Divergence intuition, part 1

All right everyone, we've gotten to one of my all-time favorite multivariable calculus topics: divergence. In the next few videos, I'm going to describe what it is mathematically and how you compute it and all of that. But here, I just want to give a very visual understanding of what it is that it's trying to represent.

So, I've got it here pictured in front of us a vector field, and I've said before that a pretty neat way to understand vector fields is to think of them as representing a fluid flow. What I mean by that is you can think of every single point in space as a particle, maybe like an air particle or a water particle, something to that effect.

Since what a vector field does is it associates each point in space with some kind of vector, remember we only show a small subset of all of those vectors. But in principle, you should be thinking of every one of those infinitely many points in space being associated with one of these vectors. The fact that they're kind of smoothly changing as you traverse across space means that showing this very small, you know, finite sub-sample of those infinitely many vectors still gives a pretty good feel for what's going on.

So if we have these fluid particles and you have a vector assigned to each one, a kind of natural thought you might have is to say what would happen if you let things progress over time, where at any given instant the velocity of one of these particles is given by that vector connected to it. As it moves, you know, it'll be touching a different vector, so its velocity might turn, it might go in a different direction.

For each one, it'll kind of traverse some path as determined by the vectors that it's touching as it goes. When you think of all of them doing this at once, it'll feel like a certain fluid flow. And for this, you don't actually have to imagine; I went ahead and put together an animation for you.

So we'll put some water molecules or dots to represent a small sample of the water molecules throughout space here, and then I'm just going to let it play where each one moves along the vector that it's closest to. I'll just let it play forward here, where each one is flowing along the vector that's touching the point where it is in that moment.

So for example, if we were to, you know, go back and maybe focus our attention on just one vector like this guy, one particle—excuse me—he's attached to this vector, so he'll be moving in that direction. But just for an instant because after he moves a little, he'll be attached to a different vector. If you kind of let it play and follow that particular dot, after a little bit you'll find him, you know, elsewhere.

I think this is the one, right? And now he's going to be moving along this vector or whatever one is really attached to him. Thinking about all of the particles all at once doing this gives a good sort of global view of the vector field. If you're studying math, you might start to ask some natural questions about the nature of that fluid flow.

Like, for example, you might wonder if you were to just look in a certain region and count the number of water molecules that are inside that region, does that count of yours change as you play this animation, as you let this flow over time? In this particular example, you can look and it doesn't look like the count changes, certainly not by much. It's not increasing over time or decreasing over time.

In a little bit, if I gave you the function that determines this vector field, you will be able to tell me why it's the case that the number of molecules in that region doesn't tend to change. But if you were to look at another example, like a guy that looks like this, and if I were to say, I want you to focus on what happens around the origin in that little region around the origin, you can probably predict how, once I start playing it, once I put some water molecules in there and let them flow along the vectors that they flow along, the density inside that region around the origin decreases.

So we put a whole bunch of vectors there, and I'll just play it for a quick instant, just kind of let it jump for an instant. One thing that characterizes this field around the origin is that decrease in density. What you might say, if you wanted to be suggestive of the operation that I'm leading to here, is that the water molecules tend to diverge away from the origin.

So, the kind of divergence of the vector field near that origin is positive, and you'll see what I mean mathematically by that in the next couple of videos. But if we were to flip all of these vectors, right? And we were to flip them around. Now, if I were to ask about the density in that same region around the origin, you can probably see how it's going to increase.

When I play that fluid flow over just a short spurt of time, the density in that region increases. So these don't diverge away; they converge towards the origin. That fact actually has some mathematical significance for the function representing this vector field around that point.

Even if the vector field doesn't represent fluid flow, if it represents like a magnetic field or an electric field or things like that, there's a certain meaning to this idea of diverging away from a point or converging to a point.

Another way that people sometimes think about this, if you look at that same kind of outward flowing vector field, is rather than thinking of a decrease in density, imagining that the fluid would have to constantly be repopulated around that point. So you're really thinking of the origin as a source of fluid. If I had animated this better, a whole bunch of other points should be sources of fluid so that the density doesn't decrease everywhere.

But the idea is that points of positive divergence, where things are diverging away, would have to have a source of that fluid in order to kind of keep things sustaining. Conversely, if you were to look at that kind of inward flow, or what you might call negative divergence example, and you were to play it, but it were to go continuously, you'd have to think of that center point as a sink where all the fluid kind of just sort of flows away.

That's actually a technical term; people will say the vector field has a sink at such and such point, or the electromagnetic field has a sink at such and such point. That often has a certain significance. If we go back to that original example here, where there is no change in fluid density, right?

What you might notice, this feels a lot more like actual water than the other ones because there is no change in density there. And if you can find a way to mathematically describe that lack of a change in density, that's a pretty good way to model water flow. Again, even if it's not water flow but it's something like the electromagnetic field, there's often a significance to this no changing in density idea.

So with that, I think I've jabbered on enough about the visuals of it. In the next video, I'll tell you what divergence is mathematically, how you compute it, go through a couple examples, things like that. See you next video!