[HN Gopher] Calculus for Mathematicians, Computer Scientists, an...
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Calculus for Mathematicians, Computer Scientists, and Physicists
[pdf]
Author : o4c
Score : 184 points
Date : 2025-11-23 16:31 UTC (6 hours ago)
(HTM) web link (mathcs.holycross.edu)
(TXT) w3m dump (mathcs.holycross.edu)
| qntty wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Thanks for the info on cheaper editions, not important to
| me but to others in USA it might be a big help.
| throwaway81523 wrote:
| Spivak's book is still good too.
| CamperBob2 wrote:
| That's a pretty diverse audience. Is this .pdf supposed to be a
| one-size-fits-all effort?
| analog31 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Usually engineering/math calc and then a much less rigorous
| business/arts&crafts calc for the rest.
| garyfirestorm wrote:
| Is there a hard copy to purchase? I can't seem to find it
| anywhere.
| kalx wrote:
| How much math skills do you need to appreciate this book?
| analog31 wrote:
| My first impression, paging through it, is that it's at a
| somewhat higher level than the typical college calculus course.
| nightshift1 wrote:
| 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._
| zkmon wrote:
| >> 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 wrote:
| 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
| kardianos wrote:
| 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 wrote:
| 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.
| elcapitan wrote:
| 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 wrote:
| 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 wrote:
| 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.
| layer8 wrote:
| Not necessarily proof-heavy, but at least with formally
| rigorous definitions and theorems.
| zorked wrote:
| Rigorous = a pain in the ass to learn from, but you gain
| imaginary points for the pain.
| nabla9 wrote:
| > 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.
| wirrbel wrote:
| is that like a Landau Lifshitz for Math?
| nextos wrote:
| No, it's just a gentle overview.
| actinium226 wrote:
| > 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.
| zozbot234 wrote:
| 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.
| bmitc wrote:
| While world class, Russian mathematics is not known for rigor,
| formality, or detail, so I'm dubious.
| mathattack wrote:
| 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 wrote:
| 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)
| anthk wrote:
| 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.
| belter wrote:
| 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.
| gbacon wrote:
| > Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen,
| so ubersetzen sie es in ihre Sprache, und alsbald ist es etwas
| ganz anderes. (Goethe)
| JosephK wrote:
| >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 wrote:
| 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.
| DroneBetter wrote:
| 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)
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