Vector Calculus · Medium problem

The integral equals $f(\text{end point})-f(\text{start point})$, and for a closed curve the start and end points are the same.

The curl of $\mathbf F$ is always a large positive number, which forces cancellation.

Closed curves always enclose zero signed area, so any integral around them is zero.

A conservative field always has zero $x$-component, so $\oint P\,dx=0$ automatically.