How to Self Study Rigorous Texts?

[camdenreslink]

I am an undergrad that will be a sophomore in the fall, and I've completed the first level class in calc-based classical mechanics, as well as Calc I and Calc II.

The math presented to me was a typical plug-n-chug, cookbook, focus on techniques rather than concepts style. In short, I don't understand anything about calculus despite getting A's in the classes.

I'm a mechanical engineering major that wants a solid mathematical foundation to take with me into higher level classes and hopefully grad school.

I'm starting with Apostol vol.1, and I've realized I haven't had a single exposure to a rigorous text of this type before. All of my required textbooks have been watered down nonsense that don't teach you much...

So...what should I do to best understand the material?

Take notes on everything I think might be significant?
Focus on proofs only?
Try to prove proofs by myself before I look at them?
Study the proofs only until I can work out the problems?

There seems to be a theorem-proof-example-problem type of style that I have had yet to encounter.

How do you guys organize your self studies?
Do you keep an organized binder with separate sections for problems and notes, or do you just use a spiral notebook and ignore keeping super organized about it?

 

[kntsy]

I also want to know some methods in self study. I am going to self study mathematical analysis, abstract algebra and complex analysis this summer.
However, i think it is quite difficult to prove theorems myself.
I also don't have much motivation in doing the exercises.
Basically i just read the books.

 

[ohubrismine]

The most important elements in mathematics are definitions, axioms, theorems, and logic. Make sure you thoroughly understand the definitions. Theorems are constructed from definitions, axioms, other theorems, and a clear logic. It is not necessary that you be able to construct all the proofs, but make sure you can follow the logic. Also note that many higher mathematics texts will take a lot of material as primitive or axiomatic. That is, it will be considered obvious and not requireing proof.

 

[morgantw]

When I study (math specifically)I find it useful to goto a book store (where you can study and are exposed to many books) or a library and find as many different books as you can on the subject your studying and find an explanation that you can grasp. People teach in different ways, books are written different ways, and people learn in different ways. If you expose yourself to different styles or methods of what your trying to figure out, your bound to find a style/method/technique that may work well or just "click" with you that you can save for your mathematical/physics tool box. It has made understanding and passing classes much easier for me...

 

[Tedjn]

My $0.02. Don't try to do too much at once. A good, challenging text necessarily means that you will work through it slowly. You need time to think, absorb, and review. Also, don't be afraid to jump around in the text as your interest takes you.

 

[some_dude]

Move at a really, really slow pace. I've been burned many times trying to rush through difficult (for me) texts. I know the feeling so well I can't stress that enough. If you rush through proofs or just trying to "get through it", you'll miss all these little insights and it will compound, and you'll likely wind up feeling overwhelmed and not enjoying it.

 

[Gold Hydra]

What you should do I go though things very slow and make any type of graph, chart, e.t.c because they can help a lot.