Is there a good book that combines Calculus I, II, III, Analysis I, II with the routine calculation exercises and that also shows rigorous proofs of the theorems? I have looked at Vladimir Zorich's Mathematical Analysis I but does not have many routine questions in it. So what are your suggestions? So the idea is I want to study Calculus with Analysis together.


closed as primarily opinion-based by YuiTo Cheng, Yanior Weg, TravisJ, Ernie060, Michael Rybkin May 9 at 19:42

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    $\begingroup$ My suggestion is you shouldn’t; the purpose of calculus is calculations, the purpose of analysis is to understand the logical foundations of calculus. It is difficult to cover both simultaneously, because the study of analysis usually assumes that you have some intuition that you are trying to put on solid ground. And calculus requires a lot of practice and problems, which would make it difficult to advance in the analysis part. $\endgroup$ – Arturo Magidin May 9 at 17:19
  • $\begingroup$ @ArturoMagidin In Serbia there is a single class for real analysis and calculus.I belive it is the same way in other Eastern European countries.But it lasts two semesters and it is only about single variable. In the second year you do the same for two variables and other parts of real analysis. Basic calculus is taught in high school $\endgroup$ – Milan May 9 at 18:07
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    $\begingroup$ @MilanStojanovic: If you’ve taken “basic calculus” in High School, then you are not truly learning both at the same time; that’s exactly my point. You should get some calculus, by itself, first, before you go on to analysis. The same scheme exists in Mexico, btw, but again, you don’t learn them both, for the first time, at the same time, any more than you would try to teach arithmetic and algebra at the same time to someone who knows neither. $\endgroup$ – Arturo Magidin May 9 at 18:17
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    $\begingroup$ No idea why this was closed. How is this any more opinion-based than the hundreds of other requests for book recommendations on this site? $\endgroup$ – Tim kinsella May 9 at 19:52

Check out Spivak's Calculus book.

From the preface:

"Every aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics. Since the foundations of analysis provided the arena in which modern modes of mathematical thinking developed, calculus ought to be the place in which to expect rather than avoid, the strengthening of insight with logic. In addition to developing the students' intuition about the beautiful concepts of analysis, it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions."

Here's an MSE post about it: Spivak's Calculus?

And then for multivariable, you might look at his Calculus on Manifolds. But that may require more background -- it's been a while since I looked at it.


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