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Problem 47 Hard 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.
$\mathbf{F}=\langle x, y\rangle ; C$ is the triangle with vertices (0,±1) and (1,0) oriented counterclockwise.




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Video Transcript

in this video, we're asked to determine to close, integral off Beth Thought dealer and, uh, curve that were given on which we're gonna find the integral It is the triangle that has, um, for two. C zero negative 11001 Enver counterclockwise. All right, so for our. So the first thing we're gonna do is we're gonna parameter rise this curve. So we see we have three line segments and we're gonna parameter trace the So our first line segment goes from 1001 So now we determine sort of the slopes. We're gonna subtract extra minus x one and y tu minus my once they're zero minus born is negative. One and one minus zero is one. So we determined our a b And then now we're gonna parameter rise are X and y so axes just x one plus a teeth, which is one plus r A is negative. One times t which is one minus t our wives. Just why one which is zero plus now, are abused one times T c zero plus one t courses. All right, so now we care. Amateur eyes are x and y in terms of T and R T ranges from zero. So are one is just one minus T and calma to teach. So our exes 11 it's tea or wise. So our primes just negative one come. So we just take the derivative one minus deal with respect to t negative one and then we take the derivative of tea with respected T. That's just now there are two were going from 01 to 0 negative. So again, we're going to determine our A and these were gonna do X two line sex on so zero by zero and why to minus y one, which is just negative one minus once we get zero negative too. And then X is just x one plus 80 or zero plus zero t which is zero and lies just why one plus BT, which is one plus negative two times t or just one minus two T and T ranges from 0 to 1. So this is our are too then our prime. So if we take the derivative of that word t the derivative of zero with respective teams to zero by all the derivative of one minus two tea with respect kids just I don't know. We do the same thing for the third line segment, going from zero negative one towards zero. And then we're gonna so I won't go through this one because we did it for first. You point your little A or little B when you find yours, then you take the derivative and find our All right. So now, to determine the line integral right over here Term in this group, it's just that you just for a repair amount or repair amateur eyes, your f and then you thought it with our prime. So now we have three different for amateur ization. So he split this integral into reports. So it's Andrew from 01 off f both are one of t. So instead of X, we're gonna put one minus T instead of why we're gonna put T and then dotted with negative one couple won for the second integral along our two or X is going to be zero while or why is one minus two teeth? Then we thought that with zero comma negative too. And then finally, for our three, our Xs, t and R y is negative one plus t and then we docked that with one comma. So since they will have the same injury, throw the internals building from 0 to 1 and every case so we can just combine them all together. So do it like this. So here we see, we have negative one minus two minus one. So we have a negative for we have t plus tty to TTY plus wheeler that 60 70 80 So it's just 18 minus war and they integral that is just four squared minus for teeth on our limit of integration is from 0 to 1. So if we plug that and we get four months for minus zero by zero so four minus four is just zero so are integral is just

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