My reading list

Have you played a RPG where you can't open a normal looking chest at the start of the game if you want the ultimate weapon at the end of it? I think that the best approach to majoring in physics is the same thing. Now, my first semester in college looks like crap.

The road to theory of Lie groups

I'm cleared to take graduate courses shamelessly as an undergraduate by the head of department, which is nice, but probably makes me the only freshman physics major doing graduate math courses for some time... I feel kind of bad for my roommate. The courses in my first semester look like this (not in their exact names):

1. Group theory
2. Continuum modeling
3. Lebesgue integral properties and measure
4. Partial differential equations
5. Applications of complex variables
6. Complex analysis
7. Differential equations (boundary value problems) 
8. Continuum modeling (graduate level)
9. Honors differential equations
10. Geometry of manifolds
11. Abstract algebra (w/ Galois theory, rings, Abelian groups etc.)
12. Partial differential equations (applications of Hilbert spaces)
13. Partial differential equations (more methods)

It looks ridiculously long, but it makes sense if you know the prequisites - a much shorter list:

1. Single-variable calculus
2. Multivariable and vector calculus
3. Real analysis
4. Tensor calculus
5. Linear algebra
6. Differential equations

Most of these fall prey to my current reading list:

1. Spivak, Calculus
2. Courant & Fritz, Introduction to Calculus and Analysis I
3. Courant & Fritz, Introduction to Calculus and Analysis II
4. Apostol, Calculus Vol 1: One-variable Calculus with an Introduction to Linear Algebra
5. Apostol, Calculus Vol 2: Multi-Variable Calculus and Linear Algebra with Applications
6. Spiegel, Schaum's Outline of Advanced Calculus
7. Spivak, Calculus on Manifolds
8. Mattuck, Introduction to Analysis
9. Kolmogorov & Fomin, Introductory Real Analysis
10. Kolmogorov & Fomin, Elements of the Theory of Functions and Functional Analysis
11. Hoffman & Kunze, Linear Algebra
12. Lipschutz, Schaum's Outline of Linear Algebra
13. Tenenbaum & Pollard, Ordinary Differential Equations
14. Gallian, Contemporary Abstract Algebra

I only have NO idea where to get a good text on tensor calculus besides the appendices of some really good general relativity textbooks. Also, I frankly prefer Spivak over Apostol. My only complaint is that Spivak doesn't develop ODEs and linear algebra such that I can attack a second course in linear algebra - and even makes it rather hard to handle Hoffman & Kunze immediately. The problems in Spivak's book are really, really HARD, too, but enjoyable. The advantage, though, is that I will find Calculus on Manifolds and A Comprehensive Introduction to Differential Geometry more accessible, both of which are great books.

The two Schaum's Outlines are very easy to understand, and it's my style to complement a heavily theoretical book with Schaum's Outlines for computational problems, examples and a more application/engineering/plug-and-chuck style.

There's Tenenbaum & Pollard and Kolmogorov & Fomin, which are really cheap. I haven't touched them, but the reviews are really positive. I also enjoyed Mathematics for Physicists by Dennery & Krzywicki, so I have no qualms about buying Dover books - they are the absolute value for money. I know that Rudin's three books on analysis are classics, too, but they're seriously costly. I liked Prof Mattuck's lectures, making his book a must-have for me.

This brings me to my conclusion. College physics is a troublesome affair. You could encounter contour integrals and differential operators in as early as your second introductory course, whereas you could be doing some 5-6 courses before you finally encounter these through a course in advanced calculus (multivariate, vector etc.) Now, there are two quick tricks to go about this. First, calculus, methods in ordinary differential equations and linear algebra are at the root of everything. If you cracked these, you'll have unlocked the next courses, e.g. group theory, partial differential equations, real analysis etc. It is quite silly to have unlocked calculus and ODEs, but be unable to proceed with modeling because you forgot to take a course in linear algebra. I also learned this the hard way: I had no idea what was going on in Spivak's "Calculus on Manifolds" even though it was supposed to proceed immediately after his "Calculus" - because I wasn't adept with linear algebra.

There is another way - if you were exposed to everything from the ground up, i.e. analysis first instead of single variable calculus (like me), then it's really messy but it's tutte le strade portano a Roma... all roads lead to Rome Lie groups anyway...  and calculus becomes a free frag. Heck, fluid dynamics made me stare at differential operators like an idiot before I even knew what a cross product was.

Secondly, I think that it is better to invest the first few semesters entirely on mathematics. Be an undercover math major if you like.

Frankly, many courses interest me. Music, professional writing, two or three computing languages, numerical and statistical courses, signals and electrical engineering courses... But I will leave those for last.