Best way to learn maths at grad level

[xcavier]

Hi guys,

I wanted some feedback from people more experienced than I. First some background, my current research area is in physics/theoretical neurosci. I'm fairly solid with maths however, I probably didn't take as many undergrad courses as I should have especially the more 'pure' stuff. Having said that, I'm very keen to pick up a lot of it on my own.

The biggest issue I'm having is that dependent on the txtbook i use some of the books seem very sparing when explaining concepts (adv. undergrad/grad level books) which is understandable. It some times takes me a long time just to work through a couple of theorems; though I find the results very beautiful and interesting due to time restrictions I never quite feel its worth the investment to then spend more time hacking my way through some problems. I know that doing problems helps to consolidate info- my biggest gripe is that even if i spent another hour or two doing problems it would drastically slow down what is already a snail's pace crawl through the text; furthermore with a lot of other research to accomplish I'm not sure if its worth the investment in time.

I should also point out much of the maths I'm talking about here is not directly relevant to my research... yet. I just want to be more comfortable with a broad areas of maths so I don't hit a brickwall one day while doing research and left scratching my head- not to mention its really interesting in its own right. If anyone else has resolved this problem all advice would be appreciated!

 

[micromass]

What kind of math are you doing?? What textbooks are you using?

There's no magical thing to do to get graduate math. It will always be quite difficult and you need some time to get through it. The best thing you can do is to find a book that makes it as easy as possible for you.

 

[xcavier]

well, as an example I'm trying to read through 'differential equations and dynamical systems' by Lawrence Perko. A lot of the ideas I've covered before but some of proofs are pretty heavy going such as stable manifold and Hartman-grobman.

More generally, I've been trying to fill in the gaps of my knowledge in areas like differential geometry, measure theory, metric spaces etc.. I just wanted to find out how other grad students are covering this stuff eg. doing a lot of questions, just reading and trying to get their head around the material or maybe some more sophisticated method?

 

[micromass]

There are many good undergrad books on the topics you mention. Maybe you should read them first?

 

[lavinia]

well, as an example I'm trying to read through 'differential equations and dynamical systems' by Lawrence Perko. A lot of the ideas I've covered before but some of proofs are pretty heavy going such as stable manifold and Hartman-grobman.

More generally, I've been trying to fill in the gaps of my knowledge in areas like differential geometry, measure theory, metric spaces etc.. I just wanted to find out how other grad students are covering this stuff eg. doing a lot of questions, just reading and trying to get their head around the material or maybe some more sophisticated method?

for physics you need to know vector calculus, Complex variables, modern differential geometry, and a lot about PDEs ... for starters.

I would take an advanced Physics book and see what math they are using then go back and learn it.

 

[xcavier]

yeah, basically I'm trying to cover some of the undergrad txtbooks but most of them are pitched at the advanced undergrad/first year grad level especially in topics like diff geometry and metric spaces. But its not so much what to learn (I've had a solid perusal through some of the physics txtbook stuff and 'know what i don't know' :) rather I wanted to ask for advice on appropriate method in learning it from the perspective of someone who wants a good b/g knowledge of the maths areas- is it more appropriate to spend a lot of time working through questions (time intensive) or read through the texts understanding the proofs and gaining a good overview of the maths (less time intensive, but also less solid in terms of reinforcing the knowledge; but the caveat is that if I'm not currently directly using the maths it'll prob go after a few weeks anyway?)

 

[chiro]

Hey xcavier and welcome to the forums.

I don't know about you, but for me staring blankly at proofs and knocking yourself out from hitting your head against a brick wall is pretty common.

In terms of research, my advice is to get away from it every now and again. Read something else and follow your instinct. You would be so surprised how the most unrelated activity suddenly gives you an idea.

I'm not sure if this is God's attempt at humor (like they are teasing us and laughing at us like we were some big brother experiment), but none-the-less, it has a habit of being this way.

[Also god if you are reading this, I ain't dissing you bro :D]

 

[chiro]

[Double post]