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Topic: Analysis books?
Replies: 7   Last Post: Nov 6, 2005 1:51 AM

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Dave L. Renfro

Posts: 4,792
Registered: 12/3/04
Re: Analysis books?
Posted: Nov 5, 2005 11:17 PM
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Colleyville Alan wrote:

> 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",

Dave L. Renfro

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