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Use Green s Theorem to find the counterclockwise circulation and outward flux for the field and curve CF= (2xy + -Sy?)i+ (2x-Sy)j...

Question

Use Green s Theorem to find the counterclockwise circulation and outward flux for the field and curve CF= (2xy + -Sy?)i+ (2x-Sy)j

Use Green s Theorem to find the counterclockwise circulation and outward flux for the field and curve C F= (2xy + -Sy?)i+ (2x-Sy)j



Answers

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field $\mathbf{F}$ and curve $C .$

$\mathbf{F}=\left(x y+y^{2}\right) \mathbf{i}+(x-y) \mathbf{j}$

Problem. 16 that we want to find the circulation. The flex. Uh, on this it is so we can use the X y e goes toe squared, too. Call science Sita science data, which is exactly the integral the pass up into pro here. So we can just use this to formula, and we no need to use the grains serum. So just into grow, see her from 0 to 2 pi and the ex on the exam should be 1st 3 Why, Right, three. Why? And, uh, the ax of the find the score to science data. You see that, Andi? Why should be? Because I don't see that. Do you see that? Right. And this is, um, the axe. The axe B minus scored two cy and thus and d y. And should be a to explain why. So two acts minus why And you Why? Cause I see that you see that? Hey, uh, this term menace is because we have a CNN asks that this becomes signed to Sita. So is this time. So that becomes by this three. Score to over two, and thus to over two. Uh, that's scored two and times to pie right, So that equals two, uh, minus scored. Two pi. Okay, that's the how can you The same. I said to calculate the bucks here. So, Andie, why took row 0 to 2 pi And does this sound two? That's real. I do. You Why do you actually be cause I and minus and DX and the extra B minus this season? And the exit B minus two. So have a place here. Sorry about that. Plastered to ask. You see that? So this time vanishes and some radishes. So we have scored two or to minus Oscar. It's over too. And times to pie. So that's zero for the bucks, right?

Okay. Uh, 18 declined The separation, the flux. So what, you want by your circulation first, use this formula here. So partial, impartial acts should be zero and minus part man. Part y I am partial y partially an impartial. Why so the minus X to Why? Yeah, I know where one of us. This. So it's just inter, bro, uh, to x y over one. Those wire square, this first the x ray and then the y and acts from minus. Why my white square two one minus y square. My wife, um I just want to What? Okay. So, no. Why times, Uh, why we're one plus y squared times x square, my justice. So stop Bo minus wise. Where? Oh, my square. Minus this. Where's zero? Right, So it's just a zero. So it's totally zero. Okay. Yes, it zero. Let's find out what? Ah, the flux is so partial. Hamper tracks. That's partially Marshall. Why? Which is too aware? One plus y square dx dy y right. Uh, calculate this. We want to first into pro acts minus this. Escort this, You know whether this so this becomes, uh, floor squirt while minus Why? Square over once. This wife's where on that. Here's a even function, right? That's a you and function. So it can change this into zero and sent this into AIDS. Do you? Why so hard to calculate this? Ah, we can use that. Why, Toby? Science data, for example. And this becomes, uh, the wise call science Sita. Do you see that? So from zero to senior from wife from, they're going. So see them from zero to buy over too. Okay. And we have a cause I square and one close. Sy, Where do you see that? Right, so this can be changed in tow. Uh, cause I square is one minus science. Where? Which is minus one minus science choir plus two. So we have this part disbarred. It's a minus y. So it's minus one times. Oh, here's a Here's a don't together eight, uh, minus one times. Eight times by over two. So it got a minus for pie was birds part, and the last part is just into grow eight. Well, the science either square, and this is a two. Right. Okay, so take a look at this. Hey, let's see that. This one we can change that wine too. Cool, science. See the square of sciences? Where? So we have. So Okay, stretch one out s. So that's why those two attention to sit square. Uh, here's until you see it. So we know this is just detained in cedar, right? Also, we got it using under one. Uh, this becomes 16 into, bro. I didn't say that from dear old. So affinity. Ah wants us to be square. One has to He where? And Yuki. Right? Hey, here's our final eso. Exactly. I think that's correct. So we can ah, let's you to be scored two tea. And this becomes, uh, eight score, too. And it's going to grow one. Does you square have a You witches are pretending to you, um, infinity to zero. So that's a pie over too. Right? So we got a four times square to pie for the last part, and it's the first part is minus four pi. So for high score to minus four by its power flux

