(Part 2) Best products from r/askmath

Anything you want to know about symmetric functions and characters is probably in:

https://www.amazon.com/Symmetric-Functions-Polynomials-Physical-Sciences/dp/0198739125


This is also a popular text on the subject that starts from more of the basics of representation theory:

https://www.amazon.com/Representation-Symmetric-Encyclopedia-mathematics-applications/dp/0201135159/ref=sr_1_1?s=books&ie=UTF8&qid=1520448586&sr=1-1&keywords=James+and++Kerber+group

Maybe this book?

Or a standard Riemannian geometry textbook like do Carmo might suit your needs.

James Stewart: 5th Edition

Arfken's Mathamtical Method

Kreyszig

Analysis I and II by Terence Tao https://www.amazon.com/Analysis-Third-Texts-Readings-Mathematics/dp/9380250649

Oh yea, interesting that they teach number theory even in CS. I guess CS is most mathematical field if you compare it other fields except math?

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I haven't gotten into this stuff very deep, I am studying through this. I am very sure I wanna pursue math but there are only limited amount of areas to have time to study and I am not quite sure how 'active' the field is on that area (foundations of mathematics).

I have no idea if its relevant to your job interviews, but there's a book by Grimmett and a Stirzaker called "1000 exercises in probability" https://www.amazon.com/Thousand-Exercises-Probability-Geoffrey-Grimmett/dp/0198572212

Just like everyone else here, I'm not really going to answer your question, but I will add that the first text on a subject is rarely the best and in many cases the terminology is completely different from modern expositions.

I happened upon an entry level college textbook on Mechanics with an 1890 copyright (I don't know exactly when it was printed, but it was old an not in great shape). I have a BS in physics and could barely figure out what it was talking about. Everything was written in terms of quaternions since vectors were pretty new at that point. Most of the examples relied on simple geometric arguments, but most of those arguments were related to topics that aren't covered as often today.

Perhaps a history of math book along with a few choice books might be nice. I have an annotated textbook called Mathematics Emerging that is sort of cool. I would not pay anything near full price for it though.

A similar problem appears in James Munkres' Topology: A First Course:

• #2, section 8-5:
  
  Show that the paths f and h in Example 2 of §8-1 are not path homotopic.
  
  [Note: those paths are given by the following. The paths f,h : [0,1] → R^(2)\{}, where
  
  f(s) = (cos πs, sin πs)
  
  h(s) = (cos πs, -sin πs),
  
  so f traces the semicircle of the unit circle in the upper half-plane, h traces the semicircle of the unit circle in the lower half-plane, both paths begin at (1,0) and end at (-1,0), and the ambient space is the punctured plane, missing precisely the origin.]
  
  Here, the problem is that you can't continuously deform the path given by f to that given by h without passing through the missing point (0,0).
  
  This exercise is in a section called "The Fundamental Group of the Punctured Plane", and it uses plenty of homotopy theory throughout the entire chapter. So while I wouldn't presume to suggest that it's impossible to find a non-homotopy theoretic proof, this is evidence that homotopy theory would present the most efficient tools for trying to tackle this.
  
  If there is some more elementary method, my first instinct was to use something like the Intermediate Value Theorem, perhaps applied to the modulus of F(θ,t). The only problem, though, is that I can't think of an appropriate function where the origin would be the unique point between different values of |F(θ,t)| or whatever other suitable alternative function there is.
  
  Sorry I couldn't provide more to help at this point.