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Use Stokes' Theorem to evaluatewhere C is oriented counterclockwise as viewed from above_F(x, Y, 2) = xyi + Szj + Zyk,Cis the curve of intersection of the plan...

Question

Use Stokes' Theorem to evaluatewhere C is oriented counterclockwise as viewed from above_F(x, Y, 2) = xyi + Szj + Zyk,Cis the curve of intersection of the plane x + z = 2 and the cylinder x2 + y2 = 144.

Use Stokes' Theorem to evaluate where C is oriented counterclockwise as viewed from above_ F(x, Y, 2) = xyi + Szj + Zyk, Cis the curve of intersection of the plane x + z = 2 and the cylinder x2 + y2 = 144.



Answers

Use Stokes' Theorem to evaluate $ \int_C \textbf{F} \cdot d\textbf{r} $. In each case $ C $ is oriented counterclockwise as viewed from above.

$ \textbf{F}(x, y, z) = 2y \, \textbf{i} + xz \, \textbf{j} + (x + y) \, \textbf{k} $,


$ C $ is the curve of intersection of the plane $ z = y + 2 $ and the cylinder $ x^2 + y^2 = 1 $

In the problem we have been given F of X, Y and Z. That is equal to do I gap plus X Z. Z gap plus X plus white Kick up So Dell cross F becomes I jay. Okay so we have the girl of F that has given us this now they're going to find the double integral that is carl F dot ds is equal to d minus of 1 -1- of -1 into one plus zero minus two. Do you? So this is equal to D zero minus one, do you? So here will substitute White plus do as Z. Now this becomes do you hear Y plus two minus one day. This is equal to double Dickel White Plus one Day. And further as there is the production of a C on the x ray plan. So so here this is the reason inside the circle X squared plus y squared equals one. So we have the integration and further the integration becomes 0- two Pi. Here it is Scientific to upon three plus half the editor which is equal to Minour cause teach upon three Plus feet up on two, they want to pie. So after solving this integration we have this as -1 upon three plus 5 Plus one upon 3. So this course each other. Therefore we have the answer as pie. Hence we can write it as F dot D R equals five. This is the answer.

Were given the Vector Field F and the Curve C and were asked you Stokes theorem in part A. To evaluate the line. Integral oversee of F So f is thieve vector field, X squared Z I plus X y squared J plus Z Squared K and C is the curve of intersection. The Plain X plus y plus C equals one and the so under X squared plus y squared equals nine oriented counterclockwise, as viewed from both. First of all, in part a the curve of intersection See that were given. This is an ellipse in the plane X plus y plus Z equals one. And it has a unit normal vector and the churches Well, a normal vector to be I plus J plus K But then we have to normalize this. So one over we're three times I plus J plus k the curl of defector field F This is zero I plus X squared J plus why squared? Okay. And we have the curl of F started with normal vector n. This is going to be one over route three times x squared, plus why square and therefore by Stokes is the're, um the line integral oversee of f is equal to the surface integral. Over our surface s of the curl of F, which is the double entry rule over X squared plus y squared is less than the nine from the cylinder of one of three times X squared plus y squared. Yeah, and converting to polar coordinates. This is the integral from 0 to 2 pi into growth from zero three of we have. See, this is of our This becomes r squared times are which is are cute the bruschetta. So actually, in between this this is actually a double integral. All right, the surface s of one of the three times x squared plus y squared yes, and then changing it to double integral over. Expert post y squared less than 49. Get rid of our one of a group Evaluating this last integral we get to pie times 81 4th or 81 pi over two importante were asked graft both plain and the cylinder with domains so we can see the curve c and the surface used in part A. To do this, I'm going to use Dez Moses Parametric surface graph for Yeah, this is a view of a plane in the cylinder and you can see the curve C and the surface contained by sea. This figure finally in part C, whereas find Parametric equations for the curve C and to use them to grab. See? So first, let's find the Parametric equations. Well, mhm. One possible parameter ization is you could have that X equals three cosign t y equals three sign t Then, using the equation of our plane, we have that Z is equal to one minus three co sign T minus three science T where t ranges from zero to keep I and I'm going to graft this using Dez Moses free Parametric curve graph for

