Real anaylsis/advanced calc text book

[doubleaxel195]

First of all, what's the difference between real analysis and advanced calculus? From what I've gathered, real analysis is more rigorous than advanced calculus?

Second of all, what's a good book for self study for real analysis or advanced calculus? I've just finished taking my first abstract algebra class, number theory, and complex variable class (this class was more engineering--hardly any proofs, more applications).

I know that learning on my own will be tough, but I don't mind. I'm taking advanced calculus in the fall and wanted something to do over the summer.

Thanks.

 

[jbunniii]

There is no standard definition of what constitutes "advanced calculus" versus "real analysis." For some authors, the two terms mean the same thing. For those of us that do make a distinction, it's not necessarily a difference in rigor but rather in sophistication.

For example, Spivak's "Calculus" is a completely rigorous treatment of calculus, but at a more elementary level than that of Rudin's "Principles of Mathematical Analysis." Some differences between these books to illustrate what I mean:

- Spivak works entirely in Euclidean space (in fact, mostly the real line), whereas Rudin uses metric spaces, with Euclidean space being a specific example.

- Spivak's proofs are much more detailed, and there is a lot more narrative explaining what he is doing. Rudin's proofs are terse, and his narrative is much less chatty.

- Spivak uses the Riemann integral, Rudin uses the more general Riemann-Stieltjes integral (and Lebesgue integral toward the end).

If you haven't worked through a Spivak-level calculus text, that might be the best place to start. Apostol and Courant are good alternatives.

If you want a bit more sophistication but don't want to plunge headfirst into Rudin, I recommend Thompson, Bruckner, and Bruckner's "Elementary Real Analysis," which is very affordably priced and quite good, or one of Bartle's books: either "Introduction to Real Analysis" or "Elements of Real Analysis," the latter being the higher-level of the two.

Which of the above to choose? I would say that if you are already pretty comfortable with delta-epsilon proofs, limits, continuity, sequences and series, and the properties of real numbers (especially suprema and infima), then go straight to Rudin. If you have seen these concepts before but are not very comfortable with them, go with Thompson or Bartle. If these are mostly unfamiliar concepts or you have never done them rigorously, then you want Spivak.

 

[hitmeoff]

I think what a lot of schools call "advanced calculus" they mean elementary multi-var/vector calc.

Analysis is to calculus what abstract algebra is to college algebra/elementary linear algebra (sort of), in that you don't do so much calculating, you do a lot of proving. Theres no "Step 1 Step 2 Step 3"...Answer way of doing problems, rather you are given definitions, theorems, etc and you are asked to do figure out a creative way to use them to get a problem done.

Though "Advanced Calculus" might refer to Real, single variable Analysis, or it could refer to multi-variable Analysis. Since you've had abstract algebra, then you know exactly what to expect in a higher level math class, so you should be fine. I am not so high on Analysis myself and like Algebra much more.

Ross's text on Analysis is generally considered the easiest Analysis book out there (and that's a good thing...lots of people will throw the name Rudin out there, but Rudin is usually a grad level book, as the poster above said...very tight, very terse and not very chatty).

 

[doubleaxel195]

Thanks for the replies!

I really don't have that solid of a background in calculus so I think I'll go with Spivak? How much would I lose if I did Spivak instead of Rudin? Like I said, I'm willing to work hard.

 

[doubleaxel195]

"Properties of the real number system; point set theory for the real line including the Bolzano-Weierstrass theorem; sequences, functions of one variable: limits and continuity, differentiability, Reimann integrability."

That's the entry for advanced calc I at my school...there is no class called real analysis

 

[jbunniii]

I would say that Spivak (or comparable) is almost a prerequisite for Rudin. While it's theoretically possible to learn rigorous calculus/analysis from Rudin starting from scratch and without strong guidance from an instructor, you would have to be a pretty exceptional student to do so.

Spivak covers all of the topics you listed. Good thing you're willing to work hard - Spivak's exercises are no joke :-) If you do as many as you can, it will be an excellent preparation for your class this fall.

P.S. You should also try to find out what textbook your instructor will be using, and get that in addition to Spivak (or instead of Spivak, depending on what it is).

 

[Sankaku]

I really don't have that solid of a background in calculus so I think I'll go with Spivak?

If you feel your basic calculus skills are ok but you are shaky with the theory, I would highly recommend Abbott's "Understanding Analysis." Think of it as catch-up for people who didn't do a Spivak-style course.

As said before, "Advanced Calculus" is a poorly-defined term. For those that think all Advanced Calculus courses are just basic Multi-Variable, take a look at Loomis and Sternberg - it is freely available here:

http://www.math.harvard.edu/people/SternbergShlomo.html

I went through a course that used Folland's textbook. I wasn't a huge fan but it did the job. In that case it was half an introduction to analysis and half a rigorous overview of vector calculus (implicit/inverse function thm in several variables, green, stokes, etc.).

 

[hitmeoff]

"Properties of the real number system; point set theory for the real line including the Bolzano-Weierstrass theorem; sequences, functions of one variable: limits and continuity, differentiability, Reimann integrability."

That's the entry for advanced calc I at my school...there is no class called real analysis

That sounds like Intro to Real Analysis so Advanced Calc I at your school is Real Analysis in my school