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Problem 63 Hard Difficulty

Suppose $C$ is a circle centered at the origin in a vector field $\mathbf{F}$ (see figure). Figure cannot copy
a. If $C$ is oriented counterclockwise, is $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ positive, negative, or zero?
b. If $C$ is oriented clockwise, is $\oint_{C} \mathbf{F} \cdot d \mathbf{r}$ positive, negative, or zero?
c. Is $\mathbf{F}$ conservative in $\mathbb{R}^{2}$ ? Explain.

Answer

a. Negative
b. Positive
C. No

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Video Transcript

All right, So this question were given your field, that is, um uh, that IHS centered at the origin, all right. And then their red arrows are vector field and were asked to determine if the line into grow along the sea It's positively negative. So if c is going to counter clockwise were born like there's we see that the path that we're taking is going opposite to the vector field blinds. So that means of CIA's CC the leaving his counterclockwise are lining to growth is, uh, negative. You're right. That bounce vote, this is naked, right? Because we see that our path is going against the years. All right, so now what would happen if we go clockwise? So for going clockwise on a car, this in green. So we're going down like this and then back up. Well, what do we notice? We noticed that the the passport taking is going with the arrows of Perfecter field or the vector field lines. So this one is because Okay, now is this vector field conservative enough? Well, let's think about it. Let's say we start at this point right here, and then we have this. So this is a this year's If we go along this path so we go from a to B in the clockwise direction we noticed that are lining to grope is gonna be negative. Right? So we go from a to B in the, um, clock eso along this path. Right here we go from a to B along this path of the upper half of the plane, we noticed that our line into girl is negative because the path we're taking is going against the line that the defector feel. But what if we go from a to B? There's in the bottom half off the surface. Well, then we noticed that we're going with the line would with the vector field. So our path are curved. The direction we're going in is with the vector field lines, So that means that's positive. So we have to line into girls both going from A to B and they're giving us one is giving us a positive value and the other is giving us and negative value now, obviously Ah, the line integral isn't serious, so one is positive and the other is negative. That means we have two different answers to different solutions. And remember, for a conservative field our the value of the line integral no matter what path we take, should be the same. So conservative. Vector fields the line Integral czar, independent of the path. But in our case, we found two different values for the line integral. So that means our field is not conservative, is not concerned.

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