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Problem 50 Easy 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.
$\quad \mathbf{F}=\langle y-z, z-x, x-y\rangle ; C: \mathbf{r}(t)=\langle\cos t, \sin t, \cos t\rangle,$ for $0 \leq t \leq 2 \pi$



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Video Transcript

So in this video, we're going to evaluate does Lining to grow, Um, for the flow Director Field and work. Also, given the parameter organization where excess co sign t y is scientist and Z's co sign t and T varies from 0 to 2. All right, so we know that the integral of the close the line integral f felt your It's just the integral of apple of our of tea dotted with our prime of TV. Okay, so rt is just co scientist signed t post 90. The derivative of bad is just negative. Sigh Inti. Course I nt negative. And wherever we have Inpex, we're gonna bug in close. I Inti, wherever we have, why we're gonna plug in sign whoever we have. Zero and rugged coastline t into our force field. Right. Okay, so this is exactly what we did. So, for F, we get signed T minus co sign T Congo zero comma coastline T minus 70. Its allies the second part right here. Syria. But look, here we have Z minus. X z is co signed the excess co scientist. So they just turns out to be. And then we're gonna thought this with negative scientific oocyte. Negative. So when we when we do the dot product, we get negative Science Square T plus signed coastline T minus scientific owes 90 plus times square. First thing we notice is this is thes cancel head positive co scientist scientist cancels with negative site. So we get is so that means our line integral that we evaluate rate your integral that we evaluate is just zero.

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