# Testing Preliminary Exams in Graduate Math

1. Aug 8, 2007

### 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:

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!

2. Aug 8, 2007

### ^_^physicist

spivak's differential geo. might be better.

3. Aug 8, 2007

### 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.

4. Aug 8, 2007

### 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?

5. Aug 8, 2007

### 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.

Last edited: Aug 8, 2007
6. Aug 9, 2007

### 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.

7. Aug 9, 2007

### 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.

8. Aug 9, 2007