proper way to approach learning higher (university level) mathematics

Question

I have been struggling to self-learn some somewhat higher mathematics- mostly university level mathematics. However, I've looked up other questions and they didn't mostly line up with what I am personally struggling with.

The problem I am facing simply put is that most of the math books at this level rely on proofs and examples more than tedious repetitive- mostly computational- exercises at the end of each section/chapter like in earlier math subjects. Problem is, I feel frustrated that I can't verify my solutions OR WORSE reproduce the proofs I analyze and study. Sure I can understand the proofs just fine (most of the time), but revising them the next day feels harder let alone reproducing them without taking a peek. I feel hesitant on whether I should take the material as is and move on to the next section/chapter or fret over the material until it becomes a second nature (probably takes weeks and highly inefficient; one page per week at worst).

How do you think I should approach this without feeling like I am skipping or just spending my time inefficiently? please advise.

Note: English isn't my first language so when I write proofs I tend to be redundant and less compliant to the common writing formats.

Answer 1

Advanced mathematics tends not to have "tedious repetitive - mostly computational- exercises at the end of each section/chapter". Rereading abstract proofs is probably not a good way to use your time. Spend as much as you can on the examples. Take each theorem and see what it says about all the examples you know to which it applies. See how the proof works in each particular case. Get several books and mine them for more examples. The ones that occur in all the books will be the important ones. The rarer examples will be instructive too.

Study counterexamples too. See how theorems fail when one or more of the hypotheses is false.

You need not master section $n$ before reading ahead into section $n+1$. You can and should circle back from time to time. The earlier material will probably become clearer that way.

(Your English here is just fine.)

Answer 2

University level...I think it depends on your current nation and your field of study as well.

For example, I learned in my engineering department in the university in Republic of Korea these subjects;

ODE: ordinary differential equation

PDE: partial differential equation

One of books widely used in the engineering course is Advanced Engineering Mathematics written by Erwin Kreyszig. This book could show you what subjects are taught in the university around the world.

Answer 3

I do think reading and understanding proofs is an important part of progressing, so this is something you should keep doing. I think spending time memorizing them is generally not as useful as understanding them once, and returning to them when you have a reason to. This might happen if you were working on a problem where you thought a method used in a certain proof you'd read might be useful.

You are right to be concerned about checking your solutions to exercises. There are a couple of possibilities that spring to mind.

The first is to get help. This could be office hours with a professor or TA (once you're in university), or tutoring by a graduate student or other person who is skilled in higher mathematics. A good teacher will often be able to look at your work and, more quickly than you might imagine, clear up some of the misunderstandings you have or help you see ideas you may have overlooked. The experience of learning one-on-one is entirely different than going to a lecture. Even a couple of hours a week could be useful.

The second is to use problem books or textbooks with solutions manuals. Of course, it's not always the case that comparing a correct solution with your own incorrect one will show why yours is incorrect. But this does happen often enough to make this a useful approach. Quite often, you will see that your own solution was generally along the right lines, but you may have missed some subtlety, left out some case that needed to be examined, or made a computational error. Not all solutions manuals are written well (especially when they're outsourced to students), so you need to be discerning.

About whether you should be 100% proficient on one chapter before moving on to the next, there is no absolute rule. It's often possible to move on, but if you reach a point where you are no longer understanding the proofs of the theorems in the text, it may be that you've gone too far. This is partly dependent on the book you're using.