Complete undergraduate bundle-pack

Question

First of all I'm sorry if this is not the right place to post this. I like math a lot. But I'm not sure if i want to do a math major in college. My question is: Can you give me a list of books that will give me the knowledge of the subjects a person doing a math major would have? I think I know all the stuff a good high school student knows. Thanks.

Answer 1

Using some of the recommendations Others gave me and the Stanford math major checklist I have made the following list: One should read all books corresponding to a subject (in order) not just one of them.. The first part is a requirement while in the second part students usually take at least 2 electives ( I give 4 examples).

Calculus:

Calculus by Michael Spivak

Calculus volumes 1 and 2 by Tom M.Apostol

Analysis

Principles of Mathematical Analysis by Walter Rudin

Real and complex analysis by Walter Rudin

Topology

Topology by James Munkres or

General Topology by Stephen Willard (harder)

Linear Algebra

Linear Algebra by Friedberg,Insel and Spence

Differential Equations:

Ordinary Differential Equations by Tenenbaum and Polland

Partial Differential equations by Lawrence C evans.

Algebra

Abstract Algebra by Dummit and Foote

Combinatorics

Introductory Combinatorics by Brualdi

Set theory:

Introduction to set theory by Hrbacek and Jech

Electives:

Algebraic Topology

Algebraic Topology: an introduction by W.S Massey

Algebraic Geometry

Undergraduate algebraic geometry by Miles Reid

Number theory:

An introduction to the theory of numbers by Hardy and Wright

Algebraic number Theory (If you also take Number theory)

Algebraic Theory of numbers by Pierre Samuel.

Answer 2

Here's one possible list.

Principles of Mathematical Analysis by Walter Rudin

Topology by James Munkres

Linear Algebra by Friedberg, Insel, and Spence

Abstract Algebra by Dummit and Foote

Answer 3

This is a blog which describes on how to be a pure mathematician. You can go through it and find out what all opportunities you have in various fields of mathematics.

Answer 4

I'm a bit unsure about this question, and its intent. But it is always important to have an idea of some ways to continue one's education.

One of my favorite, though undermentioned, resources is the Mathematics Autodidact's Guide, published by the AMS. It's a short pdf (linked here).

But FWIW, here is a list of the undergraduate math classes and their books I took and used, respectively, as an undergrad (this doesn't account for my self-study or the research bits that I did, but every budding mathematician must distinguish himself from the rest in some way or another):

Calculus (3 semesters):
Calculus in One and Several Variables by Salas, Hille, and Etgen
Vector Calculus by Marsden

Linear Algebra (2 semesters):
Carlen and Carvalho's terrible, terrible book
Linear Algebra by Apostol
Topics in Algebra by Herstein

Algebra (3 semesters):
Topics in Algebra by Herstein
Abstract Algebra by Dummit and Foote

Real Analysis (2 semesters): Intro to Real Analysis by Rosenlicht (great, though few know it)
Real Analysis by Bartle (this is intense, but flawed in that it doesn't do Lebesgue)
Advanced Calculus of Several Variables by Edwards (this was done with Bartle in one semester)

DE (2 semesters):
One of the Ordinary Differential Equations by Marsden (boring)
Calculus of Variations by Gelfand and Fomin

Probability (1 semester, thank god):
Intro to Probability by Hogg and Tanis

Combinatorics (1 semester):
Discrete Mathematics by Grimaldi

Graph Theory (1 semester):
Graph Theory by West (a great book)

Number Theory (2 semesters):
Elementary Number Theory by Rosen (doesn't require algebra)
Introduction to Modern Number Theory by Ireland and Rosen (different Rosen, famous book)
Davenport's Multiplicative Number Theory

Complex Analysis (2 semesters): Stein and Shakarchi's Complex book (part of their series on analysis) Conway's Functions of One Complex Variable

And then there were some electives in problem solving (using, e.g. Larson's Problem-Solving through Problems), game theory (Conway and Berlekamp's Winning Ways with your Mathematical Plays), additive number theory, etc. Find what interests you and follow it, I suppose.