Firstly I'd like to apologize if this is too much of an opinion-based question. I do not know where else would be appropriate for such a post, if there is such a place please let me know.
I am very interested in mathematics, particularly mathematical physics, but I currently do not have access to an institution or any professors. As a result I have mainly been self-teaching myself through textbooks. My usual routine is to very carefully go through a chapter in a book, making sure I understand every step in every proof, and then move onto the exercises. Going through the proofs this way helps me measure which theorems/results are more important, and also helps me learn common methods of proof. This part of my studying is going fine and I'm able to finish the chapter and understand most of the material, sometimes with the help of other books or online resources such as math stackexchange.
The problem I have been having is with the exercises. I use these as a way to reinforce the material and test my understanding, but also as a way to develop my problem solving skills. I very rarely am able to solve a problem in any graduate math textbook I have read (which I have already finished 2-3), and have trouble even getting started with them. Eventually if I make no progress I open a question on stackexchange, or try to find a solution online which I usually am able to follow but I would not have thought of myself. I want to get to a point where I can actually do these exercises and not just be able to understand the solutions. I am beginning to question if I'm just not cut out for it or if my approach is wrong.
In summary, I am able to read about these topics and understand proofs written by others, but I struggle on actively doing the math myself. I am looking for some advice on how to develop these problem solving skills and do better on the exercises. I have taken a look at similar questions on this site such as:
- Is solving all of the exercises in a textbook a good idea?
- What is a recommended strategy on exercises in a mathematical textbook at graduate level?
- How should I go about reading mathematics papers and textbooks as a PhD student
- Try exercises or look at the solutions for self-study
- First year Math PhD student; My problem solving skill has been completely atrophied and continues to decline
The advice given in the intersection of all these posts seems to be more practice and to get help/hints from your professor. As someone who is self-studying and often gets stuck on where to even begin, I am not sure how to apply this advice and I would love to hear what working mathematicians would advise. For example, when I don't know how to start or I am stuck what should I do? I often think about the problem for some time but get nowhere, only to find a clever trick or small lemma should have been applied.
For some reference on what kind of books I am studying, I am currently going through the following:
- Methods of Modern Mathematical Physics Vol 1. (Functional Analysis), by Reed & Simon
- Mathematical Gauge Theory by Hamilton
- Partial Differential Equations by Evans