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Let C be the top of the unit circle from (0,1) to (-1,0)_Use potential function to evaluate [ F-dr exactly, where F = 19x"87+ 9y"j:...

Question

Let C be the top of the unit circle from (0,1) to (-1,0)_Use potential function to evaluate [ F-dr exactly, where F = 19x"87+ 9y"j:

Let C be the top of the unit circle from (0,1) to (-1,0)_ Use potential function to evaluate [ F-dr exactly, where F = 19x"87+ 9y"j:



Answers

Find the work done by the force $\vec{F}$ along the curve $C$ $\vec{F}=-x \vec{i}-y \vec{j}, C$ is the upper half of the unit circle from (1,0) to (-1,0)

We're going to do problem number 12. Now, in this question, we have what we have to do. We have to just find the value off Cossack. 210 degree using a unit circle. Okay, so it would be better if you just draw union psychologist for understanding purpose. No. We will draw the action. Why? X and way excess? I guess so. Let destroyed with a different color so that they can easily figure it out. Okay, So this is the ex success on this is who I access. Okay, this is 700 thistle is the degree. This is the angle that is 210 degree. So in which coordinate to light will lie in the third quarter in the third quarter. Do less ammonia. So we'll check that at 210. Degree on the unit circle. What are the coordinates? The coordinates are okay. The coordinates here are man is under three by two. Andre Manus one by two. So these are the coordinates for X and y. So, Kosik Tita in, you know circulation. Laters. Generalized as one by way. One bite. Why? So we can write that Cossack off 210 degrees. This will be equals toe one by management by two. Okay, so this will be calls to manage to. So we got the value of Kosek. 210 degrees. That is a cause to manage to. So this is the answer. Thank you.

So in this video, we're asked to determine your crew. Well, uh, over this field right here along the curve, which goes from 00 Sure. 11 01 200 Thanks for this. Little trying What we have right here. First thing we notice is that this triangle right over here this line is why is equal to X love? It's fine. On top is by two. So again, we're gonna use greens zero. So we just write the double integral off. Thank you. By the x minus dp by you i e a So, uh, you know, so predict the derivative que, which is 10 and verse of expert perspective excess just one divided by one plus X squared and the derivative of P with respect. Why so If he is, it's Gregory of expert plus point. Well, there are no wise. So the derivative that is just now times he live My so remember our why values go for wise. He quoted xto 21 at our X values go for zero for one. Okay, great. So now it's just evaluating at the woman to grow so the integral So we need to find into role with respect to buy off one divided by one plus x squared again there are no lies, so we can treat one divided by one plus x squared as constant. So now the in general the wise just why Why value is go from X to one. So we forget those values that we get What sex right over here. They're gonna split this into two into girls to go from zero to a one off one divided by one plus x where two yes, minus the integral of X divided by one plus x squared he x from 0 to 1. So the first part is easy. So the integral well, one divided by one plus expert Sylvester's 10 adverse affects. And for the 2nd 1 we can use U substitution that new equal to one plus X squared. That means that U is equal to two x the X so x DX is just Do you divide it like till now you substitute back so you get 1/2 intervals. You divided for you just 1/2 times the natural laws of you. Please just equal to 1/2 times the natural. Log off what plus X square, So, uh, Okay. So perfect. So now we just continues. Though we have tension and bursts of X minus 1/2 Uh, the natural log. Well, one plus x square. And now this lower, integral zero. The upper integral is one. And then what we know about this is that we have This is just tension and burst one minus 1/2 financial along too minus the tangent. And versus zero is Yes, Still here. Engine inverse of 0 to 0, minus 1/2 the natural log off. One is also zero. So our final answer Arthur lining to grow Just the tension in verse of once before minus what her that look, too. That's the value for lining to grow, uh, computed using, uh, Green Spirit and just one last thing. So since that we look at our graph, we noticed that the arrows are going in a counterclockwise orientation, so there is no need to change the signs off the value of our mind. So our final answers just by over four minus 1/2 times the natural world

You know, were asked to evaluate, were given a vector field and I swear a path and were asked to evaluate the line integral in this vector field counter clockwise along this path. So the vector field is Yeah, equals y I minus X j and the path. All right, this is the unit circle given by X squared plus y squared equals one. Yeah, counterclockwise from the point 10 work. Health care saying 201 Yeah, so we can prima tries this curve. As you said, r of t equals say, co sign of t sign of t where t ranges from zero two pi over two. So that's it. Slaps. Yeah, this is how, and we have that The line integral over c r curve of f dotted with gr. This is the integral from T equals zero to pi over two of our function f in terms of t. So this is sign of T I minus the co sign of T jay dotted with D R. Well, to do this need to find our prime of t. Damn, this is negative sine of t co sign of t and so well dot This with negative sign T I plus co sign of T Jay D t Evaluating this is the integral from zero to pi over two of negative sine squared of T minus co sine squared of t d T. This is equal to this is negative one times the integral from zero to pi over two GT, which is negative. Pi over two. Yeah, you can believe the right things. The only theory

Which is force. This is why I got minus six in Jamaica. Uh, it's a squad arrest wise, but equals from while digital too. Is it a lot now this is in tradition. Why minus X and yes, they were So let's just have a bad breast Is as its peoples costea the T minus society didn't Why waas sign day do you any less costed? Et on it is just toe No, this is he gonna do the addition y dx minus six Doing this is equal to guess ended up by over two. It is a scientific into minus scientists didn t minus cost into cost The DJ This is even as and distributed through my sci fi Diddy minus causes fighting DT There is equal to minus because by true since party bless cross party Did you is giving out my ass by our go Did he just given us minus I think this is dancer


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