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Calculus for Mathematicians, Computer Scientists, and Physicists [pdf]
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- qntty 11mo agoWriting 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.
- JJMcJ 11mo agoAnyway you've already got Apostol - if it's just calculus as such get an older edition. Modern ones have extra goodies like linear algebra but have modern text book pricing (cries softly in $150/volume).
- tzs 11mo agoGetting an old enough edition of Apostol's "Calculus" to not include linear algebra might be a bit challenging. Linear algebra was added to both volumes in their second editions, which came out in 1967 for volume 1 and 1969 for volume 2. 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.
- JJMcJ 11mo agoThanks for the info on cheaper editions, not important to me but to others in USA it might be a big help.
- marai2 11mo agoi bought a compilers book that was an Indian edition. The paper and print quality was so bad (like smudgy) that I could not read it and I didn’t think I was particularly picky about this. Not sure if I just got unlucky or if this is generally true?
- SanjayMehta 11mo agoIndian editions sometimes have different question sets to prevent students from using them in other countries' coursework. 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.
- JJMcJ 11mo agoI think those restrictions apply only to retail sellers in those countries, not to purchasers or used stores.
- JJMcJ 11mo agoI know where my confusion comes from. I studied from a first edition of volume one before there was a volume two, so it wasn't marked as Vol I. Friend dug it up from his old books, since I seemed to be quick learner.
- throwaway81523 11mo agoSpivak's book is still good too.
- impendia 11mo ago> The point of calculus is... 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.
- msla 11mo agoThe rigorous form of the non-rigorous version is non-standard analysis: There really are tiny little numbers we can manipulate algebraically and we don't need the epsilon-delta machinery to do "real math". It's so commonsensical that both Newton and Leibniz invented it in that form before rigor became the fashion, and the textbook "Calculus Made Easy" was doing it that way in 1910, a half-century before Robinson came along and showed us it was rigorous all along. https://calculusmadeeasy.org/ https://calculusmadeeasy.org/ https://en.wikipedia.org/wiki/Calculus_Made_Easy https://en.wikipedia.org/wiki/Calculus_Made_Easy
- zozbot234 11mo ago> The rigorous form of the non-rigorous version is non-standard analysis 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.
- CamperBob2 11mo agoThat's a pretty diverse audience. Is this .pdf supposed to be a one-size-fits-all effort?
- analog31 11mo agoI'm probably dating myself, but at my college, there was one calculus course for everybody. But also, a lot of the students in those areas had overlapping or double majors. For instance I majored in math and physics. Perhaps the bigger question is whether it's at the right level of difficulty for the audience.
- anikom15 11mo agoI think there are usually two: Calculus for scientists and engineers which is analytical and has lots of symbols, and Calculus for everyone else which is more practical. Math majors might have their own. I also know they end up taking complex Calculus.
- analog31 11mo agoThinking about it, ours was a small college -- 2500 students. So there may have been a practical reason for everybody taking the same math courses. They were taught more as "service" courses for the sciences and engineering than as theoretical math courses. And the students who didn't need calculus typically satisfied their math requirement with a statistics course. 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.
- dogmatism 11mo ago>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!
- garyfirestorm 11mo agoIs there a hard copy to purchase? I can’t seem to find it anywhere.
- kalx 11mo agoHow much math skills do you need to appreciate this book?
- analog31 11mo agoMy first impression, paging through it, is that it's at a somewhat higher level than the typical college calculus course.
- nightshift1 11mo agothe Postscript at the end says: 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.
- zkmon 11mo ago>> 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?
- mjburgess 11mo agoNope, but mathematics research is one of the most rarefied fields being extremely difficlt to get into, hard to get money, etc. -- (this is my understanding, at least). Progress is made here by people who, aged 10 are already showing signs of capability. 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
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- kardianos 11mo agoI agree with this. But I don't see the students rejecting this, but the education degreed peoples who choose texts and the publishers want to make all learning for all people. This is foolish. Most people don't need to know calculus. And if you do learn it, do so with rigor so you actually learn it and not just the appearance of it, which is much much worse.
- DrSAR 11mo agoNot sure I agree with 'appearance [...] is much worse'. 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.
- mathattack 11mo agoSeems like a lot of different audiences. My observation is this is trying to cover 2 of the 3 common tracks: 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!
- lanstin 11mo agoI don't think they are unifiable, the aims and methods one needs to learn are just too different. Limits of covering boxes and scaling your epsilons and so on, stuff from Tao's class on analysis is far away from being able to deal either non-trivial differential equations or stability analysis. You can prove all sorts of things about dense subspaces of Hilbert space and still get totally lost in multiple scale analysis, and vice versa. (Ed: epsilon was spelled espikon)
- anthk 11mo agoGet Zenlisp running too https://www.t3x.org/zsp/index.html https://www.t3x.org/zsp/index.html and just have a look on how the (intersection) function it's defined. Now you'll get things in a much easier way, for both programming and math.
- belter 11mo agoThis 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.
- infinet 11mo agoAgreed. 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.
- infinet 11mo agoI 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.
- ocherek 10mo agoCould you please provide some examples of the good books which you mentioned?
- gbacon 11mo ago> Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen, so übersetzen sie es in ihre Sprache, und alsbald ist es etwas ganz anderes. (Goethe)
- JosephK 11mo ago>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.
- zozbot234 11mo agoCalculus was actually pioneered by the Kerala School of mathematicians in India during the European Middle Ages, several centuries prior to Newton and Leibniz popularizing it in Europe. The Indian texts were also quite well known to Europeans by that time, it was nowhere close to an independent discovery.
- danielam 11mo agoFrom [0] (emphasis mine): "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.'" [0] https://en.wikipedia.org/wiki/Calculus https://en.wikipedia.org/wiki/Calculus
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- DroneBetter 11mo agoArchimedes had functionally developed a method of integration (which was how he obtained results like volume/surface area of a sphere, or centre of mass of a hemisphere) in a manuscript that got lost to time and then rediscovered in a palimpsest (pasted and written over with a religious text)
- btilly 11mo agoWhen I saw it was for computer scientists, I briefly hoped that it would take the Big-O, little-o approach as Knuth recommended in 1998. See https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... for a repost of Knuth's letter on the topic. Sadly, no. It just seems to start with a gentle version of real analysis, leading into basic Calculus.
- svat 11mo agoRelated to your comment, not the original post/book: - On Knuth’s idea (which seems good to me), see http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-calculus-implicit-definitions-difficulty-of/ 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-calculus-i/ http://cornellmath.wordpress.com/2007/08/28/non-nonstandard-... and http://texnicalstuff.blogspot.in/2011/05/big-o-notation-for-calculus.html 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-specified-mathematical-objects-ambient-parameters-and-asymptotic-notation/ 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-couple-of-sources/ https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...)
- rramadass 11mo agoThe book does introduce/use O-notation under the section "Order of Vanishing" in the "Limits and Continuity" chapter.
- fnord77 11mo agoWhat about the Stanford Math 51 book?
- fud101 11mo agoBook looks like it could be AI generated, nothing remarkable.
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- hilbert42 11mo agoSo 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.
- SilentM68 11mo agoWithout 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.
- Surac 11mo agoThanks for the pdf. I am more in numeric but this pdf is a nice reference for things complete unreadably on Wikipedia.
- hintymad 11mo agoHonest 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.