I’m going to compile a nice list of lecture notes/books for select topics in math/physics, for anyone interested in going into pure math or theoretical physics. I will add to it periodically.

Logic, Language, Proofs, Set Theory, Functions, etc.

• “Introduction to Mathematical Reasoning” – Eccles
  This will give you a solid foundation, what we call “mathematical maturity”. Read this, and you’ll have the foundation necessary to delve into the more advanced (and more fun) stuff!

Single-Variable Calculus

• “Calculus” -Spivak
  This is a really mathematically rigorous treatment of calculus. It is heavy on proofs and uses sophisticated language, which is why I suggested reading through Eccles first.

Multivariable Calculus

• “Multivariable Calculus” -Larson & Edwards
  Basic intro to vector, vector-valued functions, partial differentiation, multiple integration, vector analysis, and related areas. Very easy to read.
• “Calculus on Manifolds” -Spivak
  This isn’t really a book on multivariable calculus, but you can get into it if you’re familiar with the concepts from Larson & Edwards and Eccles; a bit of analysis and/or topology would be helpful, but isn’t totally necessary. Really interesting stuff.
• “Div, Grad, Curl, and All That” -Schey
  A little book on some interesting applications of vector analysis; you should be familiar with all of Larson & Edwards, up until maybe the middle of Chapter 15, as well as some physics, to really get a lot out of this book. If you like physics or applied math, this is definitely a good book to read.

Linear Algebra:

• “Introduction to Linear Algebra” -Bretscher
  This is akin to Larson & Edwards. Easy to read, will give you a nice foundation in the concepts of linear algebra (more computational).
• “Linear Algebra Done Right” -Axler
  This book is really abstract, and should be used for a second course in Linear Algebra. Stemmed from an essay Axler wrote, called “Down with Determinants”; he doesn’t introduce the determinant of a matrix until the last chapter of the book, opting instead to really utilize the concept of a linear operator.

ODEs / PDEs:

As a foreword, I am not a fan of ODEs in the slightest. I won’t even provide a book, as I think it’s pointless to read a book on ODEs.

• MIT’s intro course on ODEs: http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/
  Honestly, go through this and you’ll be set. You can learn applications of ODEs in other classes.
• “Partial Differential Equations: An Introduction” -Strauss
  Once you know Multivariable Calculus and are familiar with ODEs, you can get into this book. I’m not familiar with PDEs, so I don’t really have anything else to say.
• “Partial Differential Equations” -Evans
  If you really want to get into a rigorous, high-level study of theory of PDEs, this is a good second course to read after you’ve learned some measure theory.

Analysis:

• “Principles of Mathematical Analysis” -Rudin
  The quintessential book on introductory analysis. This’ll give you a good basis for measure theory, grad-level complex analysis, and is probably good to read before Evans’ PDEs.

Topology:

• “Topology” -Munkres
  Quintessential book on point-set topology… should cover pretty much everything in your standard undergraduate course, and some graduate-level material too.
• “Topology of Surfaces” -Kinsey
  Nice intro to geometric topology, used in Stony Brook’s introductory course on topology.

  Note: I have no experience with algebraic or differential topology, so I don’t really know what books are good in those subjects. Math StackExchange should help, though.