Question (4 Points)Given F(x y) = Zxyi 2y)j. The work done by the force in moving particle from (0.0) to ( 1. 2) along the straight line joining the tWo points iS:b...

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.

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Whereas to find the work done on a particle that moves along line segments from the origin the points 100 1 to 1 0 to 1 and back to the origin under the influence of a force field f where f is the force field Z squared I plus two x y j plus four y squared. OK, so we'll use Stokes is here, Um, rather than computing work directly. That's why this is in this section. So we'll let s the the planer region, which is enclosed by the path of the particle. Now this is a plainer region because there's only four different points here and do all lie in a plane. So, in fact, they lined the plane Z equals one half. Why, Therefore s is the portion of the plane c equals one half why for X between zero and one because that's the largest tax ever gets. And then also we have y lies between zero and two because it's the smallest and largest y ever gets. And see that moves from the origin 2100 2121 20 to 1 and back to the origin from above that's a counterclockwise orientation. And so our surface this induces an upward orientation for our surface. Now the curl of our force field F this is eight. Why I plus two z j plus to why, okay. And therefore by Stokes theorem the line integral to the work done Which is the line integral over the curve C of f This is the same as the surface integral over s of the curl of f and this is the same as the double integral over the region D in the X Y plane of And because our plane is a function of X and y, you can use a shortcut. So we have the opposite, uh, the first component of our force field of the curl of our force field. Just negative eight y times the partial derivative of Z are playing with respect to X, which is zero minus thes. Second components of the curl of our force field to Z times partial derivative Z are playing with respect to why which is one half plus the third component of the curl, which is to y d a. So simplifying and plugging in Z in terms of X and y we get iterated into growth from 01 integral from zero to, um becomes two y minus. And this is one half. Why de y dx? It's really this is just three halves. Why eso taking the anti derivative with respect toe? Why beget three hats time integral from 0 to 1 of one half y squared from 0 to 2 DX evaluating We get integral from 01 times three d x, which is just three So using Stokes there and we found that the work done was just three.

So I guess in reality a power you have to use a computer or calculator to you and I'LL compute the first part. So once around the circle culture class wise so kind of standard Parametric ation Mexico's to co sign T Boy hose to society and the tea goes wrong. The other two pi So what is F or far off? T? We're simply replaced x by to society And why by two scientists Second power should be fourth scientific all scientists in our prime iss negative to sigh Inti to co sign t no so integral integral from zero to two pints a dot product of these two, it's negative. Eight sci fi coz I scurti plus a sai inti Cose I squirt ti So you find if I didn't make any mistake their teacher this who term looks like they're canceling. Therefore we should get zero Well, let me tell you actually, if I compute anything wrong ex Chris for cause I score thirteen x wise for scientific society The ribbed him is Neck deep Sigh and cose I That's right. Then we takes Ah, we take a shot for that So we're negative a psycho's I square eight cycles I sparkle. It looks like they're indeed can sew, sew, sew This this inter cross here