Let F (y + 2)i + (22 + 22)j + yk and S is the half-cylinder y = 0 and y = 13 Use Stokes' Theorem to evaluate1x2 betweenM x F) . ndo...

Use Stokes' Theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ $$ \begin{array}{l}{\mathbf{F}(x, y, z)=-3 y^{2} \mathbf{i}+4 z \mathbf{j}+6 x \mathbf{k} ; C \text { is the triangle in the }} \\ {\text { plane } z=\frac{1}{2} y \text { with vertices }(2,0,0),(0,2,1), \text { and }(0,0,0)} \\ {\text { with a counterclockwise orientation looking down the pos- }} \\ {\text { itive } z \text { -axis. }}\end{array} $$

Okay, let's go ahead and solve this problem. So we're given the vector F as X plus y squared called a white plus Z squared coma Z plus X squared. OK, and we are supposed to evaluate a line integral c f dot d. R. Where c is the curve that's drawn right here. It's an equilateral triangle with the Vergis is at 100010001 mhm. So if we try to evaluate this normally, which is something doesn't seem like it's too complicated to do. But the steps are going to be long because the parametric ization of C we can split it into three lines. So we basically have to evaluate three integral minimum in order to solve this problem straightforwardly. But we can use Stokes Terram in order to simplify the process a little bit. So let's talk about mhm. So instead of considering see as three lines a curve of three lines, I can consider it as an equilateral triangle as the surface s okay and because I know that the plane that contains this these three lines is the form X plus y plus Z is equal to one from there, we know that the normal, um, the normal vector is going to have the form at one comma, one comma one. And because when we dio ah, unit normal, when we use a unit normal vector to do the calculation, we just need to calculate the the magnitude of one comma, one common one which is equal to square root of three. So we can actually easily calculate that the unit normal vector is 1/3 off one comma, one comma one. Okay, so we're going to use that in order to solve the problem using Stokes there. Um, and let's recall that this is simply the double integral through the surface s off curl, yes dot ds, but I will use dogs in. He s okay. This is not the vector is just a surface in this particular case. So we would have to calculate what the curl of f is going to be. And that part, the calculation is not that difficult. So I will let you do this part, but it is basically we calculated like this. I j k. It's an operation to take the derivative partial derivatives with respect to X y Z and then we take the cross products. So this is going to be X plus y squared y plus Z squared and Z close X squared. I already wrote it down over here after my work. It simplifies very nicely. It'll be negative too, of Z comma X comma y Okay, so when we do curl s don't n what really happens here is that you will get negative to over square root of three and then you will have Z plus X plus. Why? But what we know about the plane here is that although we have it in different order, X plus y plus Z is equal toe one. So this simply goes away. So you will have negative to over squared of three which we can pull outside the double integral. Okay, so what we're going to evaluate is essentially negative to over three the double integral through S. D s. And then I like to use this technique all the time, but we know that this is simply the area off the surface. Okay, Now, this is an equal Atal triangle with side length, square root of two. So the area is Route 3/4 times Route two squared or, in other words, Route three over to. So when I multiply that number to negative two over Route three, you will get negative one, and that is the value that we're looking for.

Hello, everyone. Today we're going to solve the problem of Fight your functional face. That's where I that's look. The minus, like U K See, is a given but X squared. Plus why it square? Because what in the Exley, please? Or their Deco's zero? So closing the love half dot the year will be double integral or a sigma. They'll cross f dot in the so bill Cross f will be I. Nobody do. It's that square the don't by the way, two weeks. Okay, nobody does it minus like cute. So which comes to be like minus three was square I plus does that the plus toe kid so closed in Developer see her daughter They are Is it cool to double integral all the stigma minus tree? Why Square I Plus does that, plus two kids in the K delight the it's which is equals, doubling that'll over sigma to debate the X. So stigma is founded by X. Come away more or less off Hector Square. That's White Square. Let's Don Rickles what? Which is our commentator? Zero or less than recorded data less than miracles, toe by zero, less than record toe are less than recalls. Smart. So those Linda will oversee Earth door. The are will be zero toe by 0 to 1. The war the dictator which is to do are square 0 to 1, which is to put that standoff a question. Thank you.

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