Should I do all the exercises in a textbook?
The problem sets that you usually get in a university course is a small fraction of the exercises in your textbook. Which raises a question: do you need to solve all the exercises from your textbook? Maybe the author just presents a lot of them so that your professor can choose those that fit his course.
I'm looking for a reason not to do all the exercises, or even only do the course's problem sets (a small fraction). For example, Pugh's "Analysis" claims to have more than 500 exercises—that's way to much; I think the bulk of them is not essential. The main idea of studying mathematics is to learn new mathematical concepts, and not get bogged down in routine computations.
What is the reason of the problem sets? Maybe just to check that you understand the concepts well. But maybe the problem sets are so small because they are meant to be checked by some person, otherwise they would be bigger.
EDIT: Especially it relates to the courses that you need as a prerequisite for the other much more important course. For instance, I need Linear Algebra as a prerequisite for Analysis. When I get to Analysis, I will do all the exercises. But I don't want to spend a lot of time on Linear Algebra—I understand the concepts, understand the proofs, and that's it.
I would say it somewhat depends on the level, but may be not that much actually. The following is how I did it myself when in self study mode, and no professor to oversee the progress.
As an undergrad, say in vector analysis calculus, I did all the theory problems and enough many of the computational exercise to feel confident that I can do the rest. At that stage if, upon reading another problem, I could see a way to do it, then I wouldn't bother unless the problem had some intrinsic appeal (at the time I didn't know whether I want to major in math or physics, so problems motivated by theoretical or celestial mechanics would often make the cut).
As a beginning grad student it was more or less the same way, but as the exercises were largely theoretical I ended up doing most of them (they were fun actually). Later on it depended. If I only needed to get a general idea of the material, or felt eager to get to the next chapter, I would only a few exercises and try to move on. If I skipped too many of the exercises I would start feeling rather lost a few chapters further down. Then it was time to try the problems in the preceding chapter. If I couldn't, then I would go back to the preceding chapter, and so forth. Doing this iteratively worked quite well for me.
Of course, some more advanced textbooks don't have exercises. Then you need to make them up yourself and otherwise apply whatever habits have worked for you in the past.
The preceding paragraph is kinda my main point. You need to find a way that works for you. Lower level textbooks offer more repetitive work, and you can cut some of that. But at your own peril!
You don't necessarily need to do all problems in your textbook, but you need to make sure that you can do them. This usually involves doing a reasonable sample to test yourself.
The exercises I explicitly give my students in their assignments are a minimal sample, and I always make it clear that they are not enough to master the subject. I also tell them what it means "to be able to do an exercise": it means to be able to do it without help, without looking at the textbook, in a reasonable amount of time, and correctly. I have learned from extensive experience that the last sentence is not obvious to a lot of university students.
Of course, the more advanced the course the less the previous paragraph applies. For more sophisticated subjects the exercises tend to be more complicated, and not just a direct application of the topics considered.
Uh,you paid for all of them,so why not at least try them all? Or as many as time allows.
I'm only partially being sarcastic here. We learn mathematics by doing mathematics.This is particularly true of analysis, where the concepts and methods are creative and require some ingenuity in attack. And that means developing experience with solving many different kinds of exercises, from routine computations to difficult proofs. You can read 100 books from cover to cover and have total recall-and I can garuntee you won't be able to do more then pass a standard exam without working at least some of the exercises.
My advice-do as many exercises as time allows. Also-if the exercises are asking you to just do tedious computations and/or restate definitions-then chances are you're not using the right book. If an exercise doesn't make you think about the question for at least a minute before you begin attempting to solve it,then it's going to be a useless exercise to do. Period.
And in closing-Pugh's book has a truly outstanding collection of exercises and I'd strongly advise you try as many of them as you have time for.
You should do as many exercises as you need to, and you should have sufficient self-awareness in relation to the subject to distinguish between "need" and "want".
