Divergence intuition, part 2

Hey everyone! So, in the last video, I was talking about Divergence and kind of laying down the intuition that we need for it. You're imagining a vector field as representing some kind of fluid flow where particles move according to the vector that they're attached to at that point in time. As they move to a different point, the vector they're attached to is different, so their velocity changes in some way.

The key question that we want to think about is: If you have a given point somewhere in space, does fluid tend to flow towards that point, or does it tend to flow more away from it? Does it diverge away from that point? What I want to do here is start kind of closing our grasp on that intuition a little bit more tightly, as if we are trying to discover the formula for Divergence ourselves. Ultimately, that's what I’m going to get to: a formula for Divergence.

But I want it to be something that's not just plopped down in front of you, but something that you actually feel deep in your bones. A vector field like the one I have pictured above is given as a function, a multivariable function with a two-dimensional input, since it's a two-dimensional vector field, and then some kind of two-dimensional output. It's common to write P and Q as the functions for these components of the output.

So, P and Q are just scalar valued functions, and you think of them as the components of your vector valued output. The Divergence is kind of like a derivative, where you might denote it by just "div". In the same way that with your derivative, you have this operator and what it does is take in a function, and you get a whole new function. This “div” operator, think of it as taking in a vector field of some kind, and you get a new scalar valued function.

The new function you get will be something that takes in points in space and outputs a number, because what you're thinking — the thing that it's trying to do — is take in some specific point with XY coordinates and just give you a single number to tell you: hey, does fluid tend to diverge away from it? How much? Or does it tend to flow towards it? And how much?

So, this is the kind of form of the thing that we're going for. Here, what we're going to do is just start thinking about cases where this Divergence is positive, negative, or zero and what that should look like. For example, let’s say we want cases where the Divergence of our vector field at a specific point, at a specific point (X, Y), is positive. What might that look like?

One case would be where your point — nothing is happening at that point; the vector attached to it is zero, and everyone around it is going away. This is kind of the extreme example of positive Divergence. I animated this in the last video, where we have all of the vectors pointing away from the origin. If you look at a region around that origin, all the fluid particles kind of go out of that region. That's the quintessential positive Divergence example.

But it doesn't have to look like that. You could have something where there is a little bit of movement at your point, and then maybe there's movement towards it as well from one side, while vectors are kind of going towards it, but they're going away from it even more rapidly on the other side. If you think of any kind of actual region around your point, you're saying sure, fluid is going into that region a little bit, but it's much more counterbalanced by how quickly it's going out. These are the kinds of situations you might see for positive Divergence.

Now, let’s think about what examples of negative Divergence might look like. The Divergence of V at a given point—it really takes in all points of the plane, but we're just looking at specific points. If the Divergence is negative, well, the quintessential example here is that nothing happens at your point, but all of the vectors around it are kind of flowing in towards it.

This is the thing where I animated, and we took this and flipped all of the vectors and said, "Ah, there, if you start playing the fluid flow," then the density in any region around the origin increases a lot. All of the fluid particles tend to converge towards that center. But again, this isn't the only example that you might have. You could have a little bit of activity at your point itself, and maybe it is the case that things do flow away from it a little bit as you're going away, and some of the fluid particles are going away.

It's just the case that the fluid particles flowing in towards it from another direction heavily counterbalance that because then, if you look at any kind of region around your point, you say fluid particles are coming in quite rapidly— a lot of particles per time — but they're not leaving too rapidly around the other end. So kind of loosely intuitively, this is what a negative Divergence case might look like.

Finally, another case that we want to start thinking about as we're tightening our grasp on this intuition is: What does it look like if the Divergence of your function at a specific point is zero? If it's just absolutely zero, one thing this could look like is, you know, you have something going on, but nothing really changes, and all of the fluid just kind of flows in and flows out. On the whole, it balances.

If you take any kind of region, the amount flowing in is balanced with the amount flowing out. But it could also look like you have fluid flowing in kind of from one dimension, but it's canceled out by flowing away from the point in a manner that sort of perfectly balances it in another direction. So, these are the loose pictures that I want you to have in the back of your mind as we start looking for the actual formula for Divergence.

What I'll do in the next video or two is start looking at these functions P and Q and thinking about the partial derivative properties that they have that will correspond with these positive Divergence images that you should have in your head or the negative Divergence images that you should have in your head. So, with that, I’ll see you in the next video!