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I am going to be teaching calculus to first and second year students(in two separate courses) and I wanted a textbook for myself to go over the content so I can teach it better. I am a 4th year student and know advanced topics like scheme theory and connections on bundles, so really I just want to find a textbook that covers the content as rigorously as possible with as little handwaving as possible, that doesn't generalise to manifolds etc immediately. (I only mention advanced topics to emphasise that difficulty isn't a problem, but I do want it to focus on standard early concepts)

Which calculus textbooks teach first and second year content rigorously - I have heard bad things about Stewart, but I would like it to cover the same sort of level of content.

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    $\begingroup$ It was pointed out to me that the biggest mistake most graduate students make when teaching courses is being overly ambitious. Although the theory and proofs are exhilarating to the mathematician, you have to take into account how long it took you to get to the point where you are now. A rigorous text would be: "Multivariable Calculus" by Ted Shifrin (who spent years, working get the course right); however Rogawski + Lang were rigorous enough for me to teach from. Again, be warned, choosing more advanced books requires more preparation time for you! $\endgroup$ Dec 5 '16 at 6:04
  • $\begingroup$ As a side note: It would be nice if it occassionally appealed to higher mathematical content. That's something I love when reading category theory, that we see appeals to algebraic topology and algebraic geometry. (Of course this is probably not possible) $\endgroup$ Dec 5 '16 at 6:04
  • $\begingroup$ Thanks for the advice @FaraadArmwood, I am checking out all of these books listed. $\endgroup$ Dec 5 '16 at 6:28
  • $\begingroup$ Maybe Abott's Understanding Analysis? $\endgroup$
    – user170039
    Dec 5 '16 at 13:34
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Calculus by Spivak serves this purpose for single variable Calculus.

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In my opinion Apostol's Calculus (Volume 2 for Multivariable) when coupled with his Real Analysis Book is great. The former are computationally intense, and the latter offers a good foundation in the theory, I thought.

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  • $\begingroup$ Apostol's book is no doubt an awesome one... $\endgroup$ Dec 5 '16 at 6:45
  • $\begingroup$ @SchrodingersCat Indeed! It definitely stood out after Rudin $\endgroup$ Dec 5 '16 at 18:20
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"Calculus Deconstructed: A Second Course in First year Calculus" by Zbigniew Niteki makes a very good read. However, I personally recommend "Undergraduate Analysis" by Serge Lang. Hope it helps.

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