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I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus but I'm starting to wonder what the best book to continue with would be? I wouldn't want to just review single-variable calculus in rigorous form as an introduction to analysis but I'm also not sure if going straight into Baby Rudin/Apostol Vol. II or anything of that sort is any wiser. Or perhaps it is, having the physical intuitions of single-variable calculus as imparted by Kline? I'm not even sure.

LanguageDelights
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  • Why don't you want to learn it rigorously (read: properly)? Get Baby Rudin. – user85798 Jun 17 '14 at 08:12
  • If you don't want to review single-variable calculus in rigorous form, are you sure you want to study real analysis? – Silynn Jun 17 '14 at 08:14
  • Rosenlicht's "Introduction to Analysis" is pretty good and cheap. What are your goals in studying mathematics? – Spencer Jun 17 '14 at 08:17
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    Anaylsis can often be a bit of a pill to swallow for those uninitiated in proof writing. If you're intent on studying rigorous mathematics I would suggest studying another subject such as linear algebra as an intermediate step. There is a very good book on linear algebra for free online at: http://cseweb.ucsd.edu/~gill/CILASite/ . Working your way through the first part of this book should prepare you for the level of rigor expected in books like Rudin – Spencer Jun 17 '14 at 08:21
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    After reading (and liking) Kline's book, I think you may want to consider the 2-volume Introduction to Calculus and Analysis by Richard Courant and Fritz John. See the comments and answers at Difficulty level of Courant's book for some discussion of these books. Also, see the comments on these books at amazon.com. – Dave L. Renfro Jun 17 '14 at 15:09

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I moved right into David Widder's Advanced Calculus and I find it perfect, there's almost no overlap. The only thing you have to brush up on beforehand is vector notation and some multivariate topics, but Pauls Online Math Notes solve that issue easily.

Ryan
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