Exercises are not an end in themselves. They are a means of learning the subject. If you understand a topic after solving a small number of exercises you can certainly skip the rest - perhaps come back to them later for revision. If you see that you can do an exercise without writing it down, then don't write it down. (But a word of warning: it's easy to fool yourself on this point. Maybe you should write it just in case.) On the other hand, if you finish all the exercises and still don't feel that you understand the topic, seek out some more. In some cases you will be able to repeat the exercises you have already done, changing the numbers to make them slightly different; in other cases you will need to ask your instructor for suggestions. If you finish a ridiculously large number of problems and still don't fully understand the topic, it may well be that your difficulty is not with the topic itself but with something in the background. In this case you should certainly seek advice from your instructor.
One final comment: once you have mastered a topic, you will still want to do the occasional exercise so as to keep yourself "in form". Think of the world's best athletes - once they're on top, do they stop training? Of course not - they keep it up so that they stay on top. Mathematics is not really very different from that.
Good luck!
I will be contrarian and say "Yes...do them all".
Of course someone could arbitrarily double the problems in a text. Or have a problem that is unsolvable. Or say what if I did 399 of 400 problems--is all lost versus doing 400 of 400? But these are nitpicking exceptions--you want to know a practical guide in spirit: (a) do all the problems or (b) do the teacher issued and much smaller subsection of problems.
Jaime Escalante (teacher in Stand and Deliver who got 70+ AP BC passing students from a ghetto LA school in the 80s!) thought problem volume was key factor. He even assigned extra volume above the text. "Miles build champions".
Note, that an obvious disadvantage of doing more problems is time, but it is also true that with problem volume, your facility and speed increases. This is a well known training effect. In many intellectual and physical trainings. If the problems become repetitive, than it is drill. But drill goes fastr. And drill has a benefit--we are not computers to get an algorithm once and know it forever. There is "muscle memory" in piano.
It also becomes less likely that you fool yourself into thinking you know what you don't, if you do them all rather than saying "got it".
In addition, you are less likely to omit a particular trick or the like. Consider that authors may differ in how much the cover different things. One may only have one problem with Chebychev polynomials or Bessel Equation. A competitor may have several or a whole small section of the written text plus many problems. If you do all problems with the ODE text that only has one of the CP or BE problems, you at least cover it once. If you skip around, may miss it. And good luck deducting it at the time, first time, under exam pressure, with zero pre-exposure!
[You may even want to consider if your book lacks problems of a certain type. It may lack some easier problems to get familiarity. Professors and advanced classes love more difficult or project-ish problems, especially when having graded, non-drill type homework in college. But you actually get a pedagogical benefit from plug and chug drill that may be lacking if the problems are all to intricate. Also, easy problems ease you in and make the harder ones not as hard. The converse can also be true that you lack enough hard problems. Or you may be lacking problems that are word problems or that are not word problems or that cover a specific area (e.g. calc book might skimp on related rates). But again this is a caveat. In general, especially for books that sell a lot, they will have ample problems and you will get what you need by working them all.]
Also note that if you are weak in algebra (or not even weak, but just not clockwork-like accurate) that doing lots of calc problems gives practice in the algebra as well as the differentiation and integration techniques your practice. Same is true into the higher and higher level subjects.
A few practical stories of people who worked all the problems and got the benefit:
Freeman Dyson worked all the ODE problems in an ODE book over winter break, teaching himself the subject. He enjoyed it and benefited from it. (Is a video on YT of him talking about it.)
Dick Feynman felt he needed to learn classical E&M better when developing QED. He worked every problem in a conventional E&M text. He also worked every problem in calc texts and in quantum theory.
Lars Onsanger did all the problems in Whitacker and Watson.
I worked every problem in 1980s AP calculus and chemistry texts. And crushed the APs and college entrance exams. Doing better than students with higher IQs. [And I am nothing like the gods in 1-3, but would make a strong case that both for mortals and gods, there is a benefit from doing all the problems.] It really was not that bad in time...just went fast and intense every night. It was also a beautiful feeling, that feeling of mastery. Very different from courses with a B or C or even a low A. That A+ feeling where you have made it your b#$*&. Courses stuck with me for decades and were useful even after leaving business world and going back to academia later in life. Courses where I did not not do all the problems didn't give that feeling at the time or stick with me for years.