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I took my first graduate course as an undergrad last semester and it seemed that graduate student exams are much more open ended. Not only do they require you actually know more stuff (i.e. remember the theorem, the general approach) but they also require a lot more creativity, time management and general problem solving smarts.

But flipping open a text at the graduate level (Birkhauser, Springer-Verlag type), not only are most books filled with theorems, lemmas, proofs etc from head to toe, covering a very large array of concepts, but the much needed exercise at the end of the chapter is replaced with a list of references. Where are the exercises? And the course I took was sort of commonly taught I believe - stochastic networks.

I can see how this can be very problematic as one climb up the academic ladder. How would you ever find a text on "lower dimensional topology of autonomous robotic system", "Brain chaos in semi-aquatic faunas" or "Renal-sarcomere interaction theory" (three completely made up course).

How should graduate students approach a subject or an emerging field where there exercises scarce aside from what is introduced during the lecture or in existing literature?

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First, let me reassure you that many graduate classes are similar to undergraduate classes, in that often there will be regular homeworks and plenty of exercises in the books, with undergrad style exams, just harder. Of course, not all of them are like this--some are more like what you might think of as seminars, but this is sometimes true for undergrad classes as well. It depends on both the level and the focus of the course.

As for how you find exercises, learning how to ask your own questions and find your own (typically terribly naive) understanding of a subject is a large part of becoming a researcher. In graduate school, you should have an advisor who can help direct you with things and may suggest exercises or warm-up problems for you.

To give more a practical answer:

  1. If a subject is fairly well studied, you should look for other books which may have exercises.
  2. If there are no exercises, you can make your own by trying to come up with your own proof, or working out things in special cases, such as computing examples.
  3. If it is a more recent area or specialized topic, where there are no great texts, sometimes there may be good online notes from a course or summer school someone has given.
  4. Lastly, I think a great way to learn a topic is to start writing your own notes on it, and this works well for advanced/specialized topics where exercises are hard to come by. Trying to explain something to someone (even yourself or a vague nonexistent reader) forces you to ask yourself basic questions about the topic, which helps you understand both the ideas involved and larger context.
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The purpose of an exercise is to have you start applying the material to a problem, which helps you learn it.

As a graduate student, you're going to have started working on applications of your own. If you want to start applying the material you are learning, why not try applying it to work that you are engaged with? You will be in a less-constrained environment and without clearly provided answers, but, well, that's practically the definition of the difference between undergraduate and graduate education...

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