I am currently a sophomore in high school who is very interested in mathematics and (theoretical) physics, and was wondering if the diverse set of mathematicians at MSE had any suggestions as to any self study texts I could read in order to enrich my knowledge of pure mathematics. First, I suppose, you should have some quick background as to my current level and interest in the field:
I have completed the majority of Tom M. Apostol's Calculus of a single variable (I more than likely would have finished if school had not started back up before I could), and really enjoyed the emphasis of set theory and proofs in the first chapter or so. I also enjoyed the treatment of integration (definite), before differentiation (I had knowledge of both through MIT open course ware). I especially enjoyed the proofs of basic properties using the axioms of the real number system, which I attempted mostly using first order logic (although I must admit a general lack of rigor to it all).
I have also perused some linear algebra here and here, although I can in no way be said to be 'proficient' in the techniques of linear algebra. In addition I have looked at some differencial equations, but am (again) relatively clueless with respect to them. Of the two subjects, however, I found differential equations to be the most interesting, although I do not have an awfully good measure of either subject.
Through my studies of quantum field theory I have come across stochastic calculus and statistical mechanics, which (in my naive understanding) seem rather interesting, but not exactly my favorite. In my study of General Relativity I became acquainted with tensors (to some extent), but more interestingly introduced into topology. Topology is intensely interesting and is both highly abstracted (as opposed to statistics), and is conceptually (somewhat) comprehensible. Another subject I have come across, although not through physics, is analysis (I know that is very broad), and really enjoyed the strict definitions of functions and the proofs of elementary properties from axioms. Though I have mostly seen real analysis, I think the little bit of complex I have seen is also rather interesting, but I have such a shallow understanding of it that I know I can not fully appreciate it.
Lastly are my encounters with algebra, these have come in many forms from many places, but most notably in programming, number theory, and physics. I have seen (but not fully understood) rings, fields (e.g. ordered fields), et cetera, and have found it to be extremely interesting. Every time I see a new algebra concept I see the logical connections between things I never would have seen before, but a lot of it is over my head. I feel that is true of the majority of the aforementioned mathematics, I can start to approximate an understanding but I know I can not truly appreciate the mathematics. I think a good example of this is exponents: I have known and used exponent rules since sixth grade, but I had never really appreciated their power and relationships until I discovered the proof of them (while working through Apostol).
With that I ask you: What texts will help my mathematical reasoning (more than my ability to calculate) at my level? What introductory text would you suggest from the various fields of mathematics that interest you? What text would you suggest to any high school students? What would you suggest outside of math texts (i.e online resources, coding projects, et cetera)?
Best Answer
If you've worked through the majority of Apostol, then I highly recommend Hubbard and Hubbard's Vector Calculus, Linear Algebra, and Differential Forms. (Get it straight from the publisher; much cheaper than elsewhere). It is written very well, with the reader in mind. You'll learn linear algebra and some other requisites to start with. From there, you learn a bit of very basic topology and you end up doing some differential geometry-esque stuff later. There is SO much in this book... you could study it for a couple of years (especially if you dig into the appendix). Moreover, despite its rigor, there are many applications (that are actually very interesting).
As well, there is some courseware from Harvard that uses this book (look for Math 23a,b and Math 25a,b). And there is a (partial) solutions manual floating around.
You might also like to read:
-Set Theory and Metric Spaces by Kaplansky
-anything from the New Math Library (from the MAA) -Particularly Basic Inequalities, Geometric Transformations, or Mathematics of Choice.
-the linear algebra in Apostol (be sure to get Vol 2 also!)
-the whole second volume of Apostol!
-Discrete Mathematics by Biggs (get the first edition!)
-Finite Dimensional Vector Spaces by Halmos,
-Principles of Mathematical Analysis by Rudin with this and this and these awesome lectures.
-Algebra by Artin is amazing, but hard! Enjoy these lectures. Vinberg's algebra text is supposed to be amazing and in a similar flavor to Artin (but a bit more gentle).
And you might also like these (great) reading lists:
-PROMYS
-Chicago Undergraduate Mathematics Bibliography