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.
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!
ELEMENTARY
This includes “high school topics” and first-year calculus. Contents
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.
Burn, A pathway into number theory
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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.