2

The problem is, that when I am reading the proof of mathematical theorem(in my case - it is calculus), U understand the idea and every step of proof. But i can't prove the theorem individualy even if i understand all steps, that are written in book.

Should i concentrate myself on proofs of theorems or it is enough to understand the idea and process?

Daniil Yefimov
  • 1,124
  • 2
    It depends. What is your goal? It's one thing if you want to be an engineer and use the theorems (but don't really need to prove them on short notice), it's a different thing if you want to become a researcher in math, and it's an entirely different thing if you need an A or a 10 on a calculus exam so you can get into Medical School. –  Apr 16 '16 at 13:43
  • I want to work as financial engineer. And as i understood it is enough to understand the idea and the process of proof? – Daniil Yefimov Apr 16 '16 at 15:28

1 Answers1

6

I want to preface this by saying I'm a graduate student in pure math and I can only speak for pure math. I don't know what level of understanding is appropriate for applied math, and I'm sure that can vary depending on what kind of applied math.

I would say that to have a solid understanding of things at the basic graduate level, knowing just the concepts or the general idea is not enough. It is best to be able to prove most of the math that you use.

You don't need to memorize every step of a proof, that's too much and it's not really useful anyway. It's better to have the ability to quickly recollect the proof on your own. To do this, it's good when you're reading a proof to keep on the lookout for things that you wouldn't have come up with on your own, and do memorize those parts.

Besides, those difficult techniques or counterintuitive perspectives will often show up in other proofs too, or sometimes in whole fields of mathematics.

Example: in real analysis, do you remember how to prove that if $x_n, y_n$ are sequences which converge to $x, y$, then $x_ny_n$ converges to $xy$? You do so by splitting up the difference: $$|x_ny_n - xy| = |(x_ny_n - xy_n) + (xy_n - xy)| \leq |x_n - x| \cdot |y_n| + |y_n - y| \cdot |x|$$ And now you use the fact that since $y_n$ is a convergent sequence, it is bounded, and hence $|x_n - x| \cdot |y_n|$ goes to $0$ as $n$ goes to infinity.

I thought the argument of adding and subtracting $xy_n$ was elegant, but unintuitive when I first saw it. I didn't memorize that proof, but I did pay a lot of attention to that part. And, as a result, I am able to recollect the whole proof, and that technique of splitting up $x_ny_n - xy$ I have seen several times in other places in real analysis, in particular $L^p$ spaces.

D_S
  • 33,891