Complete undergraduate bundle-pack

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.

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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.

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

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.

Here is another useful list.

This is a link to the Mathematics Programs offered at the University of Toronto (St. George):

http://www.artsandscience.utoronto.ca/ofr/archived/1213calendar/crs_mat.htm

A course number with a Y indicates a full year course (72 hrs of lecture) and a course number with H indicates a half year course (36 hrs of lecture):

First Year

MAT157Y1 – Analysis I
Text: Calculus by Spivak.
Used in the past: Principles of Mathematical Analysis by Rudin.

If you have never been exposed to abstract mathematics Spivak is probably better to go with. UofT has been teaching from Spivak’s for awhile now.

MAT240H1 & Mat247H1: Linear Algebra I & II
Text: Linear Algebra by Friedberg et al.
Used in the past: Linear Algebra Done Right by Axler.

Second Year

MAT257Y1 – Analysis II

Text – Analysis on Manifolds by Munkres
Used in th past: Calculus on Manifolds by Spivak

Go with Munkres on this one. Spivak is barely a little over 100 pages in length! So you can imagine how terse it is.

MAT267H1 – Advanced Ordinary Differential Equations
Text – Differential Equations, Dynamical Systems, & Introduction to Chaos by Hirsch et al. & Elementary Differential Equations by Boyce and DiPrima

Third Year

MAT347Y1 – Groups, Rings, & Fields
Text: Abstract Algebra by Dummit and Foote

MAT354H1 – Complex Analysis I
Text: Complex Analysis by Stein & Shakarchi.
Used in the past: Real and Complex Analysis by Rudin

MAT315H1 – Introduction to Number Theory
Text: An Introduction to the Theory of Numbers by Niven.
Used in the past: A Friendly Introduction to Number Theory by Silverman.

MAT344H1 – Introduction to Combinatorics
Text: Applied Combinatorics by Tucker

MAT327H1 – Introduction to Topology
Text: Topology by Munkres.

MAT357H1 – Real Analysis I
Text: Real Mathematical Analysis by Pugh.
Used in the past: Real and Complex Analysis by Rudin.

MAT363H1 – Introduction to Differential Geometry
Text: Elementary Differential Geometry by Pressley.

Fourth Year

A lot of these courses are cross listed so they’re actually graduate courses. Check here for texts and references:

http://www.math.toronto.edu/cms/tentative-2012-2013-graduate-courses-descriptions/

Hope this helps!

Chicago undergraduate mathematics bibliography


ELEMENTARY

This includes “high school topics” and first-year calculus. Contents

  • Algebra $(4)$
    • Gelfand/Shen, Algebra
    • Gelfand/Glagoleva/Shnol, Functions and graphs
    • Gelfand/Glagoleva/Kirillov, The method of coordinates
    • Cohen, Precalculus with unit circle trigonometry, and its content is here.
  • Geometry $(2)$
    • Euclid, The elements
    • Coxeter, Geometry revisited
  • Foundations $(1)$
    • Rucker, Infinity and the mind
  • Problem solving $(4)$
    • New Mathematical Library problem books
    • Larson, Problem solving through problems
    • Pólya, How to solve it
    • Pólya, Mathematics and plausible reasoning, I and II
  • Calculus $(6)$
    • Spivak, Calculus
    • Spivak, The hitchhiker’s guide to calculus
    • Hardy, A course of pure mathematics
    • Courant, Differential and integral calculus
    • Apostol, Calculus
    • Janusz, Calculus
  • Bridges to intermediate topics $(2) $
    • Smith, Introduction to mathematics: algebra and analysis
    • Johnson, Introduction to logic via numbers and sets

INTERMEDIATE

Roughly, general rather than specialized texts in higher mathematics. I would not hesitate to recommend any book here to honors second-years, but they might not find easy going in some of them.

  • Foundations $(5)$
    • Halmos, Naive set theory
    • Fraenkel, Abstract set theory
    • Ebbinghaus/Flum/Thomas, Mathematical logic
    • Enderton, A mathematical introduction to logic
    • Landau, Foundations of analysis
  • General abstract algebra $(7)$ (difficulty: $\color{orange}{\mathscr{m}}$oderate-$\color{red}{\mathscr{h}}$igher)
    • $\color{orange}{\mathscr{m}}$ – Dummit/Foote, Abstract algebra
    • $\color{orange}{\mathscr{m}}$ – Herstein, Topics in algebra
    • $\color{orange}{\mathscr{m}}$ – Artin, Algebra
    • $\color{red}{\mathscr{h}}$ – Jacobson, Basic algebra I
    • $\color{red}{\mathscr{h}}$ – Hungerford, Algebra
    • $\color{red}{\mathscr{h}}$ – Lang, Algebra
    • $\color{red}{\mathscr{h}}$ – Mac Lane/Birkhoff, Algebra
  • Linear algebra $(3)$
    • Halmos, Finite dimensional vector spaces
    • Curtis, Abstract linear algebra
    • Greub, Linear algebra and Multilinear algebra
  • Number theory $(5)$
    • Ireland/Rosen, A classical introduction to modern number theory
    • Burn, A pathway into number theory

      Hardy/Wright, Introduction to number theory

  • Combinatorics and discrete mathematics $(1)$

  • Real analysis $(10)$

  • Multivariable calculus $(2)$

  • Complex analysis $(5)$

  • Differential equations $(2)$

  • Point-set topology $(5)$

  • Differential geometry $(4)$

  • Classical geometry $(3) $


TO BE CONTINUED

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.