Green's Theorem not working in this problem HELP

[unilquer]

I have a problem from line integrals.
My question is attached as a jpeg.
Green's theorem is not valid for mine and i couldnot find a way.
Pls take care of it.

 

[rootX]

Should you show your attempt?

 

[Defennder]

What does the last sentence mean?

Suppose you have the task of providing such vector fields on demand

 

[unilquer]

It is just a clue. It goes on with saying this kind of functions can be created without much effort and asks how they are written.
And unfortunately i don't have much idea how to approach the problem.

 

[Defennder]

I would say that the hint given is a big one indeed. It would save you a lot of computational effort if you understood it correctly. Have you learned about potential functions and conservative vector fields yet?

 

[unilquer]

Actually, i predicted that but could not be sure. Than the answer is 0 since this is a conservative field and surface is a complete circle right?

 

[HallsofIvy]

Since Green's theorem does not work here, then you need to directly integrate around the circle. What parameterization can you use for a circle about the origin with radius r?

 

[Defennder]

Yep, that should be it. Although it's more correct to say it's a line integral over a closed curve.