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As a reader of I. M. Gelfand's algebra/trigonometry and A. P. Kiselev's geometry textbooks, I am struggling to find an equally rigorous calculus textbook. I've tried Stewart's "Calculus: Early Transcendentals", but it's way too (for lack of other words) fluffy. I enjoyed Gelfand/Kiselev books because of their succinctness and rigor, where the few exercises were never the same (i.e. solving 2 quadratic equations where $a$, $b$, and $c$ are just different numbers).

I'm looking into Tom Apostol's book, but there the only editions I've seen have terrible formatting and many typos. I attempted Thompson's book (the one Richard P. Feynman studied by), but there are typos even in the answer key, which confused me severely as a self-learning student new to calculus.

Also, as a bonus, but not a requirement, if the book includes some linear algebra (like Apostol's), that would be a plus.

Fine Man
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    If you are a student new to calculus and you are a self learner, then why would you want to have such a rigorous textbook? – imranfat Nov 24 '16 at 04:44
  • Why not? I managed Gelfand and Kiselev just fine. Regarding Thompson, I just didn't like the fact that when I thought I got an answer correct (which I did indeed), the answer key would tell me otherwise, finishing my confidence in my knowledge. – Fine Man Nov 24 '16 at 04:45
  • Yes, correct, a textbook that has errors in the answer key is not good for learning experience, but if you are new to the topic (that's my assumption) and you have to learn all by yourself (not in a school setting?), then what would be wrong to learn from Stewart or any other staple calculus book first, and then move on to a real analysis textbook? It will certainly strengthen your foundation, a not unimportant matter... – imranfat Nov 24 '16 at 04:47
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    You could try Infinitesimal Calculus - Spivak and/or Problems in Mathematical Analysis - Demidovich. Alternatively, you could learn basic calculus from any less rigorous book and then begin to study Real Analysis to really understand the theory behind calculus. – Vitor Borges Nov 24 '16 at 04:51
  • @imranfat -- I feel like I waste time reading through all the fluff, and then solving dozens of problems that are too similar. I have less than a year before I go to college, and I'd like a solid foundation of calculus/linear-algebra before then. – Fine Man Nov 24 '16 at 04:57
  • @user8485 -- My father studied by Demidovich (in Russian, though). I'll take a look at it. – Fine Man Nov 24 '16 at 04:58
  • Can you read russian or german ? – Rene Schipperus Nov 24 '16 at 05:01
  • @ReneSchipperus -- Nope, I can understand Russian verbally but never learned to read it (to my grandfather's disappointment). I read translations, though. :) – Fine Man Nov 24 '16 at 05:05
  • @SirJony. Understood. However learning through exercises is a valuable experience. Standard Calc books do a good job in terms of explanation. Now I agree doing dozens of problems that are kind of same, may be a waste. On the other hand, exercises also makes you much stronger in foundation. How about odd numbered problems only? You do them right, move on to next section. Don't under estimate importance of exercises! You do know the story of a PhD candidate in Swimming who studied every stroke from the books, but never went into the water himself? One time he fell into the pool. He drowned:) – imranfat Nov 24 '16 at 05:16
  • @imranfat -- Very unfortunate story. I, for one, know how to swim (quite well), but have no PhD in it. :) On a more relevant note, how is the following combination of textbooks: Courant for theory, and Demidovich for exercises? My father used them when he was in university, and turned out quite alright :), so I think I'll try this combo. – Fine Man Nov 24 '16 at 05:32
  • Sure, and unlike in your father's time, you got the internet for reference too! Good luck... – imranfat Nov 24 '16 at 05:36
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    Spivak's Calculus is the way to go. – yoyostein Nov 24 '16 at 05:50
  • Another option might be Zorich's two-volume Mathematical Analysis. – Hans Lundmark Nov 24 '16 at 07:58
  • Does this answer your question? Good book for self study of a First Course in Real Analysis –  May 04 '23 at 00:40
  • Does this answer your question? What are the recommended textbooks for introductory calculus? –  Jun 01 '23 at 05:05

3 Answers3

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I've decided on Introduction to Calculus and Analysis by Richard Courant. So far, so good!

Fine Man
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Maybe you could consider The How and Why of One Variable Calculus, by Amol Sasane. Here is the publisher's homepage for the book: http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119043387.html , while a google preview can be found at : https://books.google.se/books?id=mHOwCQAAQBAJ&pg=PP1&lpg=PP1&dq=Sasane+Calculus&source=bl&ots=N9FS3lmhkF&sig=GzOv3aYbETCCaTH6LgEDKRaVdgo&hl=sv&sa=X&ved=0ahUKEwjwyrTtobXRAhVFOxoKHZJHAvkQ6AEIQDAD#v=onepage&q=Sasane%20Calculus&f=false

H. Severt
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Consider getting Apostol's Calculus: Volume-I. Later, if you make it through Volume-I, you can continue with Multivariable Calculus using the Volume-II of the same writer. His books are very rigorous with the proofs.

math-physicist
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