Continued from previous entry... Okay. I'm ready to continue talking about something I can barely grasp with my metaphorically soapy fingers. The next notion I'm going to need is that of a 'directional derivative'. First, a derivative is generally a fancy way of talking about 'how stuff changes'. Most people meet derivatives in the context of talking about what a function f(x) is doing - swooping up and down, playing dead and staying constant, whatever. People that take vector calculus (also known as multivariable calculus) end up messing around with functions that eat and shit vectors, as opposed to numbers. If you plot a normal function that makes its living eating one normal number at a time on a piece of paper, it's just a curve. Everyone's seen those. If you plot a hairier function, such as one that works on more than one variable at a time (think f(x,y,z) or something), it'll look like a surface (if you're lucky) or something you can't even picture profitably, being stuck to an imagination that only works for 3D. So, think of a function that when drawn, looks like a normal surface. Like a rumpled (but not egregiously so) bed cover. Now, let's pick some point on it, and talk what's going on in that area of the bed cover. Generally speaking, the bed cover will be doing different things in different directions from your chosen point. In one direction, it might be going higher, in another it might be kinda flat, in yet another it might be going down, whatever. A directional derivative makes all this precise. That is, it's a way of talking what a function is behaving like in various directions. So, in my case, if I pick a point on my bed cover, and see what it's doing in the direction of the window, it'll turn out that's rising, and fast, too - there's a good pillow under there! I just took a derivative of a function (my rumpled bed cover) in the direction of my window. In short, a directional derivative. Whew, enough of that. (I launched into it because it's always good to make sure I can still explain the things I've learned a long time ago in *simple* terms - it's a way of making sure I still get them).

Let's get back to talking about vector fields. Say we've got a vector field called v on some domain. Let's make the domain R^n. R^n is just normal, flat space, with a dimension of n. It being a domain means the vectors that make up v live on R^n. They're n-dimensional, if you will. Say further we've got a function f in there, too.

Now, we can always, if we want, take the directional derivative of f with respect to v - v's just a bunch of arrows, that is, directions! So say we do that, and let's call the result vf, just because we like to use notation as malicious as possible.

So let's write a formula for vf. (Sadly, I don't have time to explain every term after this point - I'll just trust that whoever's reading this has seen some calculus before) We'll call a point x in R^n the coordinates (x^1, X^2, ..., x^n), and by Partial_u f we'll refer to the partial derivative of f with respect to x^u, where u can take on values from zero to n. I'll also use 'Einstein Summation Convention', because I'm a bastard, and it saves typing, and it's what my book uses, and the discussion here is paralleling it. The summation convention says that whenever we see something like x^u y_u - that is, the same index, here called u, popping up both upstairs and downstairs, we're meant to 'sum over u' - that is, do a sum, x^1 y_1 + ... + x^n y_n (I'm using ^ to stuff things where exponents go, and _ to stuff things where subscripts go, if you were to deasciify everything.)

Ok, now, given all that nastiness, the formula for our directional derivative vf looks like this (v^u stands for the components of v):

vf = v^u Partial_u f

Ok? If you know about partial derivatives, this shouldn't be news - it's just slightly different notation than what you probably used in calculus class. Now, watch out. f is on both sides of the equation. Let's just take it out, then.

v = v^u Partial_u

So now we're saying v's mission in life is to differentiate stuff. It's a 'linear combination of partial derivatives.' Now, this is weird. For one thing, the partial derivative is just hanging out there on the right with nothing to differentiate. But that's not so bad, because we can always stick a function back on there, and it'll have something to do again. What's weirder is we're saying a vector field v, and a directional derivative in the direction of v, are the same. This is sloppy: according to our current definitions of these terms, they're arguably closely related, but they are not the same.

Now, you're probably thinking: "DUDE. Step AWAY from the crack pipe. You can't do that!" And, stictly speaking, you're right. The above move (just ripping f out of there) was indeed 'illegal' and sloppy. However, it's suggestive, and what my book says is we can try using it as a guide for where we want to end up, and we can redo things in a legal way, and get to the same point. That is, we're going to redefine vector fields so they can work on manifolds.

We're going to *start* with the idea that 'vector fields' are entities whose "sole ambition in life is to differentiate functions." Then we're going to build an actual definition of such entities, and *then* we're going to be in a position to show that these still deserve to be called 'vector fields'.

So the way my book defines a vector field, v, at this point is that it's something that eats smooth (meaning, not too kinky, if you were to draw them) functions that are defined on a manifold, and produces smooth functions on that manifold. I've talked about manifolds in some old posting, you can go find it if you want to know what they kinda are (feel free to replace 'manifold' with 'R^n' for now if you don't know what a manifold is) . Also, v has to have the following properties:

v(f + g) = v(f) + v(g)

v(a*f) = a*v(f)

v(f*g) = v(f)*g + g*v(f)

where f and g are smooth functions, and a is just a number. Now, the first two properties are just a way of saying v is 'linear'. The meaty property is the third one: it looks just like the product rule (aka the Leibniz rule) from good old calculus.

Whew. The above probably seems bizarrely abstract: we started with hairy balls, and now we're talking about Leibniz, and 'smooth functions', and manifolds, and other fancy shit. But the huge benefit we just got for our efforts is that in our new definition, coordinates don't show up at all! This is vitally important, because while coordinates are very useful for actual concrete calculations, the universe isn't drawn on engineering graph paper, so coordinates are arbitrary, they're up to us. There are many different coordinate systems, and they're all arbitrary - there's no best one! So to talk about the true 'nature' of stuff, it's good to avoid using an arbitrary human thing like coordinates. And the new definition does that. And my fingers are tired. And the above was largely a recapitulation of a couple of pages of Baez and Munian's far better explanations from memory. So I'm going to stop for now.

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