Developing my mathematical background

[bjnartowt]

Hi all, I have a question. I'm hoping to be a theoretical physicist, but I'm finding I really don't know the first thing about mathematics. I can plug and chug with the best of them, and I have a feel for various integration techniques, and I know enough tricks to get by (Fourier series, Fourier integrals, what a group is, how to solve a differential equation, numerical techniques, what eigenvalues and eigenvectors are, Hermitian conjugation, etc.). I'm kind of OK with complex numbers and functions too. I'm probably at the level where I could solve most all problems in Boas, but I still lack background.

To develop my background, I'm not sure what book to pick up first: topology? complex analysis? linear algebra? partial differential equations? differential geometry? group theory? quaterions and rotations, SU(3), SU(2), etc.? eeek! so much stuff I don't know, and I never know when I'll see something in research that requires me knowing stuff about it. there's a long research career ahead of me (hoping to be in theoretical condensed matter physics), and I don't know where to start. Suggestions? What field should I start with? I'm guessing higher topics in linear algebra.

 

[qspeechc]

Hi bjnartowt.

Well the first thing I want to say to you is that it is extremely difficult to become a theoretical physicist. There are very few positions and many brilliant men and women that could fill those positions. Also, to be a theoretical physicist you have to brilliant at physics but also know a fair bit of modern mathematics and be good at it. All I am saying is don't set yourself up for disappointment, just be realistic.

The main difference between pure mathematics and mathematics for physics as you would find in Boas is the rigour. Do you know how to do proofs? If not then I recommend "How to Do Proofs" by Velleman.

I am not sure how much time you have to devote to mathematics, but some experience with basic epsilon-delta proofs of analysis will be very useful if you plan of studying very important topics for physicists like Complex Analysis, Differential Geometry and Functional Analysis. There are many books that can teach you this but I am not sure which to recommend to you. Perhaps a short book like Ross's "Elementary Analysis" or Lay's "Analysis" (if you read this you do not need to read Velleman because this book teaches you how to do proofs); you don't need a rigorous calculus book because you already know calculus.
Linear algebra is a must, but if you already know some linear algera it's difficult to recommend a book. Where did you learn linear algebra from? Because it your knowledge of it may be sufficient. If you did it at the level of Strang's book then it is probably enough.
How much complex analysis do you know? I'm not sure if rigorous complex analysis is necessary for a physics student. The book I used as a math student was Conway and I enjoyed it, but perhaps more suited for you would be "Complex Variables" by Ablowitz and Fokas, which I found very good help for my applied math courses before I had taken complex analysis-- and excellent book.
For functional analysis Kreyszig's "Introductory Functional Analysis with Applications" is the gold standard for physics students.

I have given the books in order that they should be read. This short list should be (more than?) enough to get you started. You should probably see what other people who are more knowedgeable of physics than me have to say also.