I am a senior in high school who has taught myself Calculus BC and I got a 5. However, I have taken all the math I can at my school. I also taught myself multi-variable calculus and much of differential equations, and some real analysis.
I plan on finishing my study of differential equations before school starts in early September. I am currently working through Rudin's Principles of Mathematical Analysis and plan to finish that by the end of the first nine weeks of school. I have not had linear algebra, but I am a fast learner. What course progression would you recommend for the next 3 nine weeks, if I were to do a subject per nine weeks (I plan on spending a lot of time on this independent study as I am given in school time)? What textbooks would you recommend?
P.S. I am very interested in analysis and learning more about topology, but more generally pure mathematics.
Best Answer
I have to warn you that your estimate on the amount of time to finish Rudin (if done correctly) may be off.
Here's why. Up to now, you've taken the standard advanced course in high school mathematics and done quite well. This is a feat to be proud of, and unfortunately, you've done so well that you are a year ahead of the game. I say unfortunately, because the next natural step would be to take a proof based math class and learn the fundamental skill of writing clear, coherent mathematical proofs. It doesn't matter the subject through which this is done, but this is the step that should happen next.
The problem is this next step is difficult (if not detrimental) to take alone. You need someone to read your proofs, to make sure your arguments make sense and are understandable to another person, and to check that your sentences end in (goddamn) periods.
You can't do the exercises in Rudin (and for that matter learn basic analysis) without having the skills of proof writing. And for that reason, I advise you to try to find someone to help you acquire this skill. Here are three ideas.
(1) Where are you from? There may be math classes at a local university you can take and get credit for. This will have the added benefit that you will meet other people who like math. Talking about Math is a lot of fun. And while, many mathematicians learn a great deal through self study, it's typically in the context of a mathematically inclined environment. It might be surprising to learn how much of the stuff you think you know is wrong when there is someone there you try to explain it to.
(2) If that fails, try to find a correspondence course. This way you at least get feedback and keep the postal service afloat.
(3) Find a teacher at your school. Many (maybe all) were probably math majors at one point, and could read over your proofs and give feedback.
However, if none of these options are available, I would advise you to stick to the more computationally minded brand of mathematics that you have seen in calculus and differential equations. There are great treatments of linear algebra in this vein. Try Gilbert Strang's 'linear algebra and applications' which has an associated lecture series on MIT open course ware. Another option is to try to learn some programming. Java's great. And tackling a programming problem will stimulate you in a way you might have once thought was reserved only for mathematics.
If all else fails. Fly a kite, learn to surf, and prefect a secret BBQ sauce recipe. It's your last year of high school! Live It Up.