How do/should you study math? (graduate level)

[sabbthdaylake]

greetings,

when i read textbooks, i always make sure to work out every single detail of every single statement and proof.

i'm in my first year of graduate school, and i find that sometimes i can't do this, either because the book I'm working on isn't very good, or because i don't have enough time.

for example, I'm currently up to page 117 of rotman's algebraic topology text. (rotman, as some of you might know, is notorious for typos and errata in his algebra texts. I'm actually compiling a list for him for errors I've found in this text.) I'm starting to not care about some of the combinatorial details of barycentric subdivision. i want to just get to the applications to euclidean space, and frankly, the qualifying exam probably won't ask about said details.

some textbooks i can just breeze through (e.g. munkres' book on manifolds, and his book on topology) while doing most of the problems and reading everything thoroughly. other books aren't as well written, though. (i've been unfortunate to have come across some of them in my first year, wasting tons of my time. this is a whole other issue.)

my question is: how do you and how should you study? should you skip over details, or do should you grind out each and every single detail or most details? i used to be idealistic and think if i don't see every detail, then i shouldn't be permitted to use results hinging on them. for instance, if I'm skipping a couple of details on barycentric subdivision, then should i be able to use invariance of domain?

it seems to me a question of balancing efficiency and honesty.

 

[andytoh]

I take nothing for granted. For every theorem, I work out every proof and fill in all gaps. Depending on the textbook, this means writing up to 5 pages of detail for every page I read.