This may be an unusual question: I have soft question on how to finalize my course plan in my senior year. Here's a brief background:
I am an Economics and a Mathematics major who has been taking a number of physics courses, and I wish to attend graduate school in physics with a theoretical/mathematical concentration. I am currently enrolled in 4 physics courses, along with an additional non-physics course. One of these courses is a graduate course in electrodynamics, covering potential theory, magnetostatics and EM waves. If I keep all these 4 courses, I'll graduate with 13 physics courses. After taking a year off, I'd like to apply to a theoretical physics program.
I'm thinking of the dropping the electrodynamics course and going with the 4 courses: the non-physics one and general relativity, quantum optics and quantum information. I'm also informally studying some pure math with an instructor, which I continue to do over the next 1.5 years, as I eventually wish to specialize in a theoretical concentration. For instance, I intend to study Lebesgue integrals, Analysis of several variables by the end of the term, prior to starting the study of manifolds hopefully in the summer.
My concerns are:
Given I am studying material for which I am not getting any credit (self-studying), should I consider dropping the course on electromagnetism. For one, will it hurt my chances since I'm not a physics major and I'll only have 12 courses. The reason I wish to drop the course is because I'd rather first study ordinary and partial differential equations outside of class (can't take a course) and then study E&M properly.
The reason I'm asking this question because as you can see due to unfortunate circumstances, I am emphasizing self-studying material outside of the class the way I wish to study it, and which sadly isn't done in most classrooms. Case in point, studying pure math with the math instructor. But I'll hopefully get a letter from her to take that into account.
Given my slightly weird preferences (I suppose), should I consider dropping the course based on the aforementioned concerns. I'm not sure if this decision can come back and haunt me. Or to put this in another way: should I be focusing more on studying material outside of the class in my last semester, or is that a futile exercise that won't help me much -- ergo, I should sit tight and take the physics course.
I hope the question isn't out of context. I placed this question on this forum as well because as you can see, I'm self-studying a lot of math apparently! Any and all advice would be appreciated.
P.S: I chose not to enroll in independent studies based on the material I'm covering outside of class is because the pure math instructor, with whom I am studying, said as long as credits aren't an issue, then there's no need to enroll in such independent studies. I presume she said that because I told her I'll be interested in getting a letter from her anyway. So I have decided not to enroll in an independent study and hope her recommendation letter, rather simply a Pass or Fail grade for the Independent study listed in the transcript, will do the trick
First:
Dropping the grad course in E&M is fine since no one needs you to have grad school courses done to get into grad school. (Not to be mean, but...well duh!)
Getting into US grad school, even a better one, in the sciences is not especially difficult, especially if you are American born. There is a lot of money that goes into the sciences for research. And they need bodies to go work on it. Just beware the employment market after. It is shi...awful (technical term).
Allowing people into grad school, coming in off major is not that crazy (frequent in material science for instance), but it is still not as common in physics. Still, write your essay (or an accompanying letter) to explain what you plan to do. For instance, if you are going to study EE or optics type stuff and you apply to a lot of faculty that do research on this topic, they may be willing to look the other way on the gaps in your background (presumably classical mechanics, stat mech) because they have a spot where you will do fine.
Note, this implies you should look at the school you apply to, not just as a generic grad school, but in terms of what they emphasize. Yeah, every decent research university will have everything, but in reality, they end up with slants, at least on a quantitative basis, number of professors per subtopic.
As with many questions on stack, you don't give enough information to really let us help you. You don't say what physics ug courses you have taken/not taken. Knowing your gaps would help us to advise you. Even just surveying them would help you to think this out yourself.
It sounds like a pretty full schedule and some tough courses. I would drop the grad E&M course.
Self-study is fine. Can even be time efficient. But be disciplined about it. Use a text that is reasonable (classical coverage, not too difficult if possible). Do the homework problems. And then in your letter be able to say something like "worked through all the problems and chapters of book X". Yeah, there will be some hardasses that won't give you credit even for that. But it will mean something to some. Much more than if you just say "self studied" and keep it vague and general.
I'm in shock that you have not had an ODE course as a math major (and are looking at very tough physics courses and the like). Even for econ! I mean what is elasticity of supply or demand? A very simple differential equation. The school I went to (a "trade school" in Maryland) makes the English majors take ODEs.
Go get a copy of Tenenbaum and Pollard and work it from cover to cover. After doing that you should be able to figuratively beat anyone over the head with your ODE mastery (or literally, book is fat, spine is hard). Talking about really advanced calculus stuff (Lebesgue) when you don't have the basic toolkit of ODEs makes little sense.
Good luck, buddy!