Study a math course on my own, suggestions?

Question

I would like to study some math on my own. I am currently studying my second semester at a university and I have too much freetime so I would like to study something on my own, but I can't decide what to study. I have finished courses in: calculus in one and several variables, mechanics for engineers and a course that dealt with some number theory and proofs. I am currently taking linear algebra and probability theory.

So, I am wondering if anyone has any suggestions of what I could study on my own and if so, is there a book you would recommend?

Answer 1

You could make a start into analysis and abstract algebra. "Elementary Analysis: The Theory of Calculus" is a very good book for an introduction into analysis written by Ross. I don't know a good book for an introduction into abstract algebra but there are plenty. You could start with "group theory".

Good Luck!

Added: A good introduction into 'real' mathematics is: "Reading, Writing, and Proving: A Closer Look at MathematicsA textbook by Pamela Gorkin & Ulrich Daepp" which starts with basic notions of: what is a set, a bijection, how do you proof something by induction etc.

Answer 2

If you study something related to engineering I would recommend you to start a course on harmonic analysis (also known as Fourier analysis). At this moment I am reading a book which contains the very basic knowledge in this area along with a whole chapter about measure theory, which I assume you are not familiar with from what you've said, but you'll need the basics (which are not hard) to be able to study this area. The book reference is:

C. S. Rees, S. M. Shah, C. V. Stanojevic, Theory and Applications of Fourier Analysis.