Let $ \textbf{F} $ be the vector field shown in the figure.

(a) If $ C_1 $ is the vertical line segment from $ (-3, -3) $ to $ (-3, 3) $, determine whether $ \displaystyle \int_{C_1} \textbf{F} \cdot d \textbf{r} $ is positive, negative, or zero.

(b) If $ C_2 $ is the counterclockwise-oriented circle with radius 3 and center the origin, determine whether $ \displaystyle \int_{C_2} \textbf{F} \cdot d \textbf{r} $ is positive, negative, or zero.

a) Negative

b) Negative

Vector Calculus

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So here's a question I ask you to estimate whether it's like this integral or this one is positive, negative or zero on Liberia's thiss I suppose you can parallel TriStar. Then this will be the opera pines or bomb your parameter. And at the end, you you're looking at this quality, so our priority should be the direction with your your car, for a segment is going and f off Artie's just a vector, a tte that point. So if you off like, for example, if you have the textbook and you can look at the picture for the first part is from negative three to three is a line segment that goes us vertically up and along the line. You see, like most of the factors, are approximately the factor pointing down, which means they're taught product is now active. And if you Grover's an active quality issue, gives you a negative number for the second bar. If you goes, goes counter clockwise on DH, you see that you see that the factor feel giving a picture. Pre mush clothes goes clockwise, so if you cause counter clockwise, then against your you're our prime Off T and the vector. Half of our fifty are the are off the office. I'd eruption because our prime Ticos goes Because counterclockwise however you also vector along their circle sort of course are clockwise. So this also gives your negative thoughts product which, after isa great over issue, gives your nephew value.