Today we're going to solve for the number nine. Your function is. Can Britain us x Y I plus X squared, the That's That's square. Okay, sees the intersection of the parallel para Ba Lloyd that equals X squared. Let's slice quit on the plane. The D equals white with the contact clockwise orientation. So why? Because excess square plus by square the difficult X Square plus why minus half the whole square, which is equals one by foot. Go intersection is a circle, which is X squared plus y minus half. Who's core equals one by four on the plane vehicles. Is that so? They will cross f will be I the works. It's like the the light. It's a square. Okay knows that that square which comes to be zero. So closing the girl over f dot they are is equal to zero. That's the end of a question. Thank you.

Okay, let's go ahead and start this problem. We are given a three part question, but the answer is for part B and part C. I usually always do it anyways. It's basically the para militarization off the curve or the surface and the visual ization, grabbing it so that whenever you solve these kind of problems, it's just easier to see what's happening visually. In order to make sure that the dot product is taken properly, the cross product is facing the right direction, etcetera. So we're just gonna go through it exactly the same way that we have been doing it. So F is given as expert Z comma X y squared comma Z squared and the Curve C is given by Plain X plus y plus equals toe. One intersecting with the cylinder X squared plus y squared is equal tonight, so because it asks us to do the visualization, I want to go through that a little bit first. So if you imagine that Xcor plus y squared is equal to nine, is this cylinder that looks like this? We know that the radius is equal to three, and right here the center passes through the Z axis, we have a plain X plus y plus Z is equal to one and graphing that one is quite simple. If it's on the first, often it just passes through one comma, one comma, one like this. And I want you to remember that this triangular sheet, it actually extends forever. So it's more like, um, uh, she that aligns with his triangle like this. Okay, so hopefully that will help you visualize this a little bit more. So when I draw that sheet of paper so that it cuts through the cone, it will look something like this. Yeah. Yeah. And that's basically what I drew over here. Okay, so let me get rid of those Mhm. All right, Now the parametric ization. Um, using a cylindrical coordinate system would really be helpful here. So I can say that X is equal to are consigned. Fada. Why equals two r sine data Z is equal to Z, but I wanted in terms off our Insigne data. And because we know that the plane is one minus X plus y. Let's write it as one minus our times. Cosine theta plus sign data. Of course, you don't have to write it exactly the way I did. I just like Thio. Write it this way more. Okay. All right, so we're done with the primary. Parametric is ation portion and the graphing portion. So let's move on to the evaluation, Okay? All right. As usual, we are asked to evaluate the line in the group f dot d r by using Stokes there, um, we can evaluate it using the surface by taking the dog product of the curl of f with the normal. Yes. Okay. So we can calculate the curl first. Again. It's quite straightforward calculation, so I'm not gonna go through into detail what I've already done. The calculation. It turns out to be pretty nice. It's zero comma, x squared y squared. I'll put this in. Terms off are in theta later. Okay, The normal normal vector is very simple here because we are already given the plane the equation of the plane. So it is going to be parallel to one comma, one common one. So the dot product is going to be very simple here. Oh, curl of f dot n will simply be the sum of X squared and y square. So What we're going to evaluate is the double integral through s of X squared plus y squared. Oh, now I want to evaluate this double integral using thesis electrical coordinate system. So I will get double integral X is R cosine theta. Why is equal to our sign data? A Z doesn't show up, so that's good. That's good on our end X squared plus y squared does off course just simply r squared. And then we changed our coordinates from X y into our and data, So we will have r d r d feta. So the radius changes from 0 to 3. Because, as you can see, the projection off this surface onto the X Y plane is simply the circle here. Mhm. It's also given in this equation right there. Yeah, and because it's one revolution, the angle changes from 0 to 2 pi. So long story short. We're evaluating the double integral off the angle changes from 0 to 2 pi The radius changes from 0 to 3 off our cube de are de fada. So it'll simply be two pi times, um, three to the fourth power divided by four. And that simplifies to 81 pie divided by two. And that answers all of the question of this problem


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