Good books/blogs/ etc. for advancing in math

[Bob Busby]

I'm just a humble engineer. I've taken the standard slew of math courses everyone takes (Calc, diff eqs., linear algebra) and I enjoyed them but I want something more. The world of math is so vast that every time I try to branch out I get lost and give up. Do you guys have any suggestions for books/websites or topics that would help me advance past the standard. The problem I always encounter is that books are either too fruity with little rigour or they assume a lot of notation and background that I don't have. Any suggestions would be appreciated.

 

[micromass]

Hi Bob Busby!

Can you give us a little bit information? For example, what math topics would you like to learn? I might suggest some books depending on your answer...

 

[Bob Busby]

Well, seeing as I'm just reading for fun I was hoping you would suggest fields you find interesting. But here's some prospective items. Number theory and graph theory sound the most interesting. Also, I have to learn probability and statistics in the future so that counts too.

 

[micromass]

Well, seeing as I'm just reading for fun I was hoping you would suggest fields you find interesting. But here's some prospective items. Number theory and graph theory sound the most interesting. Also, I have to learn probability and statistics in the future so that counts too.

Ah, that are very interesting fields. To study number theory, it would be best to study some abstract algebra first. The book "A book on abstract algebra" by Pinter is an extremely well-written introduction on the subject. It goes quite deep but it remains elementary at all times.

For graph theory, I suppose that any discrete mathematics texts should give you a solid introduction. Check out "Discrete and combinatorial mathematics" by Grimaldi.

Probability and statistics is a very broad topic. A very good introduction to that is "understanding probability" by Henk Tijms.

As for other fields. I suppose geometry is always fun to learn. The book "Introduction to geometry" by Coxeter is a nice introduction to various fields of geometry.
A very important part of mathematics is also real/complex analysis. "A first course in real analysis" by Berberian is quite a good book!

I suppose that could keep you busy for quite some time

 

[EWH]

!#%* board ate a long post with lots of links. I'm not going to retype it. Google these on your own: "geometric algebra", "Geometric algebra primer" by Jaap Suter (best paper introduction), and Cambridge GA group publications and "part III" course and "David Hestenes", "geometric calculus" for applications. The most fun introduction to GA is with the interactive free calculation and visualization software:"http://www.science.uva.nl/ga/viewer/index.html" ", and the "conformal geometric algebra tutorial", although Jaap Suter's primer covers a simpler GA (Suter covers 2D, 3D, 3+1D, vs. CGA tutorial 3+2D).

GA is almost the only form of math you need for any physical application. It works in any dimension with any signature, handles all forms of rotation better than any other mathematical system. The concepts are very helpful for learning abstract algebra, too.

 

[DuncanM]

Following are a few suggestions you might consider.

Goodreads describes itself as a social network for readers/book lovers. If you do a search for "math", you will see some of the math books that members have read and recommend to others.

Here is a "Top 50" list I came across recently:
http://www.guidetoonlineschools.com/tips-and-tools/best-math-blogs
I took a quick look through it. Some of them are pretty good, but the list seems to be aimed more at teachers/educators at the pre-college level.

Alternately, there is Mathblogging.org, which is a directory specifically for blogs about mathematical topics.

You might also be interested in Math Overflow, which discusses topics of a post-graduate level.

Hope this helps.

 

[Sankaku]

If you want something that is both technical and entertaining, you could try Paul Nahin's books about complex analysis (and more): An Imaginary Tale and Dr. Euler's Fabulous Formula. I haven't had time to get very far in my copies, but what I read was very enjoyable.

For Algebra, I second the recommendation of Pinter.

GA is almost the only form of math you need for any physical application...

Interestingly, I just started reading Hestenes' "New Foundations for Classical Mechanics." The whole GA project looks very intriguing,

Thanks for mentioning the Suter paper. I found it here:
http://www.jaapsuter.com/2003/03/12/geometric-algebra/

 

[id the sloth]

This is a link to the Open Math Curriculum, a collection of free open source textbooks many professors have written:

http://linear.pugetsound.edu/curriculum.html

Advanced knowledge of linear algebra is useful for plenty of area's of math. I recommend "Linear Algebra Done Right" by Axler for that. It's a textbook, but I actually enjoy reading it.

I don't really know many books on more theoretical areas of math, I'm more into more applied areas myself but would be happy to recommend a few books in applied areas if you like. You seem to want rigor though and not many books I know have it.

 

[kevinfrankly]

how about this ?
http://mathworld.wolfram.com/"