Consider a vector field $$\vec{F}(x,y)=P(x,y)\vec{i}+Q(x,y)\vec{j}$$ on an open and simply-connected region. Assume $P$ and $Q$ have continuous partial derivatives. Under which conditions there exists a positive-valued function $\mu (x,y)$ such that $$\mu\vec{F}(x,y)=\mu(x,y)P(x,y)\vec{i}+\mu(x,y)Q(x,y)\vec{j}$$ is a conservative vector field?
Thanks for the help!