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Heather Z.

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

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Felicia S.

00:56

Greninjack D.

Jsdfio K.

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all right, so we're ready to talk about the final topic of this course in that Stoke storm in the divergence there. And the good news about Stokes and Divergence Theorem are that if you really understood to two different versions of grant serum with circulation version in the flux version, Stokes serum and the Divergence serum are just higher dimensional analog of those toothy Rem's. So let's talk first about Stokes there. So recall that when we learned the circulation version of Green Serum, it related the curl of a vector field in the plane to the line integral or the circulation integral on the boundary? Well, Stokes term is going to say exactly the same thing, except it's going to be for a surface in space. So here's my surface that I'll call s and notice that for a surface assuming it's bounded and oriented, that's gonna be important. We want to assume that our surface is going to be oriented. It's going to have a boundary curve that will call C Now What we need to do is we need to given orientation to see in the convention we're gonna use is that if I look at a point on surface and look at the normal direction. I'm gonna pick the orientation of the surface to be up. So in other words, the normal direction of the surface is going to be up. And then the curve I'm going to pick toe help, have a counterclockwise orientation relative to the orientation of the service. So, in other words, if I looked down straight down on the surface, it has a counter clockwise. The curve has an hour counter clockwise orientation. So, yes, So we're assuming the S is oriented and we have this oriented boundary curve. And of course, we also have a vector field that's defined over the service and the boundary curve. And so what Stokes theorem is going to say is that if we integrate essentially the curl over the surface, that's going to equal the circulation on the boundary. It's really the same idea. Like I said as the circulation version of green stare. So let me just write down what Stoke serum says. It says that if I do the surface in a girl over s of the curl of F and now the curl of F in space is going to be a vector, and what I want to do is I want to take the component of that vector that is lying in the normal direction. And so here's my normal direction, pointing upward like that, and I add up the contribution over each of each little piece of the surface or the surface elements that's going to equal the circulation integral along the boundary. And now, up till now, we really talked about circulation inter girls in the plane. But circulation into girls in space are exactly the same thing. We just have a simple closed loop in space, and I just integrate the component of the force that's pointing in the direction of the curve. So this is what Stokes Theorem says, and again, we can think about this in a picture. So everywhere on the boundary of the surface, at each point, I can measure the rotation of the vector field by how much it curls at each point. So that's the curl. So if I add up all of the curl on the surface somehow, that should equal total circulation around the boundary. So Stokes serum is thehyperfix dimensional analog of the circulation version of green serum, then the divergence serum is the higher dimensional analog of the flux version of Stokes there. And in fact, we're going to sort of redefine our notion of flux. So the flux of a vector field over what's called a closed surface. And now what I mean by closed surface is a surface that's actually the boundary of a volume region like this. So you can just think about a cube. So the flux of the vector field f over this surface that's closed. That's enclosing, Ah, volume region is given by, well, just the surface integral. But we're going to give it some special notation. Okay, so we're going to do a double integral with the circle around it to signify that the surface is closed of what? Well, what we're going to do is we're just going to take the component of the vector field that's in the normal direction. So essentially pictorially the part of the field that's going straight out of the surface like that and, of course, integrating over the various surface elements. So the flux is nothing more than the surface integral of a vector field over a closed surface, kind of like the flux in the plane was just the line integral but taking the component of the force that was pointing in the normal direction. Or, in other words, it was the amount of the vector field that's flowing out of the curve. And here we can think about it exactly in the same way. So the flux if I have a region like if I have a little piece of pipe the flux out of the pipe if the pipes carrying water is just the amount of water that flows out of that surface, and so if there's some kind of source inside this region, then you can imagine the flux is going to be great. It's going to be positive. There's going to be outlooks. But if there's a sink or a hole inside the surface, then water is going to be flowing in. There's going to be a lot of influx or negative flux. So the idea of flux over a surface really is analogous to flux over a curve in the plank. Okay, so now that we've introduced the concept of flux over a surface, we're ready to state the divergence. Dear um, and you could almost fill in the blanks because remember that the Flux version of green serum said that the flux in a girl was equal to the integral of the divergence. That's going to be exactly the same thing here. So the statement of the're, um let's introduce a little bit of notation. So we're going tohave of volume region, and I'm just drawing it as a cube. So the volume region is our the surface of the volume Reason region is s I have a vector field F. And what we're going to say is that the flux out of the surface. So that's a and then we take the component along the normal direction and then over a small surface element is equal to the divergence added up over the entire buoyant region. Okay, and the interpretation here is exactly the same as in the plane. So if I look at each point, I can measure the divergence or how much this is spreading out of at one point. And somehow if I add up the divergence at each point, that's essentially going to tell me how much of the field is flowing out of the surface of the region like that. So it really is analogous to green stare for flux. And again, you can think that inside this region if you have sources. So in the case of electric field, if you have electric charge is going to create a divergence at points inside the region that's going to create electric field. That's, uh, as an out flux of the surface. If it's water, if you have sources, that's going to create divergence in the points inside the the region, and that's going to create out flux, etcetera. So the divergence serum it's really not saying anything new. But again, it's just taking that jump from, you know, a region bounded by a curve in the plane to region bounded by a surface in space. And again, both stoke serum and the divergent serum are taking something that's a little bit more complicated. So a surface integral and converting it into something easier or vice versa, right. So this is taking a surface in a girl, turning it into a triple integral, which we like a little bit better. Remember, Stokes Theorem was turning ah, line integral into a surface integral or, you know, it could go. The other way as well. But these stadiums air really just again relating back to the fundamental theorem of calculus. It's all about relating some sort of derivative, um, interior to kind of an anti derivative on the boundary.

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