So we have. Ah, this vector field f All right, X y. That's why squared along the Eid erection. Um, X minus. Why were the chief direction do you want to? Oh, when the flux, where I'm do they curve? See that he's the X Y plane. This curve is our between the point. Syria, Syria on 11 And over here we have the curve. Why? This is why you go to X squared on this curve Here is the the reflection of duck. So all three of these, actually. So that would be X equal to y squared. So we don't think the rate where is closed, girl in the contra clockwise direction. So, like that. So, uh, buy stocks here and we call these interior. Yes, well, that flocks. He's gonna be changeable oversee over him. The x plus N b y where this is, um, my buddies and that function that, uh So I green they're green steering. Are these people doing the right side? That service? Uh, I shall offend. With respect to x minus by shuttle. What am respect? Why next? Ey, this is fresh over. So let's, uh, go ahead and see what he's myself off. And with respect to eggs, most of the bar several X minus. Why? Respect wax? And then they should be all right. Where shall over. He's with respect to why would be nobody's him. Garcelle off? Um X Why, plus y squared respect with the respect away, you need to know, um, until, uh, or partial easy respect works. That would be one personal doctor district wide would be He's gonna Reavy. Well, x on being that the to what? So you think the very dysfunction Yanks why I'm 40 minutes year. That is that these, um, move. You have the X first, so the bounds for X X goes between. Ah, there, there, there. There. Yeah, there. So this part would be y squared. Not in here. So X has to be beer down. Why squared on this side? We can. So for that, uh, for X here you take the square root. You know that I'm since excess positive not be just X because X is watching you. So, um, so that X should be between Why squared on square with the why on, uh, for why we see that y goes between. It's you on one is that point? There is one one. So, uh, to have that under? Why? Between Syria on one. So that said the bound sits the violence to be Why Squared square with the boy dump on, then here on top. We have one zero. So this one in Syria, that is for one. And for X, we would have meeting White Square. I'm squared away. So when doing this injury, uh, or for rent a week X metal off. Wanted just x with any type of minus x Early miners X squared Hafs Monday. Geof is too thin. Did you B minus two. Why Intel? Finest wise minus two x. Why? So that, uh, awaited 18 squared off? Why? I mean, why squared us? Um Square the boy. My nurse squared. Why? Squared pose minus the square. Little why still comes. Why? From being minus this evaluated y squares would be while squared. Why squared square would be Why today? For that is why the fourth power? Um miners? Why squared? I'm still attempts. Why? So that will simply find his expression Since why is positive? Why is also positive? Way so full I want to be was it you because, uh, well, why would be between zero on once? So it's positive so that I would have to be equal to screw the wife. That would be White House in the White House, minus the white screw the way I'm then or issued off minus y squared minus and miners would be plus y to the fourth and minus and minus. Um, you must y to the third. Let's to like this circles, plus soon. Why in the third, one day in Ah, you have that on the there is not in that began Cancel Teoh Movie Central You have to do right now. Not with respect to why? Why Screw eso? Forcing visited is committed in us. Why today? Agreed sometimes Why did it three house and then all these on his wife with a 1/2 so degrading? These with respect? Why? So this will harm interiority. The little boy to the 1/2 would be why they three house them's too third's that is interactive off. These irritable minus y squared would be minus. Why did the third power? So why today one engine blocks away to the second power divided by 200 element to subdivide life to I'm still squared And then here would be and the other problem miners white Three hearts would be minus. Why today, Why Huck's been Does he have a two and then not multiplied by, um, to first? I want to fix Do you like to My So that is anti for these traditional be miners. Why today? Well, we're divided by three. This will be minus flight today, Keith Power and then felt multiply by five on the bottom and then for decision would have Ah, what to the fourth power? Very by four. And so, um, a we're gonna see behind these to be just Uh huh. So when you to evaluate all these between zero on one, you want us here so that these would be equal to, uh, well, these had one would be just 2/3. You're thirds minus 1/4. Okay. Plus minus two them. Student by five. Um, miners minus 1/3 miners. Warren over tool times five. Uh, glass one, huh? And then when we evil wait that zero, we're gonna get zero. I'm serious. I'm zero, I'm sure. Thank you. Should be that minus you. And, uh we will be left with this fraction. So here we have. Ah, all these. Uh, too much. So these would be, uh, 2/3 minus 1/3 that he's 1/3 to be able to 1/3. Now, uh, he's these councils with that. Just be sensible toe to force to be able to 2/4 minus one for 2/4 minus 1/4 fourths, minus 1/4 physical 214 to these three council, we go to that plus one for I'm then, um, for these two, well, they would be minus that. But, um, all these number is equal to God, you know, like by two on the bottom. So this will be I think that's true. So, eight, you know, we're 10 being the summary. Why would then? So he's both of these fractions are gonna be nine minus eight miners once in minus nine or two times five. I'm being She was up 1/3 14 perishable to that on being well, uh, you can write, he says always through them stew, so that to get, uh, two of those fractions, we would need to You need to multiply. So this 1st 1 by for by 54 times five. It's one. Um And then this one by five times three. We five and then this one by two times three minus nine. I'm still That was three. My ideal is gonna be doctor. There. The times four times five on these one would be for M C five on that one. With you, not two times. Five times two. That's three. So that Yeah, Come on. Damn near number thing we'll have This will be 20 uh, my plus feet 18 and then his number is six. So, nine. I am six. That would be 54 minus 54. Show no to be that over for I'm spy. There's three. So because the, um, always ready why minus 54 which was the equal to on is equal to she Vieira. Mm. So 19. 19 plus duh. Said she Vehicle tow minus 19. Yeah, home minus 19. Over four times because five a school dean, I'm dizzy with people toe, sometimes $37. 16. All right. Or sushi. So minus 90/60 would be video only. Work flux. That looks

