Archive for the ‘YAKC’ Category

YAKC: Differential (One) Forms

August 11, 2011

In my previous post in this series, I introduced the idea of a derivative, and we realized a number of rules regarding them. In this post, we’re going to give some thought as to what derivatives are, look at them from from a rather different perspective, and realize several more rules regarding them. (more…)

You Already Know Calculus: Derivatives

July 31, 2011

Calculus is made to be a whole big hoopla in high school and first year university. It’s supposedly the hardest math class in high school, notoriously complicated and unintuitive.

I blame this on bad education, not just because I’ve observed so much bad math education at the high school level, but because I can’t see any other way anyone could conclude that calculus is difficult. Because I believe that everyone already knows calculus. They just never connect what they already know to the symbols they’re manipulating in math class.

(I’m writing this series of posts — yes, this is only the first of a number of posts! — in a didactic form, but I think they may be of interest to mature mathematicians. While they are (will be?)  informal and unrigorous, they provide intuitive reasons for why everything in basic calculus is true. The results in it are certainly valuable to me: they’re the result of me spending some time trying to answer ‘why’ everything is true at a compelling level. If nothing else, they may be useful in teaching calculus.)

So what is calculus? It’s the mathematical study of rates of change, nothing more and nothing less. We call the rate of change of a function its derivative. (more…)