Hello Edin_Dzeko
Just so you know I am an undergraduate (will be graduating soon enough though), and I do self-study myself in a few different areas of mathematics.
Edin_Dzeko said:
[1] When self studying, do you memorize vocabulary words and things like that? This isn't like taking a course where the teacher tells you what to memorize and what will be on the test. When self studying how do you separate the stuff? Or do you literally study the material "inside out" as in memorize and know EVERYTHING?
As for point 1, just for myself, I do not expect the master the material that I am self-studying. At the moment I get a book on a particular area and I look for the specific stuff that I need in order to work on my research.
I will not know how to prove all (or even a handful!) of the theorems, know all the terms, and any of that. The only thing that I should hopefully end up knowing, is how to apply that specific piece of mathematics to my research.
If you have different goals, then I suggest you evaluate those goals in some context. If you want to master a specific area, then it might be useful for you to either take some coursework in that area, or do graduate study and then later work on that as part of your career if possible. If however you want to study something with a particular application in mind, then by all means go that route.
[2] Suppose you're self studying at a time when school is in session and the subjects you're studying, you may not be taking that course in school. How do you juggle school and self study? It's really hard to do homework when you'll rather be reading philosophy or music theory.
You just have to put school first. I've had problems myself in the past of getting my priorities right and have suffered because of that.
What you might want to do is find a faculty member who is either in the field or knows a little about the field you are interested in. They might end up suggesting a course that is on offer, or an invitation to do research, and this would be ideal for you since you are killing two birds with one stone: it is part of your coursework, and it's what you want to do!
[3] How do you control yourself from being arrogant, a prick, and annoying everyone because of what and how much you know? I have heard stories of straight 4.0 students who knew the material inside out and would text in-front of the teacher and other stuff just because they knew that they could destroy any exam or question on the material being taught. Basically, how to humble yourself and not feel like you're above everybody else because you have learned so much through self study. "use your powers for good" :lol:
This is an easy question to answer: do research, or harder coursework.
If you end up finding really really abstract high level coursework and research easy, then maybe you deserve to be arrogant (if you do I'm sure many experience members here will put you to the test).
The best way to humble yourself is to face reality. You will find it gets hard, and that you struggle in some ways just like everyone else.
Also with research this is common. I can't count the number of dead-ends I have had, all the stupid mistakes, all the idiotic times where I have tried the same thing over and over and over again (I believe they call this "insanity").
If you get a 4.0 in your undergraduate, graduate, and qualifiers, solve all the millennium problems, and win the Fields medal by 25, I would say you deserve to be an arrogant prick telling everyone how easy this stuff is. [If you do, put in a good word for me, ;)]
Also another way to humble yourself is to be with a diverse lot of people. You'll find that in some subjects these other people will be extremely wise in some aspect of human life that isn't mathematics or science.
Find people who are diverse in their interests, and that are secure with themselves. If you want to become humble, you will want to try different things even when you know that you will probably screw up for the first time: you will value that experience, and you will never get enough of it.
You are definitely in one of the better forums to learn in a wide range of areas, and many people here are definitely good role models to analyze and observe.