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Problem 17 Easy Difficulty

A particle moves along line segments from the origin to the points $ (1, 0, 0) $, $ (1, 2, 1) $, $ (0, 2, 1) $, and back to the origin under the influence of the force field $$ \textbf{F}(x, y, z) = z^2 \, \textbf{i} + 2xy \, \textbf{j} + 4y^2 \, \textbf{k} $$ Find the work done.

Answer

$$3$$

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Video Transcript

So in this section we've been using Stokes serum has a really helpful tool. Thio, simplify The integral is that we're dealing with specifically surface integral. So Stokes Terram tells us, um, this right here where c is the boundary of a surface oriented counterclockwise And we have the freedom to choose the surface whose boundary iss see So we have as being the region inside the rectangle formed by the given points we are given f to be the vector Z squared two x y for y squared. So that means that the curl of f will be equal. Thio um the partial derivative. Why, with respect are the partial derivatives are with respect to why so that will be eight y minus the partial derivative key with respect to see um, on Ben, looking at the curl formula, we get that this will be to Z minus zero and then, uh, two y minus serum. So the curl of APS is equal to the vector eight y to Z. Why? And then we want to find the equation of the plane passing through the four given points. So we let the equation of the plane we know to be a X plus B I plus C z equals D. Um, and the 0.100 on the plane indicates that a equals D the 0.1 to 1 indicates that a plus two b plus C equals D. And then the 0.21 indicates that to B plus C equals Dean. So if we subtract equation A from a equation too, we see that, um, a is equal to zero onda. That implies that D is also equal to zero. So just by doing that simple but yet clever algebraic manipulation we can change our equation of the plane to be much simpler. So it'll be b y plus c z equals zero because both a and the R equal to zero. Then we can use Equation three true place See, with the negative to be so a guy minus two easy equals zero. So that tells us that, um, why is going to be equal to R Z is going to be equal toe y over to. That's because we can move this over and then divide by, um, divide by to be. Now that we have that, um, we can go back to our formula and what we're going to be using is we're gonna have now it's the double integral of negative eight y times zero and thats zero is coming from the partial derivative of G with respect to X um, N G in this case is Z equals y over to. So because there's no X involved, that means the partial derivatives zero, which means we can just get rid of that minus two z times one half because that's the partial derivative of G with respect to wine. And then lastly, we add Z so it'll be Are we, Adam are. And in this case, are is going to be two y. This will be d A. And this will simplify to a negative z plus to I d a. Then since Z equals y over to we can do that here, um, and simplifying things further. What will end up getting is three over to why. So we'll just be left with y inside and three over to on the outside. And with all that, we see that the bounds of integration. This will be a d y d x. We know the bounds of integration are going to be for why? From 0 to 2? Because that's what the rectangle looks like. And then, um, this will be 0 to 1. It shows us that we have the amount of work done. Will be three three units, depending on what units were in. Um, and that is as a result of what we can do A stoke serum.

California Baptist University
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