Preliminary Exams in Graduate Math

[JohnRV5.1]

Hey all,
I have to take exams in several areas--one in analysis. The exam will also include things from undergraduate analysis. Here are the specific topics that may pop up:

For advanced Calculus:
1. Integration of functions of several variables: line and volume integrals
in 2D, line, surface, and volume integrals in 3D.

2. Differentiation: gradient, curl, divergence, Jacobian. Connection
between rotation-free vector fields and potential fields.

3. Partial integration, Green’s theorems, Stokes’ theorem, Gauss’ theorem.
The consequences of these theorems for vector fields that are
divergence or rotation free.

4. The concepts max, min, sup, inf, lim sup, lim inf, lim.

5. Convergence criteria for sequences and series.

For Higher Analysis:
It will cover metric and normed spaces, banach spaces, hilbert spaces(separable spaces only), and Measure theory.

So what books would you recommend for the higher analysis section and the advanced calc. section. I'm more curious about the advanced calc topics. Would Spivak's Calculus on Manifolds be a good choice? Any others?
Thanks for the help!

 

[^_^physicist]

spivak's differential geo. might be better.

 

[redrzewski]

I enjoyed Shankar's Basic Training in Mathematics. But keep in mind that this is much more focused for pragmatic use of the advanced calc topics you mentioned, and doesn't focus on rigorous proofs. So I don't know how help that'd be.

 

[JohnRV5.1]

Thanks guys for replying. I'll take a look at the spivak and shankar texts. I am looking for something that's both comprehensive and rigorous. In addition, I'd like something with an abundance of problems, too. Any other recommendations?

 

[redrzewski]

Kaplan's Advanced Calculus will be more rigorous than Shankar, and covers everything you're looking for. I've got a really old edition, but I liked it when I used it as a reference.

Rudin's Principals of Mathematical Analysis is a standard undergrad analysis text. It'll be very rigorous. My copy is on order, but from the table of contents, it seems like it may not cover the vector calc very well. It'll use differential forms and probably not the vector div, grad, curl at all. Again, I'm speculating.

Also, I got my degree in comp sci, so I can't really speak to rigorous grad math preparation (which I'm pursuing only as a hobby).

Do you know if you need standard vector calculus, vs. differential forms? In any case, from my personal experience, it was easier to learn vector calc first, and then add differential forms instead of just jumping right to differential forms.

A good diff forms intro is Bamberg and Sternberg: A course in mathematics for students of physics

Good luck.

 

[JohnRV5.1]

In fact, the analysis I learned in undergrad came from the majority of the first eight chapters of Baby Rudin. However, I am self-studying the last remaining chapters for the exams, so I do need something with differential forms. Yet, I wonder if there is anything else I could use to provide more problems or different perspectives. As of now, I'm leaning towards the spivak texts mentioned already. I found Do Carmo's book called "Differential Forms and Applications," and there is Munkres "Analysis on Manifolds." Are these any good? Or is Rudin's coverage enough?
THank you for time.

 

[redrzewski]

I haven't tried either of those books.

I do own Lee's Introduction to Smooth Manifolds, which I've read thru, and covers what you're looking for, and is both rigorous and has plenty of real examples. Personally I wouldn't recommend it as an introduction to differential forms, though.

 

[musicheck]

One other book to consider for differential forms is "Tensor Analysis on Manifolds" by Bishop and Goldberg. Its quite cheap too! ($11)

 

[mathwonk]

i recommend spivaks calc on manifolds, it is only 140 pages, and covers partial and total derivatives, riemann integration, and the basic theorems on repeated integrals, as well as sard's theorem, as green, stokes, and gauss theorems. also differential forms are treated carefully and in detail, all at an undergraduate beginning grad level.

this is where i learned it, and i have never needed another source. this covers all your first three topics.

dieudonne is excellent for banach spaces and hilbert spaces, basic stuff only. i have not mastered measure theory so well myself but analysts i know have used wheeden and zygmund, and also royden chapters one or two?, on the basic real line case.

of course langs analysis II covers all the higher analysis on your list, in great detail.

i don't know if your syllabus includes it, but one important theorem is baire's category theorem on complete metric spaces. it allows some amazing corollaries on the existence of various surprizing objects. indeed most of the deep foundational results in banach space theory are apparently corollaries.

 

[morphism]

For higher analysis I recommend Simmons's Intro Topology and Modern Analysis. Part 2 of the book covers banach and hilbert spaces (and part 3 covers banach algebras, and some operator theory). Unfortunately, there is (almost) no measure theory in it; for this, you can try Bartles Element's of Integration.

A more thorough book on banach spaces is Morrison's An Introduction to Banach Space Theory, but it contains no "standard" exercises, so might not be useful for exam prep.

There's also another very well-written two-volume book by Kolmogorov and Fomin that covers all the higher analysis topics you've mentioned, but it's also devoid of any exercises. I just looked it up on amazon.com, and apparently it comes in two forms (both Dover reprints, so they're cheap!), presumably by different translators. I have the two volumes separately. The first one is titled Metric and Normed Spaces, and the second one Measure, Lebesgue Integrals, and Hilbert Space.

Finally, Rudin is always a good source of problems.

 

[mathwonk]

actually i used to think as a young student that anyione who worked rudins problems would always pass an analysis prelim, and anyone who worked even hersteins problems would pass an algebra prelim.

 

[JohnRV5.1]

Mathwonk, that's good to know. My goal is to certainly master the topics in Rudin, and I will undoubtedly pick up a copy of spivak's book. Unfortunately, I will have to see if my school's library has the other recommended texts--especially the ones by Kolmogorov/Fomin and Simmons. Thanks again.

 

[morphism]

You can buy them used on abebooks.com, if you like. There are hardback copies of Kolmogorov/Fomin for $6, and Simmons for $20. They also have Spivak for $20.

Good luck on your exam!

 

[mathwonk]

thats real and complex analysis by rudin, not baby rudin.