Prove that a necessary and sufficient condition that $\int_C\vec{F}$ $\cdot d\vec{r} = 0 $ for every closed curve C lying in a simply connected region R is that $\nabla \times F= 0 $ identically .
Sufficient part is straight forward application of Stokes theorem . For the necessary part , lets try to prove it by contradiction.If we assume that $\nabla \times F $is not $0$ at some point in the space . By continuity , i think that we can ensure that there exists a neighbourhood where $\nabla \times F $ is not $0$ . Can we then say that there 'definitely' exists a surface in that neighbourhood where the flux is non-zero . Intuition says that we can , but can we prove it more rigourously ! ?