Calculus for Mathematicians, Computer Scientists, and Physicists [pdf]
42 comments
·November 23, 2025qntty
Writing 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
Anyway 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
Getting 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
Thanks for the info on cheaper editions, not important to me but to others in USA it might be a big help.
throwaway81523
Spivak's book is still good too.
zkmon
>> 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?
nabla9
> Russian books doing this.
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.
actinium226
> The books in the West in general kept getting less rigorous, with time.
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.
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elcapitan
This may be a stupid question, but what do people usually mean when they refer to a mathematical text as being "rigorous"? Does it mean that everything is strictly proof-based rather than application-oriented?
actinium226
Generally that's what it means. And also when proofs are presented, a rigorous book will go through it fully, whereas a less rigorous one might just sketch out the main ideas of the proof and leave out some of the nitty gritty details (i.e. it's less rigorous to talk about "continuity" as "you can draw it without lifting the pen" as compared to the epsilon-delta definition, but epsilon-delta is pretty detailed and for intro calculus for non-mathematicians you don't really need it).
mike50
This is the reason that everyone at my university said to just take the Applied version of Calculus 1 and 2 t avoid the proofs.
zorked
Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.
layer8
Not necessarily proof-heavy, but at least with formally rigorous definitions and theorems.
zozbot234
If you care about getting all the nitty gritty details of a "rigorous" proof, maybe the quicker approach is to install Lean on your computer and step through a machine-checked proof from Mathlib. What you get from even the most heavyweight math books is still quite far from showing you all the steps involved.
kardianos
I 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
Not 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.
mjburgess
Nope, 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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bmitc
While world class, Russian mathematics is not known for rigor, formality, or detail, so I'm dubious.
mathattack
Seems 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
I 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)
gbacon
> 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)
CamperBob2
That's a pretty diverse audience. Is this .pdf supposed to be a one-size-fits-all effort?
analog31
I'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
I 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
Thinking 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.
beezle
Usually engineering/math calc and then a much less rigorous business/arts&crafts calc for the rest.
garyfirestorm
Is there a hard copy to purchase? I can’t seem to find it anywhere.
kalx
How much math skills do you need to appreciate this book?
nightshift1
the 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.
analog31
My first impression, paging through it, is that it's at a somewhat higher level than the typical college calculus course.
anthk
Get Zenlisp running too 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.
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belter
This 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.
JosephK
>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.
DroneBetter
Archimedes 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)
zozbot234
Calculus 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.
null
When 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-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.