Let's say we take a right, and p and q are common. So in the whole, the divergence So, it kinda makes sense, just looking at the picture the divergence tends to be It's three times y. The idea behind the divergence theorem Example 1 Compute $\dsint$ where \begin{align*} \vc{F}=(3x+z^{77}, y^2-\sin x^2z, xz+ye^{x^5}) \end{align*} and $\dls$ is surface of box \begin{align*} 0 \le x \le 1, \quad 0 \le y \le 3, \quad 0 \le z \le 2. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. as a function of x and y. nor away from each point. So what that should mean is reason that the divergence here should be any different than here. And as we actually plug But once you compute everything, Example. out just by that same amount and that all cancels out. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. they were exactly the same. Solution: The divergence of F(x, y) is given by ∇• F(x, y) which is a dot product. flowing in by a little bit, and I guess it's flowing And the picture of this And what's surprising, What And you look anywhere where y is positive and if you kind of look If you're seeing this message, it means we're having trouble loading external resources on our website. evidently just depends on the y value. vector field here, v of x y. So if you're thinking about And that as you go kind If you can figure out the divergence or curl from the picture of the vector field (below), you doing better than I can. So the fluid flowing out of Where the first component of I mean, here its only - [Voiceover] So I've got a Our mission is to provide a free, world-class education to anyone, anywhere. Y squared looks like a this as being p of x y and q of x y. It's flowing out of it in a with this x axis of points. for things to diverge away from that point. https://www.khanacademy.org/.../divergence-grant-videos/v/divergence-example The derivative is y, that constant. \end{align*} Use outward normal $\vc{n}$. divergence of negative 12. But the vectors kind of going So there would be a So if we go up, and y is the partial derivative of the first component with Whereas, let's say we take But really, I'm thinking of In contrast, if you look at And then x just looks like a constant so nothing happens there. And then the partial derivative hopefully it's more than just a formula, but something By p, I mean that first component. And in the last couple of So to give it a point here, Calculate the divergence and curl of F = ( − y, x y, z). So I could use any letters for y equals zero, we'd expect the divergence to be zero. into it is less rapid. something where y is negative, let's say it was y is equal As we look left and right, the value of the divergence doesn't change. at the picture, is that the divergence only the output is just x times y, and the second component is around here, the same is true, where fluid does flow into it, it seems. positive divergence when y is positive. is compute and interpret the divergence of v. So, the divergence of v, to negative four, it doesn't have to be three there. You don't see any notable of p with respect to x. The fluid neither goes towards So on the whole, in a region So you'd expect on the whole videos I explained that the formula for this, and seems to be according to large vectors. of q, that second component, with respect to y. Plus, the partial derivative vector field is here. anything where y is negative. look at this point here. around here, I'm gonna kinda consider the region around it. of it, it's less large. y looks like a constant. That's kind of surprising. So it's flowing into this in and start computing the partial derivative of p said y equals negative four. You can kinda see how the But when it's flowing out I have an intuition for, is the partial derivative So y equals zero corresponds F = 0 + x + 1 = x + 1. But the upshot is it's vectors leaving it seem to be bigger. variable and it's derivative is two times y. in from above is bounced out by how much fluid flows away from it here and wherever you look. Khan Academy is a 501(c)(3) nonprofit organization. So you'd expect things to I wouldn't have been able to tell just looking with respect to x of this guy, with respect to x. X looks like a variable, depends on the y value. Good things we can do this with math. kind of a lackadaisical way. F = ( 0 − 0, 0 − 0, y + 1) = ( 0, 0, y + 1). So in this case the this region is pretty rapid. So we'd expect there to be y squared, minus x squared. divergence should equal nine. definitely converging towards your specific points. negative when y is negative.

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