Do exercises in a theoretical reference book need to be solved when doing research? I am a first year graduate student in a computational math program. Based on my background (I just finished a one-semester graduate real analysis course), instead of reading a specific textbook, my supervisor suggested me starting reading papers. And if I find unclear concepts, I can refer to some books in library, learn the specific knowledge and come back to the paper.

In general I agree with this method since I think this is the most effective way to learn a new technique, that is, applying the new knowledge directly to my research. But I am not sure what I should do if it's a pure math concept, instead of a numerical scheme. For example, say the existence of weak solution of a particular PDE. After reading the relavant chapter or chapters of a classic book which I borrowed from the library, should I try to do the exercise after those chapters before moving back to the paper? Based on the suggestions here, I should try to solve as many as exercises in that book to make sure I understand the theorems and techniques, and this is what I usually do in my undergraduate study. 

But I have several concerns about this approach. Firstly, it may be time-consuming and may delay the research process. Secondly, unlike reading an undergraduate textbook, I started the reference book in the middle, while the exercises may require some previous chapters' techiniques,  which I may not know and may not be directly related with my current research. 

So may you share your experience about how to deal with this senario? Do you come back to the paper immediately (say after knowing the statement of a theorem) or do you spend some time solving exercises? If the exercises involving previous chapters' concepts, do you usually read previous chapters as well or do you just skip those exercises? I know it's good to learn more things, but given the time constrains and tons of things I need to learn, sometime it may not be practical.
You are definitely looking at the wrong factor here. The question for you should not be "Should I solve the examples?", but rather "Do I understand the technique well enough to confidently apply it to my own research?".

If the answer to the second question is yes, you don't need to waste your time on doing more examples. If the answer is no, you need to study the technique more before applying it, and doing the examples may or may not be a good way to do it. However, in the second case, doing the examples cannot be "too time-consuming", as you will need to spend more time on learning the technique anyway. At the end of the day, it isn't about "solving as many examples as possible", but about understanding the material well enough. As a young researcher, you should be advanced enough to tell when this is the case.

That being said, the way how you phrased the question makes me wonder to what extend you actually understand the technique. Specifically:


  I started the reference book in the middle, while the exercises may require some previous chapters' techiniques, which I may not know and may not be directly related with my current research.


Of course one needs to know the concrete example to be sure, but if you are unable to do the examples because you have not read the previous text book sections, it seems to me that your understanding of the overall area is not all that great yet.