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Anshu R.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

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Masoumeh A.

01:34

Scott N.

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Felicia S.

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all righty. Now we're gonna apply the divergence serum. And the good news is is that the divergent serum is a little bit easier. Thio apply because you're not converting a line integral to a surface integral. You're actually converting a surface integral thio. Just good old fashioned triple zero. So let's see how it works. So recall that the divergence theorem here says that we can replace this sort of crazy surface integral. Where s is gonna be the boundary of this cube? Or in other words, this is the flux of the vector field Out of the Cube. We can replace this with the triple in a girl over the volume region. But the volume region is a cube. So actually, my limits just go 0101 0 to 1. But now, when I integrate is actually the divergence of the vector field. Okay, so what's the divergence of a vector field? Well, I just take the partial of this with respect to X, which is hero. I take the partial of this with respect to why which is nine x squared z squared and I take the partial with respect to this Z and at that. But that's also zero. So this is just the x d y DZ and look at that. This is amazing. I mean, the Stokes theorem applications just involved, you know, some technical details. We had parameter isa curve, etcetera, etcetera. Here. We don't have to do any of that because we're just converting the flux out, which is a surface integral to a triple integral over the divergence. So the divergence serum is really just a beautiful serum. I mean, if you tried to do the flux over this cube, it might take you a few hours to dio. I mean, you should never do that. This is the point. You should converted to a triple Integral. Okay, so what do we get? I mean, this is not so hard. This is just gonna be X cubed over three. So that's just gonna leave me with three execute Z squared from 0 to 1, which is just three. And then I integrate c squared. That just leaves me with one c cubed, evaluated from 0 to 1. But that's just one. And then I just have this. This is just why Evaluated from 0 to 1. So the final answer is one, and that's really cool, because we can think about that's the flow rate or the flux of the Specter field out of the queue and when you found it by just doing the surface are the triple integral of the divergence and some. So Somehow we were just adding up kind of the you can think about it like it's the charge distribution inside the Cube or the gravitational distribution on just measuring that as a divergence, and it was just giving us the flux out of the region, which is just one.

04:53