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Tutorial ExercisoEvaluate the surface integralfor the given vector field and the oriented surface 5_ In otherwords; find the flux of F across 5_ For closed surfaces...

Question

Tutorial ExercisoEvaluate the surface integralfor the given vector field and the oriented surface 5_ In otherwords; find the flux of F across 5_ For closed surfaces, use the positive (outward) orientation;F(x, Y, 2) = xy | + yz j + zx k 5 is the part of the paraboloid 2 = 3 ~ x2 _ y2 that lies above the square 0 $ x $ 1, 0 $ Y $ 1, and has upward orientationStepSince 5 is part of the paraboloid 2 = g(x, Y) = 3 - x _y, it is the graph of function, and SO we know that f ds = Fls (~pBk Thus, we ha

Tutorial Exerciso Evaluate the surface integral for the given vector field and the oriented surface 5_ In other words; find the flux of F across 5_ For closed surfaces, use the positive (outward) orientation; F(x, Y, 2) = xy | + yz j + zx k 5 is the part of the paraboloid 2 = 3 ~ x2 _ y2 that lies above the square 0 $ x $ 1, 0 $ Y $ 1, and has upward orientation Step Since 5 is part of the paraboloid 2 = g(x, Y) = 3 - x _y, it is the graph of function, and SO we know that f ds = Fls (~pBk Thus, we have the following. fI ds J ' J ' [~xt-zx) - yz( - 2y) + Zx Jo' J " [zxky_ 2y2(3 - x2 _ v2) + x(3 _ x2 _ y2)] dA Jo ' Jo " (2xy+[6y2 2x2y2 _ 2y4 + 3x - x3 6y- 2912 2y'+30 - r dy dx Step 2 Now, we have the following (Zx2y 6y2 Zx2y2 _ 2y4 + 3x x xy2) dy 3x Suhmit Skip (you cannot come back)



Answers

Evaluate the surface integral $ \displaystyle \iint_S \textbf{F} \cdot d\textbf{S} $ for the given vector field $ \textbf{F} $ and the oriented surface $ S $. In other words, find the flux of $ \textbf{F} $ across $ S $. For closed surfaces, use the positive (outward) orientation.

$ \textbf{F}(x, y, z) = xy \, \textbf{i} + yz \, \textbf{j} + zx \, \textbf{k} $,



$ S $ is the part of the paraboloid $ z = 4 - x^2 - y^2 $ that lies above the square $ 0 \leqslant x \leqslant 1 $, $ 0 \leqslant y \leqslant 1 $, and has upward orientation

We're trying to calculate the flux of our surface and were given Ah, see, in terms of x and y as these ecology of x of y, which is equal to the spirit of X Square plus vice where and this is important because since he's entirely in terms of x and Y, we can use the formula that the flux of our surface is equal to negative p times the G partial with respect to X minus cube times the G partial with respect to why plus our d A, um and where f is equal Teoh p i plus q j plus RK And we're also given that f is equal to negative X I minus real NJ plus Z Cube K. So the reason this formula is really nice is because we already have almost all the variables in this case, P is equal to negative. X que is equal to negative. Y and R is equal to Zeke cute. So all we're missing out to solve this problem is the limits of integration and the two partials with respect to X. And what So I'm just going to rewrite this that updated. So this would mean that are integral is equal to X multiplied by X over sclerotic X squared plus y squared um, times why started? Plus why times Why over square roads x squared, plus wise work plus a cute dear and this will give X squared plus y squared over square red X squared plus y squared When we combine the basis and weaken. This is the same as expert plus y squared to the power off one versus the power of 1/2. So this would be equal Teoh X squared plus y squared to the power of 1/2 when we cancel So playing in that means that we're solving the integral of square right x squared plus y squared plus c cubed do yet and now we want to integrate purely in terms of x and y. So now we need to get rid of thes e in this problem and we can do that by in fact, we were given with Z is equal to in terms of x and y you were given that Z is equal to square root expert plus y squared. So if we plug this in instead of Z, we will get that Zeke huge is equal to Ziese Square Times E, which will be equal to the square root of X squared plus y squared times X squared plus y squared. Supplying this back into our problem we'll have our integral is the square root of X squared plus y squared plus x squared plus y squared multiplied by the square root of X squared plus y squared D A. Now from here we have to find the limits of integration. So we know that Z goes from 1 to 3 from one, 23 Um so the best thing we can do here is to swap this over two polar sense. Since he is equal to square it X squared plus y square that looks a lot like Z squared equals X squared plus y squared, which is polar. So we can support this over the polar and set's equal are So this will give, um, to co centric circles with Radius one and three centered at the origin. And that tells us that our goes between one and three and fada goes between 01 to buy zero and 25 Now, from here we can go self, so this will be equal to then a girl from 1234 from 0 to 2 but of our squared plus one Times Square, it r squared. And because we converted to polar, don't forget that there's an R d r defeat. My apologies. You put 1/3 on the outside, so this will be defeated. Er, not the RDC. And this is just taken by taking the fact that Z is equal to our which is equal to square it x squared plus y squared and substituting out. That's how we got our square pulse one multiplied by square root of R squared and now going on from here This will be equal to the integral from 0 to 2 pi because these we have purely theta and purely our terms inside the integral. Although the fate of terms are non existent right now, um, we can break it apart into the feta. Integral multiplied by the are integral. This would be the integral from 0 to 2 pi of di fada multiplied by the integral from 1 to 3 of art of the fourth plus r squared they are, and this will be equal to two pi times one the three of artist fourth plus R squared D R. As the integral. This has just become a theater, which will become to buy. Now we just have to solve her final integral to get the flux of our problems, and that would be equal to two pi multiplied by are the fifth over five plus art of the third over three from 1 to 3. Now the equal to two pi times 2 43/5 plus 27/3 minus two pi of 1/5, plus 1/3. And this will be equal to if the mold fight over finds by freeing over threes by five to get the base of 15. And this will be equal to 1712 pie, and that ever since the flux of the surface for this given problem,

