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The figure shows a vector field $\mathbf{F}$ and two curves $C_{1}$ and $C_{2} .$ Are the line integrals of $\mathbf{F}$ over $C_{1}$ and $C_{2}$ positive, negative, or zero? Explain.

Line integral over $C_{1}$ is positive

Line integral over $C_{2}$ is negative

Vector Calculus

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Johns Hopkins University

Oregon State University

Harvey Mudd College

University of Nottingham

{'transcript': "So this problem similar to the previous problem, you have to look at a textbook for the picture, both for C one. Um, basically his ideas you integrate. This is the cmas degree, which is up upon lower buff who apparently her. And this is your attention. Sorry, This is your attention, pectoral kerf. And this is fact. Or giving the vector field. So you're basically looking at whether they're top product is a positive or negative. That means if they're if they're about the same direction, that is possible. If there about the opposite direction there is active for C one, I'LL say looking from the picture is it's not exactly the same, but he's about the same direction. Um, so the vector dot product should be positive. And there for the second one, I'Ll say all the doctor always not necessary. Exactly. The case, I will say, is pretty close to the case where, where this is always affect her is sort off so perpendicular to his attention that throws across and, uh, and probably us at the end of curve There some negative parcel. I won't say it's exactly zero papa for CTO. Oh, sorry. For sea to it should be somehow small negative number that is close to zero. And again, this one usually should have the guy's picture, and that should be clear."}