> I will be taking Calculus I, II, and III with the > first class beginning this coming January. My goal > is to study enough analysis (concurrent with the > Calculus courses) that when I complete Calculus III, > I can enroll in an analysis class that while still > undergraduate, would be an upper-division, rather > than an introductory, class. > > I've seen in the sci.math archives Rudin recommended > for graduate-level work and some of the authors of the > Dover books (Kolmogorov, Shilov, Rosenlicht) recommended > for undergraduate, but this is self-study only for > the purpose of avoiding a lower-division class and > having an instructor for the upper-division stuff. > > Given that, what would you recommend?
I'll assume you're talking about a typical U.S. Calculus I, II, and III sequence.
It sounds like you simply want to omit the now common sophomore/junior level "transition to advanced mathematics" course. This should take far less work than it sounds like you're planning. For one thing, this type of course only developed within the past 30 years or so. (I think partly in response to pushing higher levels of abstract algebra and other pure mathematics courses into the undergraduate curriculum in the 1960's, and partly because of the increase in the percentage of the U.S. population that began attending college in the late 1960's and the 1970's.) In some colleges, this type of course serves a dual role as a discrete math course (including combinatorics and mathematical induction) for computer science majors. Thus, this course is probably not even necessary for a reasonably good mathematics student
If your college offers honors versions of the elementary calculus sequence and you're able to get into the honors sections, then you can probably skip the elementary analysis course without missing much. If this isn't possible for some reason, here are three texts that you should try to get a hold of and read through while you're taking the calculus sequence:
Michael Spivak, "Calculus", 3'rd edition, 1994.
Tom Apostol, "Calculus", Volume 1, 2'nd edition, 1967.
Richard Courant and Fritz John, "Introduction to Calculus and Analysis", Volume 1, 1965. [This has been recently reprinted, I believe.]
Spivak and Apostol are widely used in honors classes and at places like CalTech, MIT, Univ. of Toronto, etc. Courant/John is a classic that is in some ways pitched at an even higher level (the problems are harder, for one thing), but it gives an outstanding coverage of *everything*, from theoretical to applied, that someone planning to continue studying mathematics ideally should be exposed to in an elementary calculus course.
Incidentally, Apostol and Courant/John each have a 2'nd volume covering multivariable calculus, but I think for what you're looking for the 1'st volumes will suffice.
Finally, for what it's worth, the best (very) elementary real analysis text I know of for someone looking for something to read along with a traditional calculus text (not the three texts above, but the sort of text U.S. colleges usually use for their calculus sequence) is
Victor Bryant, "Yet Another Introduction to Analysis", 1990.