tameree wrote:I am looking to develop my mathematical mind as much as possible. The "highest" classes I have taken are Calculus I & II as well as linear algebra. I'd be looking for either a bunch of books or some sort of guide (similar to the guide "How to become a good theoretical physicist" for physics) that ranges from algebra up to topology, complex analysis and partial differential equations.
For an absolute starter I really like Biggs' "Discrete Mathematics", from Oxford Press. The first time you go beyond arithmetics and calculus, and maybe linear algebra, into a bigger field of mathematics, this is exactly the kind of stuff you will need to learn. Apart from covering some basics of logic, combinatorics and algebra (stuff that is used in practically ANY other course in ANY kind of math,) it is also an introduction into more formal theorem+theorem=proof reasoning, and more like that. It's very basic, but covers a LOT of ground. A lot of IMPORTANT ground. I'll give a little overview of the awesome stuff it contains:
chapter 1, statements and proofs. Yeah, this is basically a chapter on how to actually do formal axiom and proof based mathematics. Stuff the professors will assume you know, or at best impart to you little by little.
chapter 2, set notation. Once again stuff you'll be assumed to know in EVERY other part of mathematics.
chapter 3, the logical framework. Explains things like contrapositive statements, equivalence of statements, and logical notation.
chapter 5, functions. Formal concept, bijections, composition, etc.
The above chapters are absolutely crucial for all other mathematics, and if you're a uni student you'll notice these things are used in EVERY math course you EVER take.
It also introduces the integers and rationals, and the usual operations on them, in a formal (algebraic) way. This includes induction.
Then combinatorics, especially enumerative combinatorics, i.e. how to count the number of elements in a set (binomials and stuff like that).
And finally an introduction to algebra. Groups, rings, etc.
And this is just what I think is the most important. If i had to characterize it, I'd say it was a crash course in formal math. And it's well-written
. Yeah, I'm a fanboi. I can still enjoy some of the exercises. If you already know these things don't buy it though, it will not serve as a reference book, it is very basic.