Books for Reviewing Undergraduate Mathematics Before Grad School

[PieceOfPi]

Hi,

I will be starting my graduate study in mathematics in August 2011, and I was thinking of reviewing mathematics that I have learned while I was an undergraduate as soon as my summer vacation begins in June.

The topics that I am considering to review include algebra (abstract and linear), analysis (real & complex), topology, and geometry. Of course, if there are other subjects that I should review before beginning my study at the grad school, please let me know.

I am mostly interested in reviewing analysis, since I took both real and complex analysis sequences last year (I studied all the other topics that I mentioned above this year). For real analysis, my class Baby Rudin for single-variable stuff, and Spivak's Calculus on Manifolds for the multivariable calc. For complex analysis, my class used Brown/Churchill.

In the past, I asked a similar question, and one person recommended me Anthony Knapp's Basic Analysis for reviewing real analysis. While I have not studied from this book before, the topics covered in this book seems reasonable, and I also like the idea of having solutions to the exercises (great for self-study).

Let me know if you have any recommendation of books and topics to review over the summer.

[slamminsammya]

I am in a similar situation. Does anyone have any recommendations for a good refresher book in multivariable calculus? I.e. one that will cover the important concepts but in a manner that is perhaps more sophisticated than what you encounter in a first year undergrad course?

[anonymity]

I am in a similar situation. Does anyone have any recommendations for a good refresher book in multivariable calculus? I.e. one that will cover the important concepts but in a manner that is perhaps more sophisticated than what you encounter in a first year undergrad course?

I have heard good things about "Div, Grad, Curl, and all that"

http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

I am not sure if this matches up with the rigor that you are looking for (how much more sophisticated?), but it is very well received both on these forums and elsewhere.

"
I had three years of higher-level calculus between my BS and MS in mechanical engineering, and none of these classes have explained the concepts in this book with such clarity and accessibility. The sample problems at the end of each chapter cement the concepts just learned. For me, they were just challenging enough to test and hone my skills, but not so crazy that I felt like I was stroking some intellectual ego instead of learning practical concepts.

I highly recommend it to people of similar backgrounds as myself--people with already decent math backgrounds, but who need to hone their vector calculus skills to enter the world of physics, electrical engineering, fluid mechanics, continuum mechanics, or anything else along those lines (lines! Hah! Pun!). I feel like this book was written just for me! Are there really that many of us?

"

(From amazon -- the first comment that I saw..)

[jbunniii]

Have you seen this book?

All the Mathematics You Missed: But Need to Know for Graduate School (http://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071)

Also:

The Linear Algebra a Beginning Graduate Student Ought to Know (http://www.amazon.com/Algebra-Beginning-Graduate-Mathematical-Sciences/dp/1402054947/ref=sr_1_3?s=books&ie=UTF8&qid=1306207261&sr=1-3)

On life as a mathematics graduate student (and beyond):

A Mathematician's Survival Guide: Graduate School and Early Career Development (http://www.amazon.com/Mathematicians-Survival-Guide-Graduate-Development/dp/082183455X/ref=ntt_at_ep_dpt_3)

This one is a good source of problems with solutions, taken from the Berkeley preliminary exam, which covers undergraduate material:

Berkeley Problems in Mathematics (http://www.amazon.com/Berkeley-Problems-Mathematics-Paulo-Souza/dp/0387008926/ref=sr_1_1?s=books&ie=UTF8&qid=1306207447&sr=1-1)