Hello, I am currently a senior in High School (America). I am enrolled in AP Calculus BC, AP Physics B and C: Mechanics. I have taken 6 previous AP exams, (most importantly AP Chemistry with a score of 5). I am enthralled with the study of mathematics and physics and maintained a grade of 100 in both my physics and Calculus class (The average grade is a D to a C). Due to my success in these courses and my adoration for the subjects, I desire to continue my exploration in these topics. I self studied Calculus I and II in my junior year (thus leading to my success in AP Calculus). Despite my academic success in these areas, I fear that the academic courses in universities will be overwhelmingly more difficult than my AP courses. Therefore, I decided to begin self studying Multivariable calculus, Differential Equations, and Linear Algebra. I attend Differential Equations lectures at a local community college and have been watching the DE and Linear Algebra lectures at MIT via the OCW. My question is primarily this: What can I do to better prepare myself for a double major in mathematics and physics? Am I well enough prepared now? How well versed in mathematics and physics should I be before entering college? My physics course covers calculus based mechanics, noncalculus based electricity and magnetism, thermodynamics, optics, and waves, plus other areas. My Calculus BC course covers the typical areas of Calculus I and II (save for hyperbolic functions, a topic that I will be self studying.) What should I do to further prepare myself? I've heard stories of students with 5's on AP Calculus and AP Physics doing miserably in college. Any specific courses I should take? I ultimately plan to go for a Ph.D. in mathematics or physics (depening on how well I do in my undergraduate studies.) Lastly, are there any textbooks in particular I should have and study from? I have been studying Calculus out of Larson's Calculus I-III and physics out of Serway and Faughn. My physics teacher has a copy of Apostol's Calculus and Walker and Resnick's Physics; I'm sure I could borrow those for study if they are superior study aides. Thank you for your time and any advice.
I would consider you to be more than prepared for university. Students who do well in high school, but still fail university is probably because they never "understood" what was going on. If you feel like you understand what's going on and not merely memorizing problems and what not, I would say that you are fully prepared and shall have no worries of failure.
Just a tip. Try not to race forward into new subjects, but instead ground yourself more firmly in things you already know. You will find this to be far more useful once you start your university courses. For example, you say you've already covered Calculus I and II, then go get a rigorous calculus textbook like Spivak or Courant and try to go through it by proving all the theorems and doing a few of the exercises. (Doing all the exercises is by no means a simple task, especially with Spivak.) Apostol is also a good text, but I'm not too fond of it. Of course my advice is only good if you're considering going into a decent math program which emphasises theory much more than application.
Thank you for your advice. Indeed, I have strived and worked hard to elevate myself above mere memorization of "formulas", unlike my classmates. However, my biggest downfall is I have very very little experience with formal proofs. The only "proving" i've done is proving limits, proving derivatives (such as tan(x)^x or ln(x), e^x, sin(x),cos(x), etc.) I have no been tasked with formal proofs nor have I been exposed to any deep theory in Calculus, merely the information provided to me by my Larson's textbook, an old version of Calculus with Analytic Geometry by Thomas, and any work in my class. I will attempt to find a copy of Spivak's Calculus and attempt to do the proofs and all of the exercises, however, I am unaware if I will be able to find a copy in my area. The only Calculus texts I have are Larson's, Stewart's, Thomas's (printed in the 70s, so it's quite old), and I may be able to get a hold of my teachers copy of Apostol's Calculus. I wholeheartedly desire a strong comprehension of the Calculus, and am willing to do extra studying and exercise with single variable calculus. Would Apostol's text by equivalent or near close in rigor to Spivak's? One additional question: I wish to attain a Ph.D. in either the field of PHysics or Mathematics, and I hope to get a position as a professor. I became a physics and calculus tutor last year and learned that I love teaching people concepts in math and science, so the idea of a professor sounded like a perfect match. Any suggestions on how I could get to this dream job? Everyone, thank you for your advice.
Just remember in order to double major you will need to take twice the liberal art credits. So if you are required to take 26 credits of liberal arts to satisfy one major, you will need to take 52 total to double major.
I'm still an undergrad but from what I hear, if you want to get into a good grad school (since you wanna go for a PhD), keep your GPA up and Get some undergrad research experience under your belt. My university requires a course in undergrad research and I'm sure may others do too. Anyway, best advice I can offer: You are doing a double major in two of the hardest fields I can think of. If you keep you work ethic where it is, and keep your priorities straight (LOTS of people have trouble with this as freshman in college, you'll see)you'll be fine. Good luck to you!
I can't even imagine what kind of high schools some of you guys are coming out of. My high school didn't even offer AP classes, and about 50% of the class didn't even make it through all four years. Hell, we didn't even have a formal trigonometry class .
I can speak first-hand that this certainly is not the case for all schools. Nor most schools, from what I gather. It's usually just a core curriculum (sometimes not even that), and then a certain sequence of courses to complete each major.
