13.3#11

Hello Professor! I'm a bit stumped here. I am not really sure where to go from this point. My understanding of a conservative vector field is that it is path (in!)dependent, and should turn up the same value for work for any path between two points.  (true!)

However, I don't see what I am doing wrong. I have been drawing out the fields and attempting to estimate paths, but I cannot get past these answers. Could you please enlighten me as to what
paths I am thinking of incorrectly?

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Trying to guess about wha their line integrals will be for various paths is *NOT* very helpful, because there are so many possible paths.  There are two keys required for these problems.  The first is remembering that being path independent/conservative is the same as (if the domain is reasonable) being a gradient.  The second is to have a gallery of examples of what gradient fields look like.   For example, the gradient of a linear function is always a constant vector field, that is at every point you have the same vector with the same magnitude pointing in the same direction. Do you have any of those?  I suggest you look at your favorite quadratic functions and sketch out their gradient fields, there are some of those in the collection above.  You should also be able to look at a vector field sketch above, and guess what are the functions F_1 and F_2, and from that understand if
∂F_1/∂y=∂F_2/∂x is  possible.