Divergence intuition, part 2

- [Voiceover] Hey everyone. So, in the last video I was talking about divergence and kind of laying down the intuition that we need for it. Where 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 in that point in time and as they move to a different point the vector they're attached to is different so their velocity changes in some way. And 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? And what I wanna do here is start kinda closing our grasp on that intuition a little bit more tightly, as if we are trying to discover the formula for divergence ourselves, because ultimately that's what I'm gonna 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, you know, feel deep in your bones. So a vector field like the one I have pictured above is given as a function, a multi-variable function, with a two-dimensional input, since it's a two-dimensional vector field, and then some kinda two-dimensional output. And it's common to write P and Q as the functions for these, the components of the output. So P and Q are each just scalar value functions and you think of them as the components of your vector valued output. And the divergence is kinda like a derivative, where you might denote it by just div, and in the same way that your derivative, you have this operator and what it does is it takes in a function. And what you get is a whole new function. This div operator you think of as taking in a vector field of some kind and you get a new function. And the new function you get will be scalar valued, it'll be something that just 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, the form of the thing that we're going for. So here what we're gonna do is just start thinking about cases where this divergence is positive, or negative, or zero and what that should look like. So for example, let's say we want cases where the divergence of our vector field at a specific point XY is positive. What might that look like? So one case would be where your point, nothing is happening at that point and the vector attached to it is zero, and everyone around it is going away. This is kinda the extreme example of positive divergence. And I animated this in the last video where we have you know, all of the vectors pointing away from the origin and if you look at a region around that origin, all the fluid particles kinda go out of that region. And that's the quintessential positive divergence example. But it doesn't have to look like that. It actually, I mean, 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, and vectors are kind of going towards it, but they're going away from it even more rapidly on the other side, so 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." So these are the kind of situations you might see for positive divergence. Now negative divergence, negative divergence. Let's think about what examples of that might look like. Divergence of V at a given point and you know, really it's something that takes in all points of the plane but we're just looking at specific points, so 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. And this is the thing where I animated, where we took this and we 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 "you know, 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, and it's just the case that the fluid particles flowing in towards it from another direction heavily counterbalance that. Cuz then if you're looking 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 round the other end. So kind of loosely, intuitively, this is what a negative divergence case might look like. And finally, another case that we wanna start thinking about as we're tightening our grasp on this intuition is what happens, or what does it look like, if the divergence of your function at a specific point is zero, right, if it's just absolutely zero? And one thing this could look like is, you know, you have something going on but nothing really changes and all of the fluid just kinda flows in then it flows out and on the whole it balances. You know, 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 cancelled 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. And 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, you know, 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 next video.