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Just curious about how people usually self study these subjects.

1) Is it the most efficient to read through the chapters, and write down theorems, definitions, and take notes of important parts and work out all the proofs and examples? And then make sure you can re-prove everything after finishing a chapter? Like for example, studying analysis, topology, etc. And the same with physics. Or is it better to just do all the problems and move on? What would be the most efficient method of studying considering time and being able to understand it 100%? Like, I know it's good to use multiple books too, and the optimal method would be to work out everything, but it seems too time-consuming to be able to learn enough and go into the research field and publishing faster.

2) And how long does it usually take one to finish a textbook? What are the advantages to taking a class than self-studying assuming one would put in as much work as needed if one had to take a final exam?

3) And where exactly does talent play in the process? Is it just about understanding things quicker and applying it more efficiently? I find that I can do math and physics with relative ease; are there people with varying degrees of talent and how pertinent is it to have the "most" talent in terms of being one of the top in math and physics research career?

4) By the way, just one more question, how important is like physics or math competitions to physics or math careers in research? I only got interested in these subjects very recently, so although they were my best subjects in school and I understood them easily, I never really practiced for olympiads or anything. And the only thing I did regarding competitions was math team and contests like ARML.

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3) Talent can help a little, but research is more about hard work: terrytao.wordpress.com/career-advice/… 4) Contest math is almost completely irrelevant to research. Research takes a long time, nobody currently knows the answer to your research problem, and you are free to consult sources. Contests have a short time limit, you are typically allowed no references, a known answer exists, and moreover many contest problems tend to revolve around some very particular clever trick. –  Henry T. Horton Aug 19 '13 at 3:25
    
So contests are mostly about practice and experience? Does talent mostly help in understanding and being able to apply new concepts easier? What do you think about my other questions? thanks. –  Benny Chang Aug 19 '13 at 4:18
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1 Answer

  1. The most efficient way to learn something is to teach/explain it to a layman.
    (e.g. answering question on MSE). The process forces you to organize your thoughts and identify the key ingredients and connections for the topic at hand. This leaves a deeper imprint on your neural network. Creating your own notebook, revisit it frequently and regularly drawing pictures to summarize the relationship also helps.

  2. The time to finish a textbook strongly depends on your age. When I'm a teenage, I can finish a math/physics textbook in a day but now it will probably take me more than a few weeks. The key is to learn math/physics as early as possible when your brain is still fresh. The advantage of taking a class is it help you to identify the important pieces in the material and help you to remember it longer (assume you are lucky and don't have a sleep-inducing professor).

  3. No comment.

  4. Once you get into research, math/physics competitions becomes completely irrelevant. People judge you by what sort of result you can produce. However, it does help you in getting accepted by a University. It also give you an edge when you apply for a non-academic job that value math skill (e.g. financial industry).

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Is it really possible to finish a text in one day? I've been working on Hubbard and each section takes at least 3-4 hours to write all the important things out and go through all the problems –  Benny Chang Aug 19 '13 at 15:33
    
@BennyChang it depends on the thickness of the book, the level of the material and how good does the author present it. One advantage of youth is one has a much better short-term memory. For elementary/introductory text book, a lot of statement/problem becomes obvious if you remember the hints in previous Chapter. As we age, we lost that advantage. That's why creating your personal notebook is useful. It help you to mark down the important concepts which you will need in later part of your reading. –  achille hui Aug 19 '13 at 15:59
    
+1 I deleted my answer, as this is better. –  user83622 Aug 19 '13 at 21:49
    
Thanks for the thorough answer. Is there a specific method to studying these subjects in general that's usually more efficient and helps to understand and retain the most? And any other opinions on 3? –  Benny Chang Aug 19 '13 at 23:06
    
@BennyChang, Nothing specific I can think of. If you are lucky and can find someone around same level of sophistication as you, then frequent exchange of ideas / intuitions / understandings will make the process more interesting and effects long lasting. No comment about 3. Independent of how much talent you have, one should really do math/physics simply because you find it interesting. –  achille hui Aug 20 '13 at 9:13
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