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Is there a problem in studying analysis before calculus? Most people say that analysis is rigorous calculus, the university I'm studying teaches calculus first because they believe it's better for the student to have a intuitive background (which is obtained with calculus) and the go to analysis but I've seen some other universities that teach analysis first, for undegraduate levels.

I decided to study analysis first and I'm being able to understand it, I'm just not sure if any loss is made by not studying calculus. What could also be a nice alternative would be to study some Springer books on calculus and analysis, what do you think?

EDIT: One of the doubts I also have is if I'll be able to use calculus if studying only analysis.

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Spivak is an excellent "middle of the road" book. I would like to point out that calculus is only enriched by its numerous applications; they aren't merely some sort of baggage to be avoided. While there may not be a problem with skipping it, I wouldn't go so far as to avoid the 'calculation and application' side of things completely. – Robert Mastragostino Apr 16 '13 at 6:44
Indeed. Spivak's book is a gem. – dezign Apr 16 '13 at 7:13
I voted to reopen this. I'm not sure I understand why it was closed in the first place. It's an "advice question", but it's asking for advice particular to the study of mathematics, which IMO makes it on-topic and reasonably appropriate. Possibly CW-ification would have been better than closure? – Pete L. Clark Apr 21 '13 at 14:43
up vote 6 down vote accepted

I have always held that the best way to learn mathematics is by doing computations and working out concrete examples, thus taking an abstract theorem based course without the intuition gained from working out the motivating examples I think in general is not a good idea. Proving theorems will improve your mathematical reasoning and maturity, but generally speaking it doesn't help you solve problems. Once you've gained a feel for things based on the examples you have worked out you gain more of an intuition for proving deeper theorems as well. Moreover, I took analysis years ago and I don't remember anything, yet basic calculus comes up all the time (even in my research).

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In most American universities, Real Analysis subsumes Calculus. Therefore, there is no loss in skipping straight to Real Analysis-- provided that you can handle it, i.e. provided that you have the "mathematical maturity."

I personally skipped calculus, and the only problem I ran into was the following: I found that people who took calculus tend to develop proficiency at problem solving techniques that I didn't develop during analysis. Thus, they had lots of homework sets that covered routine problems which required them to learn problem solving techniques, whereas analysis was focused more on "theorem proving" and not as much on "problem solving." However, this is easy enough to solve. Just crack open a calculus textbook and do a hundred or so exercises.

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I have to agree with this. If you're able to understand a "Real analysis" proof, don't waste your time with calculus as it's taught in a normal university in the U.S. You can pick up anything that is taught no problem in a matter of no time. – Suugaku Apr 16 '13 at 6:48

They are basically the same subject, but if you cannot judge whether a series converges, how to do a double integral, how to apply Stoke's formula, etc, then you may benefit some studying some basics first. After all analysis is in place to make calculus rigorous, and if you do not know calculus well, chances are if you study higher level analysis you will get into numerous difficulties because they are all based on calculus. For example in elementary ODE, PDE, complex analysis, etc. I do not see the point to rush studying a subject.

A good mathematical analysis text to recommend is Zorich's book. The book I is more elementary and may be substituted by Rudin if you wish.

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