Dif. Eq.
Starting with Cal II, I have had a vague feeling that I’m missing my basics. That is, I can understand how to use the tools of calculus, and even understand some nontrivial results, as well as reach them independently. But with every new thing I learn, I feel that much further away from my mathematical basics. It is no longer possible to look at an equation and see what it means and how to resolve it intuitively. I’m too far away from vanilla derivatives and integrals for me to doubt my grasp of them; I should be able to use the without thinking about it. In a perfect world, I would remember all the reasoning that went into bringing me to the level of mathematical involvement that I’ve reached. But of course, I don’t; otherwise, I wouldn’t be writing this, would I? Now I’m in differential equations, and I find myself in a quandry. There is so much more I would like to know, both in the world of math and outside. And all that knowledge seeking will of course require reading. Should I take the time to secure my foundations before I move on, or should I keep pressing ahead hoping that my mathematical understanding doesn’t fall apart around me? I think it is time that I force myself to review the basics. Even if only the not-so-basics, like the techniques of integration, and the logics of limits– the two things I seem to have the most trouble remembering. I would hate to find that after 4 years of college, all I succeeded in doing was turning my mind to a mush of unconnected and arbritrary techniques. Hence, I will return to the theory and not study any other math (other than school-related) until I have it where I want it. Indelibly imprinted on my brain.
Which brings forth a related question that has been nagging me for quite a while, and I only take this occasion to bring forth? Is it true as it seems to me that the type of undifferentiated, random, cursory technical reading I do actually is negatively affecting my ability to concentrate on, think about, and remember anything for more than a few seconds? I think that this is a true effect– perhaps caused because when I read, I can glance over a pivotal argument or theorem and see how it was done, without having to put any effort into deriving it myself. When I in turn have to solve a problem of my own, two things happen: 1) I get frustrated, thinking it should be easier than it is, and 2) my brain feels full with the wool of others’ thoughts. So much so that I can’t differentiate them from one another. The answer to my question, whether I have it right or wrong, is of course to stop reading for a while, give my brain a rest. Maybe that is all it needs, time to recuperate and integrate new information. But can I do that?
Possibly relevant posts:
- Uniform Planar EM Waves (9/7/2003)
- Sci-Fi as the last bastion of philosophical writing? (2/18/2008)
- Differential Geometry (9/16/2004)