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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?

Eva
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    Depends what your university already offers. I'd recommend something it doesn't. –  Apr 15 '15 at 11:37
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    An important question too : What part of maths do you like? – Tryss Apr 15 '15 at 11:38
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    Do you feel comfortable with mathematical proofs? Are you being shown proofs and asked to prove statements yourself in your linear algebra course? If not, since proofs are a huge part of mathematics, I recommend you find a text on elementary proof writing and familiarize yourself with some standard elementary proof techniques. Such a text would also have exercises where you can prove statements on your own. – layman Apr 15 '15 at 11:39
  • I really enjoyed the calculus courses and I like the ones I study now. But since I haven't studied that much yet I don't really now what I would like to study. My university doesn't offer that many courses, it's not a big university. – Eva Apr 15 '15 at 11:43
  • You might enjoy studying complex analysis. If you're self studying, I would go with a less rigorous approach to get you started. –  Apr 15 '15 at 11:46
  • I'm being shown a lot of proofs in linear algebra, but so far I haven't had to do any on my own. I don't feel too comfortable doing proofs on my own. I took a course were we did a lot of them and we had to do some ourselves, but that was the first course I took and I felt really insecure – Eva Apr 15 '15 at 11:46
  • @Eva That's why I think reviewing a text on elementary proof techniques would help. It would review logic first, and then show you the standard elementary techniques that you no doubt come across often. I think spending time on this will help you feel more secure about not only understanding proofs, but writing them on your own without any help, too. – layman Apr 15 '15 at 11:49
  • I think you're right. I will probably have to do a lot of proofs in future courses so it would be a good idea to learn more and feel confident about writing proofs. Is there a book or author you could recommend? – Eva Apr 15 '15 at 11:57
  • @Eva Check the document linked below out. If you think it is helpful and kind of an easy read, then you should check the author's book out (it's on amazon; titled Bridge to Abstract Mathematics: Mathematical proofs and structures. http://www.mhhe.com/math/advmath/rosen/r5/student/data/proofwriting.pdf – layman Apr 15 '15 at 12:39
  • In your place I would freely go through a lot of results in form of theorems Lemmas etc. out of several well written books. Then it would be " why should that be so?" in some cases, an area to probe details of connecting arguments .. – Narasimham Apr 15 '15 at 14:05

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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.

Nescrio
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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.

Alberto Debernardi
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