Complete undergraduate bundle-pack [closed]
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!