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the TLDR is I am looking for a book to relearn Calculus. I finished Calc 3 past semester, but despite high nineties in all my Calculus classes, I don't feel like I learned much motivation for what I am doing and with my Calc 1 and Calc 2 far in my mind (took a leap year), I often find myself having to google how to do improper integrals or to wonder the reason why something works. I can do problems all day with relative ease, but the why of it all is mostly a mystery.

I did some research and Spivak, Apostol and Courant were the textbooks that seemed to fit the bill. Still, I am not settled yet on which one I should go with or if I should go with them at all. I also don't plan (currently) to read all of them, so if I go with Spivak, I would not bother reading Apostol volume 1.

Looking forward to your recommendations, and thanks in advance.

J. W. Tanner
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Solar
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  • If you want to know the why behind calculus, I recommend "Analysis I, II" by Terence Tao (the fourth edition has recently been released), where he literally starts from $0$. – lorenzo Feb 26 '23 at 16:31
  • Let me suggest the-somewhat misleading by the title-"Calculus, Basic Concepts for High Schools" by Tarasov. As you read, very soon, "basic" translates to basic for 1-st year Uni students. Then Apostol, Spivak are great choices. – MathematicianByMistake Feb 26 '23 at 16:47
  • What are your plans for further undergraduate work? Do you think you'll attend graduate school (what subject)? What are your career plans? If you plan to stay in mathematics (especially pure mathematics), then probably Spivak's book would be better than Apostol or Courant, at least for Calc 1 & 2 (Spivak does not deal with multivariable calculus), for one reason because it's a lot shorter than trying to get through both volumes of Apostol (see also this answer). (continued) – Dave L. Renfro Feb 26 '23 at 16:48
  • Hey @lorenzo, thanks a lot for the recommendation. I looked it up and it seems it doesn't have any solutions for the problems in the book. If it doesn't bother you, how did you confirm your answers? – Solar Feb 26 '23 at 16:48
  • However, if you're planning to study engineering or physics or some other math-intensive science field, then I recommend carefully and thoroughly reading Morris Kline's book. And in either case, if you have a lot of time and very strong motivation/drive, the 2 volume calculus series by Apostol is hard to beat. Several decades ago it was fairly standard (in the U.S. at least) to take a 2-semester sequence of "Advanced Calculus" (continued) – Dave L. Renfro Feb 26 '23 at 16:48
  • (and many books with this title were published in the 1950s-1980s for this purpose; even much earlier than this, but before the early 1950s you'll probably find that the approach and topics are too different from the present perspective to be optimally useful for your purposes) in the 3rd undergraduate year, and this course tended to refresh most calculus topics while also preparing one for real analysis courses, so you might consider one of the standard advanced calculus texts as a possibility (e.g. Taylor/Mann, Buck, Kaplan, Widder, etc.). – Dave L. Renfro Feb 26 '23 at 16:49
  • @DaveL.Renfro First thanks a lot for the highly detailed answer! I am currently math & comp sci double major, so I am lining up to take Real Analysis in not too long, and will have to get used to mathematical rigor sooner or later. As for master I am not sure yet, I am definitely interested in AI and and are leaning towards that, but a master in Math also sounds like an interesting option (and still allows for a career in AI). – Solar Feb 26 '23 at 16:56
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    Hey @MathematicianByMistake, thanks a lot for the recommendation, will give it a look! – Solar Feb 26 '23 at 16:58
  • @Solar By posting the solutions I wasn't sure of on MSE or by asking the author directly on the pages dedicated to the books on his blog. – lorenzo Feb 26 '23 at 17:01
  • I am currently math & comp sci double major --- Then probably Spivak, because you don't need the engineering or physics or history extra that Apostol, Courant, Kline provide, and I doubt trying to tackle a 600+ page advanced calculus book is a reasonable goal. Spivak will give you plenty of practice with computational calculus stuff (finding derivatives, integrals, etc.) as well as interesting side-detours (mostly in the problems) that will enrich your knowledge-base, and Spivak also provides a nice on-ramp to real analysis. – Dave L. Renfro Feb 26 '23 at 17:13
  • And finally, Spivak is so well known here that you can ask a question about any problem in Spivak and many in MSE will have had first-hand experience in working the problem! – Dave L. Renfro Feb 26 '23 at 17:15
  • @lorenzo the idea of interacting with the author/asking for the solutions is one I had never even considered and it's cool ash, thanks a lot for the insight and the recommendations. – Solar Feb 26 '23 at 18:01
  • @DaveL.Renfro I see, it seems Spivak is the way to go then. Thanks a lot for the recommendations and the in-depth breakdowns, it's really appreciated! :D – Solar Feb 26 '23 at 18:04

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