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I wonder if anyone of you have an experience that you take a course like linear algebra or multivariable calculus in a semester and when you go to the second semester, you find many special detail has faded out and just the general idea remained. I have such experience and i indeed take linear algebra and multivariable calculus 2 semester ago and in the coming one, i am about to take a more advanced course, something about some differential form ( we will use the Analysis on Manifold as the textbook) and it seems learning differential form involves a lot Linear algebra and multivariable calculus so i want to ask how you guys review the content you need? would you just borrow a book and read again all the contents? If you don't even have that experience, do you have any way to suggest the way how you study which may help us to refrain from forgetting those content easily.

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From my experience, there is always an unjustified overconfidence in material that has previously been learnt that makes me take revising less seriously than I should. I've done this occasionally but I think the best way to review is to try and write out a complete set of notes directly from memory, with little consultation as possible. I find myself stumbling almost immediately when doing so. And I try to do lots of questions from textbooks related to the subject. It really takes a lot of time though. –  E.Lim Dec 23 '12 at 7:52
    
@E.Lim Do you mean to say "revising" or "reviewing"? –  Doug Spoonwood Dec 23 '12 at 10:43
    
@Doug Hmmm, there's probably a nontrivial distinction that can be made between those two terms, but I'll just say that I see no problem (barring time constraints) in reviewing the same way as one revises. –  E.Lim Dec 23 '12 at 16:55

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It is of course a very personal question. Here are a few things that I think are important.

When you study, focus on understanding rather than on memorizing. Whatever you memorize will be gone shortly ater the test is over. What you understand will stay with you forever (or at least almost that long). When that is done, encountering past material in more advanced courses becomes itself a kind of review as well as shedding new light on old subjects.

Of course, from time to time, there is a need to review material (whether if forgotten or because the details were intricate and you forgot a technique or two). That is easily done today by googling whatever it is you need and finding a wealth of resources to allow you to refresh your memory.

If you anticipate (correctly) that you will need lots of linear algebra in multivariable calculus and differential forms and you wish to review linear algebra, try to find a text specifically on multivariable calculus and/or differential forms that includes a lengthy introductory section on the linear algebra content needed. That way you will actually be reviewing the relevant material.

Reviewing for an exam is a different thing of course and greatly depends on the course and the form of the exam. Again, it is very personal and you will need to develop your own strategy after figuring out your strengths and weaknesses. I found that solving tons of problems and going through two of three relevant books worked well for me for most course I took.

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