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00:06

Jeffery W.

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

02:07

Fangjun Z.

00:50

Masoumeh A.

00:38

Amy J.

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All right. So we want to apply Stokes theorem to compute, to compute this line integral, or this circulation integral in space. And so we see that, uh, sees the boundary of the surface given by the piece of the plane given going within the cylinder X squared plus y squared equals four. So our region is gonna look like in a lips. It's going to be, you know, kind of taking a plane and cutting a cylinder. So our boundary curves c is going to be here and in the region inside, with surface, it's gonna be s. And now the orientation, it's going to be upwards. So from view viewed from above, the curb is gonna have a counterclockwise orientation. All right, Okay. And so the first thing we need to dio is remember what Stoke serum says. So Stoke serum says that we can convert this line in a girl circulation integral to a surface in a girl of what? Of the curl of the vector field. Let me take the component of the curl that's pointing in the normal direction of the surface over all of the surface elements. So first things first, let's parameter rise the surface. Now notice that X squared plus y squared is four. So I'm living within a circle of radius to in the X y plane. So that's kind of a hint to tell me. I probably want X and y to be s cosign t and s 70 essentially just polar coordinates and then Z is X plus four. Well, x s cosign t. So this is going to be s ko 70 plus for And this will be for S t in the rectangle, 0 to 2. So that's just the radius of the circle next by plane and then t is like data. So going from zero to two pi. So let's find the partial derivatives. So that will be Constantine Scientist E and then goes on t partial with respect to t will be minus s sci fi s Constantine and then minus s scientific. And then the cross product is going to be as sine squared t minus s coastlines. Great teen. Then we have minus but really plus s scientific Oh, Santee. And then this is gonna be minus But plus as scientific Oh Santee again. So that's gonna be zero then Finally we'll have s cousin square team and then plus s science. Great. Okay. And that looks pretty good. Let's see. Science crew t thing is, actually I'm sorry. This is gonna be minus two. Just double checking. So this was minus s sign security and then minus esco since 14. So this actually simplifies. So this is just s And this is minus s. So we have minus s zero s. All right. And then what about the curl of the vector field? Well, let's write what it is, so we can practice. So we have I j Okay, partial with respect, X. Why? NZ. So I'm just gonna put partial X partial y partials e for short. And then we have the X component, which is to Z than four x than five y. Okay, so this is going to be partial with respect to why the Five Wise? That's five minus partial with respect to Z of for X zero. Then we have partial with respect to exit five y zero and then minus and minus partial with respect to Z A to Z, which is to and then finally we have partial with respect to X. So four and then zero so forth. All right, so we have minus s zero s and 5 to 4, and we just need to take the dog product. So are in a grand is really going to be for our surface in a girl minus five s and then plus for us. So this is just minus s. Well, that's really nice. So now we're ready to set up the surface integral. Which is just gonna be the double integral over S and T. So s goes from 0 to 2. T goes from 0 to 2 pi. So this is what our original line, integral, is equal to the Inter Grant is minus s. Then we have yes. Do you think so? This is minus s squared over two evaluated from 0 to 2. So that's just going to be negative for over two. So negative, too. And then this is just negative. Two times two pi So negative pork pie is equal to the value of that line integral. And we evaluated it using soap steer. We converted the line integral to a surface integral of the curl and found that the values negative four by

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