We've already explored a two-dimensional version of the divergence theorem.
And hopefully, it made intuitive sense. The way that I drew this vector field right over here, you see everything's kind of coming out. You could almost call this a source right here, where the vector field seems like it's popping out of there. This has positive divergence right over here. And so because of this, you actually see that the vector field at the boundary is actually going in the direction of the normal vector, pretty close to the direction of the normal vector, so it makes sense. You have positive divergence, and this is going to be a positive value. The vector field is going, for the most part, in the direction of the normal vector. So the larger this is, the larger that is. So hopefully, some intuitive sense. If you had another vector field-- so let me draw another region-- that looked like this, so I could draw a couple situations.
So one where there's very limited divergence, maybe it's just a constant. The vector field doesn't really change as you go in any given direction.
Over here you'll get positive fluxes. I don't know what the plural of flux is. You'll get positive fluxes, because the vector field seems to be going in roughly the same direction as our normal vector. But here, you'll get a negative flux. So stuff is coming in here. If you imagine your vector field is essentially some type of mass density times volume, and we've thought about that before, this is showing how much stuff is coming in, and then stuff is coming out. So your net flux will be close to zero. Stuff is coming in, and stuff is coming out. Here, you're just saying, hey, stuff is constantly coming out of this surface. So hopefully, this gives you a sense that here you have very low divergence, and you would have a low flux, total aggregate flux, going through your boundary. Here you have a high divergence, and you would have a high aggregate flux. I could draw another situation.
So this is my region R. And let's say that we have negative divergence, or we could even call it convergence. Convergence isn't an actual technical term, but you could imagine if the vector field is converging within R, well, the divergence is going to be negative in this situation. It's actually converging, which is the opposite of diverging. So the divergence is negative in this situation. And also the flux across the boundary is going to be negative. Because as we see here, the way I drew it, across most of this boundary, the vector field is going in the opposite direction. It's going in the opposite direction as our normal vector at any point.
So hopefully, this gives you a sense of why there's this connection between the divergence over the region and the flux across the boundary. Well, now we're just going to extend this to three dimensions, and it's the exact same reasoning. If we have a-- and I'll define it a little bit more precisely in future videos-- a simple, solid region. So let me just draw it. And I'm going to try to draw it in three dimensions.
And then you sum it up. That should be equal to the flux. It's completely analogous to what's here. Here we had a flux across the line.
We had essentially a two-dimensional-- or I guess we could say it's a one-dimensional boundary, so flux across the curve.
And here we have the flux across a surface. Here we were summing the divergence in the region. Here we're summing it in the volume. But it's the exact same logic. If you had a vector field like this that was fairly constant going through the surface, on one side you would have a negative flux. On the other side, you would have a positive flux, and they would roughly cancel out. And that makes sense, because there would be no diverging going on. If you had a converging vector field, where it's coming in, the flux would be negative, because it's going in the opposite direction of the normal vector. And so the divergence would be negative as well, because essentially the vector field would be converging. So hopefully this gives you an intuition of what the divergence theorem is actually saying something very, very, very, very-- almost common sense or intuitive. And now in the next few videos, we can do some worked examples, just so you feel comfortable computing or manipulating these integrals. And then we'll do a couple of proof videos, where we actually prove the divergence theorem.