How to learn pure math and other stuff

[pawan28612]

hey guys,
iam an engineering student taking a year off (after my first year) and i was trying to teach myself linear algebra and set theory from a mathematical point of view
( the way that linear is taught to engineers is frankly an insult to the subject) and i find that i keep getting stuck at specific theorems.

now i believe that if u get the theory u can then apply it the problems and it should follow that no matter how hard the problems are they are merely a variation of the theory which is basically a generalization of the problem.
am i right ? or just confused?

i was wondering how to deal with theorems. for example it is useful to translate theorems into words and then think about them?

also i would wondering generally wat are the undergrad topics i need to know to study the following subjects on my own. ?(i study great on my own i can easily bet that all the math that i have learned over the years is because iam motivated by some strange for to do math in my spare time didnt learn a thing in first year calc pissed me off)


1) quantum mechanics:!)
2) Functional analysis
3) Complex Analysis

also which of these topics is the easiest to tackle first ?
__________________________

 

[leon1127]

complex analysis is the the easiest. It merges with basic Calculus beautifully.

Quantum mechanics requires a lot of linear algebra, Complex Analysis, Differential Equation, Vector Calculus and etc. I think the maths of it is beyond most engineering programme that i know of.

Functional Analysis usually requires knowledge of topology, analysis, intergral equation. You really have to know something about real space b4 getting into that.

 

[mathwonk]

check out some of the posts in my thread who wants to be mathematician.

 

[pivoxa15]

I am a novice with regards to pure maths. But from my little experience so far, the best advice (which I got from my lecturer) is to 'think logically and work hard'. This probably goes for other stuff as well but it seems especially relevant to pure maths as it has no empirical content.

 

[Hurkyl]

i was wondering how to deal with theorems. for example it is useful to translate theorems into words and then think about them?

Yes... and no!


There are theorems that tell you things you already know. It is important to translate these theorems into words, but not for the reason you expect! When you "know" how to "prove" something in words, you need to know which theorems to use to turn your words into something rigorous.

For example, consider the triangle inequality for vectors:

$$||x + y|| \leq ||x|| + ||y||$$

When you translate this into words, it says that the length of one side of a triangle is shorter than the sum of the lengths of the other two sides.

But the usefulness of this translation is not so that you can understand the theorem!

Suppose you're trying to do a proof/calculuation/whatever, and you "know" that something ought to be true because the length of one side of a triangle is shorter than the sum of the lengths of the other sides. Well, then you know you that the triangle inequality will be a useful thing for making your work rigorous!



Sometimes, theorems tell us about new things. In such a case, translations are only analogies, and can sometimes be very misleading. But as you use these new things, and these theorems, you will eventually intuit these new concepts and new words to go with them, and then this theorem will become a theorem about things you already know!



Sometimes, it is not useful to try and translate something into words. Theorems often tell you something you can "do", and when you're used to abstract reasoning, it can be surprisingly efficient to forget what things "are", and just manipulate them algebraically.

This is also a useful approach to theorems you have much difficulty understanding -- if you just take it abstractly, you can often still use the theorem to good effect. And if you use it often enough, it will eventually make sense. But I want to emphasize that this approach is useful even for things you understand!


For example, the triangle inequality for vectors!

$$||x + y|| \leq ||x|| + ||y||$$

Sometimes, you'll be doing a proof/calculation/whatever, and you will have written down ||x + y||. It is frequently useful to obtain an upper bound on whatever you're calculating, and that's precisely what the triangle inequality does.

The triangle inequality tells you things that you can do -- one such thing is to take an expression such as ||x + y|| and immediately write down a (possibly simpler) upper bound ||x|| + ||y||. It is extremely useful to simply throw that into your bag of tricks, without the attached baggage of a geometric interpretation.

(once you've finished your calculation, it may or may not prove fruitful to go back and reinterpret everything geometrically)