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Problem 48 Medium Difficulty

Line integrals of vector fields on closed curves Evaluate $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ for the following vector fields and closed oriented curves $C$ by parameterizing C. If the integral is not zero, give an explanation.
$\quad \mathbf{F}=\langle y,-x\rangle ; C$ is the circle of radius 3 centered at the origin oriented counterclockwise.

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Video Transcript

Okay, So in this video, we're given the field staff, which is just why call a negative X and were asked to determine the line in general. On the curve for curve is a circle of radius three centered at the origin. And first thing we're gonna try to do is for amateur Isa skirt. So this is pretty simple because we know that we not a pro amateur circle. So since the circle has a radius three, the X is just three Ko Sai inti and the lie is three scientist E where are T ranges from 0 to 2 now. What we're gonna do is we're gonna take the derivative with respect ity of X and y So the derivative of three co scientists just negative three scientific, the derivative of three scientist is just three co society. No, we're going to determine the integral along the curve of f dot your well, that's basically the integral from 0 to 2 pi. Because now we're taking the derivative of the integral with respect team. So we Yes, I would respect to t. So our limits are from 0 to 2 pi half of our city dotted with our prime of TV. So we see that our efforts given in terms of X and y so now we have to. Her amateur eyes are field with X being three ko sai nt on our lives being three scientist. So if we plug that in, we get three sign t Common Negative three co sign T and our prime of these days Negative three. Sign teen cover three coats like No, we take the dog product so we get negative nine sine squared minus nine co sites. Where and first thing we notice is that both terms have a common factor. It's negative. Nine. So if we allowed a negative nine what's left on the inside of science where keep was co size Burti well, sine squared plus co sign Square is just one. So we have negative nine times the integral from zero to buy off one. Now the integral of one D tea is just tea on our limits of integration are from zero to pine So we have negative nine times to prime i zero, which is negative. 18. All right now what's the reason that this line integral is not equal to zero? Well, the reason for this is that our field F or force field is not a conservative. And how can we make sure that it's not a conservative field? Will we take the derivative of little left with respect to why? So our little F is just why so this is our little left that this is our little G. So we take the derivative of apply with respect to why we get worn. And if we take the derivative of little G with respect to X or just negative x of respect for X. We get negative points now, since these two are not equal or field is not conservative and so you are integral is not serious.

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