If "liberal arts credits" means what I think of as "general education requirements", this is definitely not true everywhere. Where I teach, a student with a double major is subject to the same general education requirements (English, foreign language, history,etc.) as a student with a single major. In fact, a double major in math and physics is in principle fairly easy here, because a physics major requires so much math to begin with. At least it's easy in terms of number of hours. It's not like, say, physics and business, where there is practically no overlap in major requirements.
Yeah, we don't need to take them twice here either. I thought of doing a Physics and Mathematics major. It didn't seem difficult, but I went down the road of Mathematics because I enjoy it much more.
I received a double major in math and physics at UCLA, and I can definitively state I did not have to take double the number of general ed classes, liberal arts classes, or whatever one desires to call them. My only recommendation is that, as you progress through your classes, try to take your the following upper division math classes before going too far into the upper division physics: linear algebra and differential equations (ODE and PDE) - these will help immensely in both your EM and QM classes, as well as any fluid mechanics or acoustics; differential geometry - good for relativity; take a probability class - good for statistical mechanics; your vector calculus will give you a good grounding for analytic mechanics; I didn't take optics or solid state, but since EM is required before these (at least at UCLA), you should a solid footing for these classes. While a solid class in abstract algebra is always good, it didn't help (much) with any of my physics classes except for 2 graduate classes I took my final year.
I agree that are you exceptionally well-prepared for college already. In fact, I'd say you're in at least the 99th percentile, if not the 99.9th. Keep in mind that you're not going to get into multivariable or diff eq for a few semesters anyway, and you don't need to know the material beforehand to do well in the class! It certainly can't hurt to have some familiarity with the subject, but don't bother attempting to do it all before the class even starts. The reason why some people who make 5's on AP exams go on to do poorly in college is largely a matter of study skills. College courses are often faster-paced than high-school classes, and require a greater degree of self-sufficiency. Students who are poor self-learners or otherwise cannot bring themselves to do homework regularly will find college quite difficult. In many higher-level college courses, your only grades are a midterm and final -- the homework is not graded, yet, if you don't do it, you'll probably fail the class. Many students leave high-school with a notion that homework is trivial and unimportant, while in college it is most likely your primary learning tool. This doesn't sound like you (at all), however, so I would not be concerned. Congrats on your achievements in high school -- keep up the good work, but recognize that you're so far ahead of the curve that there's little you can do to get even further ahead. - Warren
This is true. I remember when i took Structural Analysis I, the teacher assigned all the problems in the book and never graded them, but if i hadn't made them, i bet i wouldn't know as much as i do now.
Thank you for your advice everyone. Thanks to AP, I should have around 50 credits or more by the end of this year! Hopefully this will allow me to focus on more math and physics in college rather than classes of less importance (in my opinion.) Are there any "must-get" books for Undergraduate math or physics? I was wondering what would be a good physics book for General Physics II (Electricity and Magnetism), Linear Algebra, and Differential Equations. I'm sure my college classes will suggest a book, but I generally have more than one book and study and do practice problems out of the better of the two. Thanks for any suggestions!
If you want to focus on mathematics and physics, shouldn't you be doing the humanities and social science courses you need to take? That's what I would do.
Quite so, I already planned this. I've taken AP Bio (5), AP US History (4), AP Stats (4, stupidly), AP World History (5), AP Language (5), and AP Chem (5). I'm currently taking (besides Calc and Physics) AP Macroeconomics, AP Politics, AP Literature, and AP Psychology. So, I shoudl have most of my humanities and social science credits done with. I already got English out of the way (Yay!). Assuming I get 4's and 5's on those other tests I'm taking this year, I think that should exempt me from all of my less important classes (humanities and other nonsense courses.)
Holy cow, that's a lot of AP's! I thought I had a lot with five of them! Are you trying to win one of those AP scholarships? If you haven't had experience with proofs I recommend that you take a course called Discrete Mathematics or its equivalent early on. And maybe you might want to add in a Logic course on the side, or get a beginning textbook in logic and study the formal part of it. That's what I did last year and it has served me well. Get into the habit of trying to prove every statement you can that the author makes, or at least understanding his proof. If you can prove everything the author says then you probably know the material.
You're telling me: I thought that my 9 AP tests were impressive. I took 3 my junior year of high school: US History (5), English Literature and Composition (4), and Physics B (5). When I was a senior, I took Latin Literature (4), Chemistry (5), Computer Science A (5), Macroeconomics (5), Microeconomics (5), and BC Calculus (5). Interestingly enough, the hardest of those was, by far, Latin. Basically, as preparation for the test, you have to translate the Aeneid in almost all of its entirety.
Wow; those are some nice AP Scores! From what I hear, the Latin test is indeed amongst the most difficult. Very impressive. With respect to proofs, I have been doing exactly that: when I see a book say "Proof", I just read the opening statement, and attempt to prove it myself. So far, I've been quite successful. I've been doing this out of my Calculus book (Larson) and my brother's "Advanced Engineering Mathematics" (O'Neill.) So far, so good. However, I lack the true "rigor" of a proof sometimes and I hope to pick up a book on logic and things of that form. Any suggestions on a good book for Set Theory, Analysis, and Proof learning? Thank you!