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I would like to study university level mathematics on my own. I found rigorous books on single variable calculus and topology and learned those. But what would one recommend for several variable calculus? I found a lecture notes but there were many details omitted and for example Taylor polynomial was given only a special case and there was no material included to address how to estimate the error in Taylor's polynomial.

Previously I have read this book (the link is to a .pdf file) and some lecture notes by Olli Martio. I was unable to find those note from the Internet. Also I read the book Topologia by Jussi Väisälä. Those are Finnish books but English material would be fine.

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    $\begingroup$ Welcome! Including information on what books you learned from before, especially for single variable calculus, would be very helpful to anyone trying to answer your question. Also the set of lecture notes you found that you did not like could be linked and described better. $\endgroup$
    – Chris Cunningham
    Sep 25, 2019 at 12:34
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    $\begingroup$ Most any book with the title "Advanced Calculus" should be suitable, such as Advanced Calculus of Several Variables by C. H. Edwards. A little more advanced than the dozen or so well known advanced calculus texts is Advanced Calculus: A Differential Forms Approach by Harold M. Edwards. (continued) $\endgroup$
    – Dave L Renfro
    Sep 25, 2019 at 15:31
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    $\begingroup$ A lot more advanced than standard advanced calculus texts is Advanced Calculus by Loomis/Sternberg and Advanced Calculus by Nickerson/Spencer/Steenrod (both books freely available on the internet, at least where I live). $\endgroup$
    – Dave L Renfro
    Sep 25, 2019 at 15:32
  • $\begingroup$ I'm not sure I would call my current set of notes rigorous, but I do care more about exploring error in Talyor polynomials for examples. I don't think I derive the error bounds, and I suspect you might want to find a good numerical methods course for that sort of analysis. That said, you might find supermath.info/OldschoolCalculusII.pdf helpful. $\endgroup$
    – James S. Cook
    Feb 27, 2020 at 23:30

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Possibly not helpful to you since to my knowledge it is only available in French, but I have a marvelous recollection of the "Petit guide calcul différentiel à l'usage de la licence et de l'agrégation" by François Rouvière.

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I like the various lecture notes by William Chen a lot. Clear and reasonably complete, if introductory.

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    $\begingroup$ Wow, I don't see how he has time to do much more than writing all these books! $\endgroup$
    – Dave L Renfro
    Feb 28, 2020 at 9:50

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