
> but is the thinking caveman really doing the same?
> It's like a musician saying they see music in every action, but to most non-musicians (even if the stated thing is kind of true) that doesn't make a lot of sense etc.
The musician, like the mathematician, understands that every sound is musical. If you want to see this in action it's quite enlightening[0]. I'm glad you brought up that comparison because I think it can help you understand what I really mean. There is depth here. Every human has access to the sounds but the training is needed to put them together and make these formulations. Benn here isn't exactly being formal writing his music using a keyboard and formalizing it down to musical notes on a sheet (though this is something I know he is capable of).
But maybe I should have quoted Picasso instead of Poincaré
Learn the rules like a pro, so you can break them like an artist.
I know this is confusing and I wish I could explain it better. But at least we can see that regardless of the field of expertise we find similar trains of thought. Maybe a bridge can be created by leveraging your own domain of expertise
Maybe I can put it this way: gibberish is more intelligible when crafted by someone who can already speak.
People should have at the minimum a conceptual idea of Calculus. A good motivation is Everyday Calculus: Discovering the Hidden Math All around Us by Oscar E. Fernandez - https://press.princeton.edu/books/paperback/9780691175751/ev...
> And if you do learn it, do so with rigor so you actually learn it
This is not strictly necessary for everybody. The conceptual ideas are what are important; else you are merely doing "plug-and-chug" Maths without any understanding. You need to focus on rigor only based on your needs and at your own pace. Concepts come first Formalism comes second.
A good example; In the Principia Newton actually uses the phrase Quantity of Motion for what we define today as Momentum. The phrase is evocative and beautifully captures the main concept instead of the bland p = m x v definition which though correct and needed for calculations conveys no mental imagery.
In Mathematics one should always approach a concept/idea from multiple perspectives including (but not limited to) Imagination, Conceptual, Graphical, Symbolic, Relationship, Applications, Definition/Theorem/Proof.
There's not much need for a large amount of PhD places, and funding, for pure mathematics research.
Likewise, on the applied side, "calculus" now as a pure thing has been dead alone time. Gradients are computed with algorithms and numerical approximations, that are better taught -- with the formal stuff maintained via intuition.
I'm much more open to the idea that the west has this wrong, and we should be more focused on developing the applied side after spending the last century overly focused on the pure
My pet peeve about calculus books is that they almost always overlook the importance of continuity. In some extreme cases, they even start with infinitely small sequences, with some rather gnarly theorems like Bolzano–Weierstrass theorem about converging subsequences.
I think this is a mistake. It's much easier to start with continuous functions and build from there. Modern readers then can visualize the epsilon-delta formulation of limits as "zooming in" on the function. The "epsilon" is the height of the screen, and the "delta" is the "zoom level" at which the function fragment fits on the screen.
And once you "get" the idea of continuity and function limits, the other limit theorems just fall out naturally.
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!
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.
Sadly, no. It just seems to start with a gentle version of real analysis, leading into basic Calculus.
- On Knuth’s idea (which seems good to me), see http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-ca... but also the last two comments by David Speyer.
- And see also http://cornellmath.wordpress.com/2007/08/28/non-nonstandard-... and http://texnicalstuff.blogspot.in/2011/05/big-o-notation-for-... that seem to have tried teaching along somewhat similar (but different) lines.
- See https://terrytao.wordpress.com/2022/05/10/partially-specifie... for a further formalization of O notation.
(These links via comments I left to myself at https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...)
Perhaps the bigger question is whether it's at the right level of difficulty for the audience.
Math majors might have their own. I also know they end up taking complex Calculus.
Complex analysis and real analysis were among the higher-level courses, attended mostly by math majors, with the proviso that there were a lot of double majors. That was where it got interesting.
The requirements for the physics major were only a handful of math credits shy of the math major.
lol, that's how I ended up with a math major. Got lost in the physics (realized I had no intuition for what was actually happening, just manipulating equations) took a couple extra courses, and boom! Math!
While not every student is expected to read the book sequen- tially cover to cover, it is important to have the details in one place. Calculus is not a subject that can be learned in one pass. Indeed, this book nearly assumes readers have already had a year of calculus, as had the students of MAT 157Y. I hope this book will grow with its readers, remaining both readable and informative over multiple traversals, and that it provides a useful bridge between current calculus texts and more advanced real analysis texts.
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.
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.
"Bhāskara II (c. 1114–1185) was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function.[18] In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. [...] In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics stated components of calculus. They studied series equivalent to the Maclaurin expansions of [redacted] more than two hundred years before their introduction in Europe. [...] however, were not able to 'combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today.'"
"Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus (c. 390–337 BC) developed the method of exhaustion to prove the formulas for cone and pyramid volumes.
"During the Hellenistic period, this method was further developed by Archimedes (c. 287 – c. 212 BC), who combined it with a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems now treated by integral calculus. In 'The Method of Mechanical Theorems' he describes, for example, calculating the center of gravity of a solid hemisphere, the center of gravity of a frustum of a circular paraboloid, and the area of a region bounded by a parabola and one of its secant lines."