Mathematical intuition?? Where are students expected to learn all the little algebra tricks that can turn unsolvable looking diff EQ's, integrals, laplace and inverse laplace problems into cakewalks? Things like adding 5+(-5) to the numerator or multiplying by just the right x/x to nudge a nasty looking equation into something of the right form. Every time I encounter a problem that requires a trick like this, I usually spend hours researching on the internet or in other math books trying to figure it out, assuming that I'm making a mistake somewhere rather then just missing a trick. When I finally do figure it out, I feel like such an idiot because I didn't pick up that little trick along the way in my math classes. I'm Just curious, I've never considered the time I spent looking up a certain math subject and studying it in more detail than I normally would to be a waste of time. I've always gained a deeper understanding the material, but I've rarely actually found the trick on the web, I usually figured it out on my own, :LOL, usually in the shower. I have noticed though that a lot of my classmates are much better at spotting when these kinds of tricks are needed and seem to already have knowledge of them. Are their brains just geared more towards math then mine is? Did they have better algebra classes and calculus teachers?
That's a good question. While taking differential equations, I usually found a trick quickly enough. It was just a "hmm, I think that doing this will simplify it", and it does! Maybe if you were to examine why multiplying by some form of x/x or something works, you may be able to over time learn to use tricks like these efficiently... If I absolutely can't solve a problem, after a little bit of not thinking about it, I figure out what would probably work.
Here is a nasty little integral i found in Spivak, out of the 300 this was the only one that took longer than 30 mins to solve. Hope you guys have fun with it! [tex] \frac{dx}{dy}= x^4+1 [/tex]
kdinser, you just reason your way to the answer. Most tricks are really not that obscure. With enough practice, you can almost sense them immediately. For example let's look at SeReNiTy's problem. (Note: don't look further if you were gonna attempt to solve this on your own! ) dx/(x^4 + 1) = dy The right-hand side looks hopeless the way it is. So let's think about how we usually handle integrands of rational functions. First thing that pops into my mind is a trig substition. We know that 1+(tan(u))^2 = (sec(u))^2. Since we have an x^4, let's try x = sqrt(tan(u)). (Although you should be skeptic this would lead to anything useful, but you never know!) Then dx/du = (sec(u)tan(u))/(2 sqrt(tan(u))). The RHS now becomes (1/(sec(u))^2) * (sec(u)tan(u))/(2 sqrt(tan(u))) du Which simplifies to (tan(u))/(2 sec(u) sqrt(tan(u))) du This doesn't look too helpful. So let's move on. The second thing that popped into my mind was partial fractions. Since x^4 + 1 is irreducible (over R), I don't think this method will yield anything helpful -- but it might, so try it for yourself. The third thing that struck me was x^4 + 1 *almost* looks like (x^2 + 1)^2. In fact: x^4 + 1 = (x^2 + 1)^2 - 2x^2 Hey, that's the difference between two squares! Let's see where this goes... (x^2 + 1)^2 - 2x^2 = (x^2 + 1 - sqrt(2)x)(x^2 + 1 + sqrt(2)x) Split this into partial fractions, and you're off to the races.
This isn't irreducible over R, you factored it a few lines later...no polynomial over R of degree 3 or higher is irreducible over R. You can always break them up into a product of linear and irreducible quadratic factors (then apply partial fractions).
You're right of course. What I meant to say was something to the effect of "it isn't immediately obvious what this can reduced to"! I guess I got sidetracked and said that instead.
I found myself asking the same question in my first few years of university. I think the latter. It is often a matter of being exposed to the tricks and other formalities. Certainly, most high school math teachers are not necessarily exposed to these methods, unless they have degrees in mathematics.
I have to say, the 2 tricks that I mentioned no longer hold me at bay, so I guess I can say that I'm learning. My complaint is, that I had no chance to learn them before they showed up a calc 2 exam or a diff EQ final. Both times, they made the difference between a 3.2 and a 2.8. I don't worry to much about gpa because I know if I'm learning the material or not and I know that overall, I'll finish with a 3.4 to 3.6 gpa in my engineering classes. But it is frustrating to lose points on an exam because you were never exposed to a certain algebra trick or because you forgot some obscure trig identity.
Hello kdinser, not too long ago I read this article (scientific american). In it, it details the possible thought proccess of a chess grandmaster. Aparently( and I agree with it) chess masters dont actually calculate all posible moves like a computer does. Instead, for a paticular arrangement of the pieces of the board they know based on experience what are usually the best paths to follow. Math is the same way, there are many paths that you can follow but only a few will get you to the end with relative ease. I think only experience and lots of practice will get you better at it. The more you practice the fundamentals, the deeper you are going to "see" the problem at a subconscious level. For example instead of readind (x^2-1) you read (x+1)(x-1). This is a very basic example but I bet it applies to much deeper processes. I would assume that the problem you are having is due to lack of practice of the fundamentals. I encounter the same problem that you do, but since I've become a pre-calc tutor I get more than enough practice and the paths to follow are becoming ever simpler. If I were you I would try becoming a homework helper for pre-calc level exersices. You ought to find more than enough practice there and you will be helping others. Just my sugestion.
I pick it up algorithmic procedures/short cuts very fast. I think maths, like evrything else in life, is something which some people have an immediate aptitude for. Of course, you can train hard for it - read lots of books/do loads of exercises - but for some, it can also just come naturally.
Well, there is the "use it, or lose it" factor. If one learns the various 'tricks' or 'short cuts', one might forget them if one does not use them periodically. One option - when one learns a 'trick' or 'short cut', or comes across a handy piece of information, write it down in a personal notebook/handbook. I wish I had been more diligent as a student, particularly as a grad student. Lots of my notes have been boxed away, and periodically I see a problem on PF, particular ones involving PDE's and EM Theory, where I have done the solution of the problem or one similar, but I don't have those notes at hand. Somewhere I have an excellent set of notes on PDE's for various applications in heat transfer, wave and vibration problems, and stress analysis, not to mention diffusion problems, EM Theory and Schrödinger's equation.