So we're gonna be using this formula right here, which is f dot De ass is equal to the double integral over the region of negative p gx minus que g y plus r d a. So first thing we'll dio is calculate our G setbacks and g sub y given that g of X y is equal to the square root of X squared plus y squared. So when we evaluate it, we get RG sub x x over the square root of X squared plus y squared and RG sub y is the square root of or is why over the square root of X squared plus y squared So when we multiplying everything together we end up getting the double integral over the region d of the square root of X squared plus y squared plus z cube d A, um and keep in mind d a is dx dy y or d y d x so we can't have the Z. So instead, we're gonna replace it with this because this is the same thing and see, So when we do that, um, we'll put it in place. And what will end up getting as a result when we substitute it is this double integral of x squared plus y squared plus one times the square root of X squared plus y squared d a. Then we can convert this to polar coordinates eso When we do that, we recognize that the radius is the race of the region is between one and three and our state of values between zero and two Data Um so what we'll do is we'll put that here zero to pi that is eso We'll have our squared plus one and we should recognize that when we see these X squared plus y squared, we should definitely convert it. Then what we'll have is the square root of r squared, which is just our times r d r d theta eso multiplying everything how we end up getting the integral from 1 to 3 of the in a row from 0 to 2 pi of our to the fourth plus R squared D r D theta we can move this d theta over here just giving us two pi times the integral from 1 to 3 of art of the fourth plus R squared er this is just gonna be part of the fifth over. Five. This will be our cubed over three. When we evaluated at three and one, we end up getting two pi times 1/5 plus one third, and our final answer is going to be, um, way wouldn't just get that. It's also two pi times 2 43/5 plus 27/3 eso When we multiply all that and that everything together we end up getting 17 12 pie over 15 as our final answer and remember, that is flux.

We were asked to calculate the flux of the given surface and we're given that Z is in forms of G of x y, merely that Z is equal to X and y eso we will be have to use the equation that thes surface of our flux integral is equal to negative p times the partial with respect to X minus. Q. Times The partial with respect to why, plus our d a, um, plan. In this case, F is going to be equal to our equation, so F is equal to p I plus q J plus Archaic and we're given that f is equal two x y i plus y z j plus z x k. So since we're also given that Z is equal to G of X, why, which is equal to four minus X squared minus y squared? Uh, we're starting to plug into our formula. We actually already have everything we need besides the X partial and the white partial What the X partial of C is going to be d g D acts which is equal to negative X squared drive, which is negative two x and it will be the same for wise, so D g d y is equal to negative. To what? So now we have both the partials we have P Q and R because P Q R are the terms attached i, J and K And that is enough to plug in for integral. Um, because we're also given the X goes from 0 to 1 and why goes from 0 to 1? So playing into our internal, we'll have the integral from 0 to 1 into girlfriends. Yodo One of negative X y cause that's a negative p multiplied by negative two x minus Boise I won't buy by negative two y plus z x d a. And this will be equal to go from 0 to 1 and a girl from 0 to 1 of two x squared y plus two y squared because he plus z x d y d x In this particular problem, the order really won't change anything. Dx dy Why would also work? Justus? Well, as do I. D x and now substituting out Z because we know ZZ Goto four minus expert minus y. We need to get rid of the Z in order to make this a integral in terms of all the X and y. So doing so well give the integral from 0 to 1 Girlfriends, you know, one of two x squared y plus two Why square multiplied by four minus X squared minus y squared plus tax multiplied by four minus X squared minus y squared of the white checks. And this will be equal to the same integral 0101 of two x squared wind plus eight y squared minus two X squared y square. This isn't going to be extremely pretty, by the way. This is 23 term foils when it's two x squared y squared minus two y to the fourth plus four x minus. X cubes X cubed minus X y square, D Y d backs And now we're good to do the first. Integral with respect to what? And this will be equal to the angle. From 01 of X square. Why square plus 8/3 Swyche Cute minus two X squared. Why cubed over three minus 2/5 wide of fifth plus four x wine minus X cubed Why minus X Lies cubed over three d x, and that will be from 01 so sentence firms Jodl one. We're just going to rewrite this both every Why changed to a one just rather nice for us. This will be equal to the integral from 0 to 1 of X squared, plus 8/3 tu minus two X squared over three minus 2/5 plus four x minus X cubes. So there should be a why on this term my four x cubed y, um minus X over three decks. And this is a early integral, just with many different X terms in it's all of these are give me just ah, integrated using the normal, um, rule of integration. So doing this will give the integral from first let's simplify and go from 0 to 1 34/15 plus X squared over three plus 11 x over three minus X cubed d x, and this will be equal. Teoh 34 x over 15 plus X cubed over nine plus 11 x squared over six minus X to the fourth over four from 0 to 1. So again we'll just be swapping out all the exits with ones. So this would be 34 15th plus 1/9 plus 11 6th minus 1/4. From here, we have to find a common denominator. So we need a denominator that has, um if you ever need to find a nominators on, like, quickly, One way to do it is if you just factor them into prime. So in this case, five and three, three and three, three and two. And to and to, um, as long as you make sure each prime is accounted for Ah, you confined your denominator quickly. Highway. So in this case, five times, three times, three times, two times two. Because the most we have is two twos in the four term and two threes in the nine term than a five on the 15. This will give 15. 45 91 80 Denominator is 1 80 I'm from there. Um, I will make the 34 turned to a 408 The one turned into a 20. The 11 will turn into a 330 and the last one will turn into a 45. And then this will give a answer of our flux to be 713 over 180. And this problem was more tedious. than complex right? It was a lot of computation. That's also part of the reason why our answer is so gross. But this would be how you calculate the flux for this particular surface.

