Word of warning: I don't expect this particular 'posting' to make sense to anyone. Basically, I need to try to write out my current impressions about some stuff I've been reading, to figure out what I *do* understand, and what I don't. So, err, without (much) further ado...

I'm still reading my books on real and complex analysis, but I've also got a new book called *Gauge Fields, Knots, and Gravity*, by John Baez and Javier Munian. Very good book so far, very highly recommended, yada yada yada, more on such things some other time. I'm going to be talking about my attempt at digesting the chapter on vector fields.

The chapter starts off with a hilarious quote from Heavyside, where he heaps delightful scorn on opponens of vectors, and takes off from there.

To start with, think of a vector as everyone normally does: as an arrow. That is, a pointy thingy of a certain length and direction. Now, spiraling upward in abstraction at a dizzying rate (if you've never seen vectors before, you'd now talk about all the neat things about them and their uses), we talk about 'vector fields'. The way everyone normally conceives of a vector field is pretty simple. We have a 'field', in the sense I'm interested in, when we assign a number to every point in space. To be more careful, that's a *scalar* field. A vector field assigns an arrow to every point in space. If you want, think about hairy balls: each hair is a 'vector', and if a ball is really, really hairy, it'll have a hair coming out of every point on the ball. (Actually, you'll probably want to keep in mind that this makes the most sense if the hairs are really short - how the bloody hell would you define a 'direction' for a big, long, curly hair?)

Hmm. Ok, actually, I've got to go contemplate some of this stuff before going on.* More in a few minutes, hopefully.

Yes, I'm probably going to have to think about arrows and hairy balls and other things, but get your mind out of the gutter, damn it! I'm talking math here!

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