Advice in Bachelor Degree

Question

First of all, I´m very sorry for my bad english, especially writing. Ok, for differents problems i´m studing a Bachelor degree in Mathematics. These degree is online.

Now, the problem with my school is that it is new (created in 2009 for the Mexican Government ) and that implies that it is very bad in quality. Let me explain, they don´t have researchers, a decent teachers, the content (I mean the content in the subject is very low in quality if you compare with more prestiges universities in Mexico like ITAM, UNAM). Besides, they assign you a "teacher" and when you ask him a question (by email) never respond you, so it is on you to learn mathematics.

After that, i´m here to ask for help,tips and recommendation about books, pages or materials that help me to learn very well, with the condition that these materials have to be very well explained, remember I´m learning myself. I dont care the difficult that implies learn yourself mathematics because I like and it is my passion. My advantage, if you let me say that ,I sort of know english. Right now, I now about MIT opencourseware, Harvard video classes an so on, and i´m taking these videos like my courses to learn complementing with reading, examns. As you can see i´m interested in learning a decent international level in mathematics because im planning a master in economics in USA or England, so the point is prepare myself for that kind of level.

Actually these is my list of subjects of fourth-mester (it is a course during four months, sorry i dont know how called it):

Linear Algebra I - watching course MIT opencourseware Gilbert Strang. Book: Linear Algebra, G. Strang.

Probability I : Watching course "Statistics 110" with professor Joe Blitzstein from Harvard. Book: Statistical Mathematics and Data Analysis. J. Rice

Discrete Mathematics: Watching course "Mathematics for computing Sciencies". Book.: Discrete and Combinatorial Mathematics. Grimaldi.

For clarification, when I say "watch course" it means i already got the lectures, homeworks and exams for the three subjects.

I´m taking other subjects but it is about economics and history in Mexico, so for this purpose it doesn´t matter.

If any could help me about where i can get introductory materials about this subject I will be really thankful. What do you think about these courses and books? may be a better introductory book or video course?, advices about how to learn mathematics?. I read that some people is in Harvard, MIT and thousands of universities top rank in the world, and i would really like that those people tell me some tips because they have a better experience and I think the internet is the best way to catch that. I have to say that any advice it would an enormous help.

Thank you very much indeed for any suggestions. JAPS.

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Update: sorry, my mistake. My bachelor degree is formed by twelve "fourth- mester". I mean twelve units of fourth months each one. right now im in the unit four and these is For now, I already took diferencial and integral calculus both with Apostol "Calculus; Vol. I and II", introduction to logic (no book), Geometry I (Book: Wooldrige " a modern approach to geometry"), and other subjects of different kind. Before i go too far, I prefer ask my doubts and catch all kinds of tips.

in the future, I will take Mathematics Analysis I and II, differential ecuationsI and II, partial ecuations, Geometry NO-Euclidean, Estocastic Process, modeling estocastic process, estatistics I, II and II, Topology I, Complex variable I and II, Linear ALgebra I and II, Modern Algebra I and II, Combinatorial ANalysis, Numerical Analysis I and II.. and I think that´s it.

Once again, I really appreciate to take the times for respond guys. JAPS

Answer 1

These are my suggestions for books. I have taken courses or studied extensively from each of these:

Algebra: Dummit and Foote, $\textit{Abstract Algebra}$

Analysis: Rudin, $\textit{Principals of Mathematical Analysis}$

More advanced analysis: Fitzpatrick, $\textit{Advanced Calculus}$

Linear Algebra: Anton, $\textit{Linear Algebra and its Applications}$

Answer 2

As an aspiring economist I imagine you are familiar with the economist Adam Smith right? He attended the University of Glasgow, and Balliol College, Oxford. I was interest in Adam Smith as an undergraduate, and during some independent research I came across a book in our library. The story goes that Adam Smith thought that Oxford was a place where a greater part of the public professors had basically given up on teaching (Wikipedia). However, Adam Smith enjoyed the great library that the had at Oxford and spent endless hours in the library reading the work of other great intellects. Regardless, of whether this story is true or not it inspired me, and hopefully it will inspire you as well. You do not need to attend an Ivy League school to be a great intellect, all you need is curiosity, diligence, an appetite to learn, and great books written by great teachers!

I would say that the most important subject to focus on for an Economist is Analysis (and lots of it), because analysis incorporates so many other branches of math a solid back ground in Analysis will assure that you know the other. Of course, Advance Probability Theory and Stochastic Calculus are necessary as well (but again a solid background in analysis should help you through this, at least this is my approach).

Here are some free (open source) books which I have found helpful:

Basic Analysis

Introduction to Real Analysis

Abstract Algebra

There are many other good books which are available via the internet, which may or may not be open source. Also, the Federal Reserve posts great amount of economic articles available on many different economic topics, some of which are for educational purposes; the National Bureau of Economic Research also post some free articles as well.

Some good books that are worth purchasing are:

Foundations of Abstract Analysis (Second Edition) By Jewgeni H. Dshalalow

Abstract Linear Algebra by Morton. L Curtis

Real Analysis with Economic Applications by Efe A. Ok

A First Course in Probability (Any Edition) by Sheldon Ross.

Economics Books:

Intermediate Microeconomics by Hal R. Varian.

Macroeconomics by N. Gregory Mankiw

I have personally added each of these books to my own library and our goals are similar, so I hope all of these will be of help to you. Also check out the course below at MIT, Good Luck! Oh yeaa and http://math.stackexchange.com is also a great source!

MIT: Microeconmics