Last May we started talking about linear algebra, with a little aside into complex numbers and another into power series along the way. Before that all, long long ago, we were talking about single variable calculus. Specifically, we were studying functions which took a real number in and gave a real number back out, and the two main aspects to this study: differentiation and integration.
The first part studied how a function changed as its input varied near a fixed point by coming up with the best linear approximation of the function near that point. Now that we’ve got an understanding of linear functions between higher-dimensional real vector spaces, we can work towards extending this idea of differential calculus into multivariable functions.
The second part studied how to “add up” a continuously-varying collection of values, each with its own (infinitesimal) weight. Again, our new understanding of higher-dimensional analogues of linear spaces and functions will help us find the right way to generalize the integral calculus.
I’m trying to get access to some of my references again, since I no longer have even as much of a mathematical library down the hall as Western Kentucky University provided. So I’ll pick up when I can.