Problems, Everything. We want to find the separation that bucks. So we can use this from the so part man part. Drax should be two acts. Two X cube. Why? And partial on partial axe is also to ask you. But why? So? Separation should be his girl. Right? We can used this woman. Yeah. So the intro, uh, you grow ex from's with the two. And why From lex. Why? From X square, minus x two x Hey, part of I am partial X. Oh, it is three x square ice where and thus partial y. And that's just one of two ex Forest, right? Andi y yes. So that's equals two. About this town becomes Mr to Rex wars, um, two x minus Access where? Like and plus Oaks Square. White Cube. So this minus while this Because thanks. Minus three x by and three thanks. Four. And Linus killed. And your ex. So we have Hey, after the fifth minus one over to experts steaks and plus. And this becomes too after Fitz, Right? Ah, minus three extra fee. Thanks. And nice. So the extra date on minus axe to the oh, He had the cell in sorry. Yes. Okay. Right. So this becomes three, actually. Fifth no one. There were two. 2 to 6 on minus towards having so nowhere to two to the seven. Okay, this time. God. And for this Oh, it is minus. So we're eight. I'll plus through our age 2 38 and minus one. The one I to deny. So that becomes 32 minus 64. Ah, thus, uh, 32 times three. So there's become 64. And Linus, uh, 64 times eight. Right. So it is becomes ni nine times before minus eight times a day for riches. 64 overnight. Well,


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