I've encountered this exercise in my textbook, which I'm not really sure how to approach. Help would be appreciated!
Let $a,b,c,d$ be constants, and $P(x,y)=(ax+by)/(x^2+y^2), Q(x,y)=(cx+dy)/(x^2+y^2)$. We define a vector field $F=(P,Q)$ and $A=\{(x,y)|(x-3)^2+(y-2)^2<1\}, B=\{(x,y)|1
. What are necessary and sufficient conditions on the constants (a,b,c,d) for F to have a vector potential in A? What about B?
Edit: since A is a "star set" (not sure what the English term is) we can find necessary and sufficient conditions by comparing the (x and y) derivatives of P and Q. I'm not sure about B though.
Thank you for your assistance.