Complementary self-studies during my undergrad

[TubbaBlubba]

I've just made it through my freshman physics year (covering introductory Linear algebra, Mechanics, Single- and multivariable calculus, Thermodynamics, Wave Theory, and Scientific computing). Due to some health issues, I am essentially unable to attend lectures, but I am, to be frank, a gifted student and I have passed every course with top marks and commendations with at best moderate effort (although I do take it very seriously and put a lot of work into reports, projects and exams). I expect things to get a bit more difficult hence, but I suspect that I will feel that there is a lot of slack I could be picking up, and I want to make the most of my time (and I do have a lot of time!). Thus, I really want to do some indepth self-studies to prepare myself as well as I can, and I would really like some suggestions, both for areas of study and specific works.

The areas I'm most interested in, and depending on career viability hope to focus on, are on the theoretical side - general mathematical physics, quantum field theory, general relativity, etc. Currently over the summer, I'm working my way through Spivak's massive 5-band textbook on differential geometry, and also reading Feynman's second Lectures volume (since I have a lot of EM Field Theory next year, I might as well be prepared). I'm considering a more indepth look at multilinear algebra, complex analysis, and partial differential equations, and I feel like developing solid abilities in pure mathematics are the most sound use of my extra time for the moment, since that will make it much easier to handle more difficult ventures into physics later on.

Suggestions, comments, experience? Is my approach reasonable? Should I alter my priorities?

Thanks in advance!

/Leonard

 

[axmls]

While it doesn't have to be your main focus, be sure to get some experience programming if you haven't yet, particularly (from what I hear) if you plan on going into theory.

 

[pmr]

The content of your post already makes it evident that you're very smart. Explicitly stating that you're gifted and have top marks with "moderate effort" is, at the very best, superfluous.

 

[TubbaBlubba]

While it doesn't have to be your main focus, be sure to get some experience programming if you haven't yet, particularly (from what I hear) if you plan on going into theory.

Yeah, I've done a lot of Matlab, and I know a bit of C. I definitely intend to learn Fortran as well.

The content of your post already makes it evident that you're very smart. Explicitly stating that you're gifted and have top marks with "moderate effort" is, at the very best, superfluous.

Well, I just want to make it clear that I'm not neglecting my studies or barely getting by in favour of some kind of vanity project.

 

[cgk]

Yeah, I've done a lot of Matlab, and I know a bit of C. I definitely intend to learn Fortran as well.

Do not learn Fortran. That will do more harm than good.[1] Also, if you know one or two other languages with strong imperative elements, picking it up on the fly is a matter of one day to one week. I am not exaggerating.

The trick with programming is not to learn lots of languages, but to become really good in applying two of them. Good choices for these are a "glue" language like Matlab or Python (with scipy) and a "brick" language like C++, Java, and possibly also C. Extending your horizons by looking into different language classes (like functional ones) and learning the actual comp sci background (data structures and algorithms, programming theory, etc) will be required at some point to become good at programming, but for a start, just actually doing projects to solve actual problem, and actively working on improving the programs will be much more effective (ask yourself: "how could I have made this program better? Would a different structure have served it better?" "how could I have arrived at this program faster?" "should I have thought about the program or problem in a different way to make it easier?" etc).

[1] You never want to use Fortran in a new project. And in an old project, you need to keep to the style of using Fortran which they do. And there is a chance that this is a unholy mix of a few million lines of F77 and F90, possibly even with "fixed form" source and 6-letter variable names.Anyway, regarding your original post: Your choice of topics looks good. If you are into that, a bit of functional analysis or group theory might also be helpful later on (e.g., just do one Differential Geometry tome instead of five and put in some elementary group theory or functional analysis instead). But be prepared that in physics, the math you will see will often not be very strict, and sometimes simply wrong. You will need to either ignore this (concentrate on finding and understanding the physics in there, let the math follow), or fill in the gaps by yourself. Some techniques may look inelegant at first, but will turn out to work spectacularly well (e.g., Dirac notation in quantum mechanics or component/Einstein sum writing of multi-linear equations with "epsilon and delta tensors" (even just writing the word "delta tensor" pains me)). Also, in most cases high level math is not used, at least not in undergrad. E.g., it is quite unlikely that you will see differential forms and exterior products in an EM lecture, even though this is the most elegant way of formulating the base equations. And do not ever expect a check if a series summation is actually convergent...

Why am I saying this? Just be aware that the approach to math in math and physics is very different. I have seen very bright students picking up on high level math first, then not taking physics seriously due to the math slip ups (instead of concentrating on the physics in physics), and then failing physics due to this. Do not fall into this trap.

 

[TubbaBlubba]

Do not learn Fortran. That will do more harm than good.[1] Also, if you know one or two other languages with strong imperative elements, picking it up on the fly is a matter of one day to one week. I am not exaggerating.

The trick with programming is not to learn lots of languages, but to become really good in applying two of them. Good choices for these are a "glue" language like Matlab or Python (with scipy) and a "brick" language like C++, Java, and possibly also C. Extending your horizons by looking into different language classes (like functional ones) and learning the actual comp sci background (data structures and algorithms, programming theory, etc) will be required at some point to become good at programming, but for a start, just actually doing projects to solve actual problem, and actively working on improving the programs will be much more effective (ask yourself: "how could I have made this program better? Would a different structure have served it better?" "how could I have arrived at this program faster?" "should I have thought about the program or problem in a different way to make it easier?" etc).

[1] You never want to use Fortran in a new project. And in an old project, you need to keep to the style of using Fortran which they do. And there is a chance that this is a unholy mix of a few million lines of F77 and F90, possibly even with "fixed form" source and 6-letter variable names.

Thank you! Interesting, then I will put more time into C++ and maybe Python as well instead.

Anyway, regarding your original post: Your choice of topics looks good. If you are into that, a bit of functional analysis or group theory might also be helpful later on (e.g., just do one Differential Geometry tome instead of five and put in some elementary group theory or functional analysis instead). But be prepared that in physics, the math you will see will often not be very strict, and sometimes simply wrong. You will need to either ignore this (concentrate on finding and understanding the physics in there, let the math follow), or fill in the gaps by yourself. Some techniques may look inelegant at first, but will turn out to work spectacularly well (e.g., Dirac notation in quantum mechanics or component/Einstein sum writing of multi-linear equations with "epsilon and delta tensors" (even just writing the word "delta tensor" pains me)). Also, in most cases high level math is not used, at least not in undergrad. E.g., it is quite unlikely that you will see differential forms and exterior products in an EM lecture, even though this is the most elegant way of formulating the base equations. And do not ever expect a check if a series summation is actually convergent...

Why am I saying this? Just be aware that the approach to math in math and physics is very different. I have seen very bright students picking up on high level math first, then not taking physics seriously due to the math slip ups (instead of concentrating on the physics in physics), and then failing physics due to this. Do not fall into this trap.

Yes, this is something I have noticed a lot already (e.g. differentials in thermodynamics). I think a rigorous understanding of the mathematics will ultimately help me understand and accept the non-rigorous techniques used in physics, but I also know that at times math can obfuscate physical intuition. Hopefully, learning advanced math will allow me to concentrate on the physics in physics, and perhaps understanding the math will take a bit of the mystery out of the more unintuitive aspects of it. I find that Feynman's books, for all their flaws, are very good for learning physical intuition; his thought experiments and explanations of things like reversibility helped me a lot in thermodynamics

Thank you for your extensive and thoughtful reply!