Defining the flats. We have the double integra of mps. To be equal to this, where this M. Is a unique normal vector. So, you know, it's that's our T. S. It's equal to the magnitude of our you. Because our RV D. E. So the flags through the surface is always defined only if the surface is worth oriental herbal, so that if the surface is oriental herbal, then there will be two normal units vectors at every point. So we can therefore right the S. To be equal to end the A. Which is equal to platts or minutes. Are you codes RV divided by the magnitude of our you? Because R. V. Times Are you because you are free? The A. Which is equal to plots or mine? It's are you cause RV mm mm. So to find the flags we always use and which corresponds to the positive orientation. So in view of that, it is given that the upward orientation is in the positive orientation. So according to our size five. So according to according to ex exercise five, we have uh to be called to you blood V. I plus U minus B. T. Plus one plus. Do you bloods fee? Okay so the interval of U. And V. You have you from 0 to to envy Is from 0 to 1. So you first find the durability of I with respect to you. And this would give us I plus G plus two K. And that's of RV. RV will be equal to I minus G. Blood's key. So this implies that the course brew. That's that is are you because RV it's going to be vehicles. So you have I. G. Thank you. You have one one. So 1 -1 1 and this is equal to three. I lot G minus two. Mhm. So if given that the parallelogram is oriented in the upward direction, that means that the upward direction is a positive one. So the components of the S. Must be the city so it must be positive. So what it means is that so this implies that our D. S. It's going to be minus are you? Because our RV E. It's equal to -3. I my name is key last two K. The mm. So giving our F. So give him our F. To be equal to F. In the X. Y. Zero components To be equal to zero. You wasted the power X. Y. I minus three. Z. Mm. Yes. Yes. Three said he was the power S. Y. J. Components plus xy. Okay then so we have it's in place we have F. Are you the Should be equal to 1-plus 2 U. Plus V. You have E. To the power you blood speed. U minus fee. I minus three. One plus Sue. You plus V. You have each to the power X. Y. That is U. Plus V. U minus B. G. Then you last fee U minus V. Okay. And this is equal to 1-plus 2. You plus fee. You have eaten the power U squared minus V squared I -3. 1 Plus two. You plus fee. It's the power U squared minus V squared G. Lads U squared minus V. Squared key. So therefore therefore the floods. We have the w in secret to s of F. The S. Is going to be equal to this two U squared minus B squared the mm notes that are you is we deal they ain't ever zero su then fee is also within the intervals everyone. So this then it's going to be equal to two. You have the integral from 0 to 1. 0 to 2 of you squared minus V squared the you d fee. And this is equal to suit n. c preserves one of You have Yuki over 3 -3 cubed minus U. V. You U V squared 0- two d. V. Which is equal. Soon Sue The n. c. girl from 0 to 1. This will give us, it's divided by three minus two V squared T. V. And this is equal to two. You have it be divided by three minus To be cubed divided by three day in server from several to one. So the lower interval uh goes towards zero and this is going to give us to, It's divided by 3 -2, divided by three Which is equal to two off, divided by three which is equal to four. So hence our flats is equal to full


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