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QuestionUse Stokes' theorem t0 compute (f (curlF N)dS for the vector field F (T,V,2) = 21+Wj+rk where $is the triangle with vertices (1,0,0), (0,1,0).and (0,0,...

Question

QuestionUse Stokes' theorem t0 compute (f (curlF N)dS for the vector field F (T,V,2) = 21+Wj+rk where $is the triangle with vertices (1,0,0), (0,1,0).and (0,0,1) vath counterclockwise orientation~Somy; thats Incorrect Try Yagaln?

Question Use Stokes' theorem t0 compute (f (curlF N)dS for the vector field F (T,V,2) = 21+Wj+rk where $ is the triangle with vertices (1,0,0), (0,1,0).and (0,0,1) vath counterclockwise orientation ~Somy; thats Incorrect Try Yagaln?



Answers

Use Stokes' Theorem to evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ $$ \begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k} ; C \text { is the triangle in the plane }} \\ {x+y+z=1 \text { with vertices }(1,0,0),(0,1,0), \text { and }(0,0,1)} \\ {\text { with a counterclockwise orientation looking from the first }} \\ {\text { octant toward the origin. }}\end{array} $$

In the problem. The equation of the circle Is obtained by substituting y equals zero equation of the hemisphere. That is x squared plus Y squared equals one. The paramedic representation of the boundary. C. Is given as this. And further we have this integration that is dead after dear. That is given as integration two by 20 cost to T plus one upon to DT. So further solving the integration we have therefore we have after they are equals minus pi. Now we have to compute that is currently F dot Ds. So this is equal to if which is why I gap plus J Jacob plus X K cap. And this is described as P cap plus. Kill the cap plus are okay cap. So first of all we have to find there girl and then we have to solve them integral. After solving the girls we have this as -1 -1 and -1. Therefore we will put the value of curl And other parameters in the integral and 12 it. So further we have this as they did well now after so I'll be in the integration. We have the limits as this has given us to listen to go to one 0-2 pi and it is minus are caused tita divided by foot under 1- Our Squire minus one minus odd sign. T divided by but under one minus R squared are D D D D R. So this is equal to integration 0-1. So we have they double integral curl F dot ds equal to Have.dear. That is equal to -5. So this is the answer.

Okay, let's verify strokes Theory. Um, one more time. F is given to be why commas, aecom x and s is thehyperfix fear with radius one. Why being positive here? So I oriented bit in the direction so that why is the vertical access x comes out and Z goes left. And right now, the orientation off the surface is supposedly in the positive y direction. So sitting up the parameter so that it works out in that particular way is going to be very important here so that your values are going to be correct. So we're going to go through that park carefully. Okay, so let's start with the curve. C are of data, the parameter ization I choose. So that X is going to be my sign. And my Z is going to be my co sign here because we want the the vector go towards the positive wide direction. The normal. Okay, so I will say excess sign data. Why is located at zero? That's where the curve is. And Z like I mentioned its coastline. So the derivative, our prime it is going to be cosign common zero comma sign. Negative sign. Okay, so according to the parliamentary ization f can be rewritten as zero comma co sign comma Negative. Why is sorry. Sorry. Why is zero on Z is cosign and X is sign? Yeah, got to be consistent there so f dot are prime. It is equal to negative sine squared. Okay, so the integral from 0 to 2 pi off negative sine squared we've done. There's so many times we should know by now that this is going to be negative. Pine. Okay. All right. Now let's take a look at the surface. Integral. We want to be careful with the para mature ization again. I will choose fee and data. I'd like to use the spherical coordinate system. And again here. Why is the one that Onley goes? Um, half of it. So I will choose why to be my co sign of fee. Um and then we want the X to be the sign of data and easy to be the coastline of data similar to the previous situation. So I am going to be very careful here and say X is sign fee cause I'm PETA. Why would be casino fee and Z would be signed vehicle assigned data no X should be signed. Feast signed. Data and Z is the one with the cozy right. These should look rather similar to each other. Okay, All right. Moving on. Let's take the derivative with respect to feed our Sophie. It is equal to co sign fee. Sign data negative. Sign of fee co signed fecal science data. Let's be very careful. Here is well, because we have to Trigana metric expressions and the angles are different. So be careful with which drew that if you're taking okay, our self data. So let's see two derivative with respect of Fada. Now sign. So it's going to be co sign to sign Fecal sign data. This doesn't have a data in it. So it's zero and the last one is going to be negative Sign. So you will have negative sign fee Sign data. Let's take the cross product. I chose the feet and the fetus so that I know that the cross product will actually go outwards like this. So I did the calculation already. When you simplified, you will get sine squared feet Signed data. Now the middle term, you will have a, um sign co sign fee sign fee terms, so I always like to convert that into a sign of 2 ft. And because the cosine squared fada plus sine squared data shows up, it'll simplify significantly. So just as a side note, you can keep that in mind. But the middle part, it'll be half off. Sign of two fada or 2 ft. My God. Okay. In the last term, the the Z is going to be sine squared. Fee call sign data. All right, now let's focus on the curl. Interestingly, this is going to be negative. One common negative, one common negative one. So we don't need to worry about the privatization here. Curl f dot the normal vector. You will end up having negative sine squared of fee. I factored it out, and then you will have two terms science data in a co sign data. Then you'll have minus one half sign of 2 ft. Okay, so the double integral that we're going to calculate you will have this input. Let me just call this a and the B D finger. Now, what's important here is that you gotta be careful with the with the angles again. We know that Fada is going an entire revolution, so it's going to be 0 to 2 pi. But the fee It only goes from top to the X Z plane, so it's going to be 90 degrees, so it's going to be zero to pi over two. Mhm. It's quite straightforward calculation, even though it's a little bit long. But you can also confirm that this will be negative. Pine. There you go. So we were able to confirm that the integral Czar exactly equal to each other, and it fireflies Stokes there.

In the problem we have the strangle that is given as this. So this is our desire triangle. Having the coordinates 00 and one 010 100. You can see from the carve. So this is our triangle desire triangle. And now they have to find out the girl of F. So further this is Dell cross F. And this is equal to I. Z. Kick up. And yeah, this is equal to. So here we have The desire to Wrangle as this one. This is the desire triangle. And this is formed by these coordinates. So further we have the karloff. If that this that is equal to as the triangle is formed here. So we can see the base and height is one and one hence the area that is equal to minus one. Therefore we can say that integration F dot d r is equal to minus one. And this is the answer.

Olivia. Our problem. Your function of his ex way is that is equal Latest three by square I Plus for that the plus six six kick. So see if the triangle in the plane their decor toe one by two. White with what? Jesus Two comma, zero comma zero zero comma to Colombia. And they conduct look with orientation so close, Integral oversee. If not, they are Because doubling the global sigma the Cruz have door in B s, which is double integral over our they'll cross have dot the Lord the the So where are is the production off the surface Sigma On that side play d equals the off X y zed Half into my mind is that they cross f is equal to I deluxe minus three wide square The do away for that key does it 66 So it comes to be minus 40 when a 60 day it was three late Q The deep is it going toe zero this half day minus kick. So are is a function and ex commonly which is zero or less than record tracks less than a recorder to zero less than record by less than a record too. Tu minus six Closer didn't policy after the articles. The interview zero toe zero to minus X minus. Florrie minus 60 day. Because three very kick mine escape Hindu Dele the X If we saw we get minus three by two into a text minus three X square plus executed by three 02 which is my last day. Now another question. Thank you.


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