3D divergence theorem
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Explanation of example 1
In the last video we used the divergence theorem to show that the flux across this surface right now, which is equal to the divergence of f along or summed up throughout the entire region, it's equal to 0. And what I want to do in this video is think a little bit. When you get an answer like 0, you want to think about why is it? This is saying that there's no net flux across the surface right over here. Or if you sum up all of the divergences in this volume, you are getting 0. So why is that? Well, the simple way to think about it is, when we took the divergence of f, this vector field f is hard to visualize, but the divergence of f is fairly easy to visualize. The divergence is equal to 2 times x. So over here you're going to get, as you go further and further in this direction, as x becomes larger, your divergence becomes more and more positive. So you have kind of a divergence of 2 right over here. You have a divergence of 1 along that line. And you have a divergence of 0 right there. And that's also true, obviously, as you go higher. Because you're just changing the z. You're not changing the x. So all over here you have positive divergence. Over there you have positive divergence, and not just along that plane. But if you go in the x direction as well, this whole region of space, you have positive divergence, I guess you could say, in the positive x side of our octant. But then as you go on that side, on the other side, you have negative divergence. And this diagram is symmetric with respect to the zy plane. And so those divergences cancel them out. You would have had a positive flux across the surface or a positive value right over here. Instead of calculating it for this region, we had calculated for region that was just between x is 0 and 1. So let's just think about that region. So that region that would have been cut off right over here. That would have been cut off right over there. And so the back-- I guess you could say the back wall of that, the back wall of this, would have been the zy plane. Now if we care about this volume-- so we're essentially eliminating the rest of it. So let me try to eliminate it as best as I can, change the colors. So if I eliminate that part of it over there and all maybe even what we see, all of that-- I should have deleted that first. So if we eliminate the back part of it and we're just dealing with it when x is positive, then our entire solution that we did in the last video would have been the exact same, except now x is going to vary between-- instead of negative 1 and 1, it'll vary between 0 and 1. And so our bounds of integration, x is going to go between 0 and 1. And then in that situation, our final answer-- this part, this would be between 0 and 1. That would all be 0. And we would be left with 3/2 minus 1/2. 3/2 minus 1/2 is 1 minus 1/6, which is just going to be 5/6. And so when you just think about this part of it-- this side of it, this one that I've just drawn-- you had a positive flux of 5/6. On the other side you had a negative flux of 5/6. And then they canceled out. One way to interpret that is, if we thought about the entire surface, like what we saw in the last video, then you had an aggregate inward flux in the last video. That's why it was negative. And then it's completely offset by an outward flux right over here, the positive. You had a negative divergence in that other region, and you have a positive divergence that completely offsets it in this region right over here.