Given the function $F(x,y,z)=(y,x,x)$, I have to show it isn't conservative and find a close curve where it isn't equal to 0.
It was easy to show that $D_1F_3\neq D_3F_1$, but I can't find an easy curve that isn't 0. I tried unit circle but it equals to zero.
Watching the plot I can't figure it out. In general, I don't know how to solve this when I'm in $\mathbb{R^3}.$
Consider the circle $C : t \mapsto (\cos t, 0, \sin t)$.
You have $C^\prime(t)=(-\sin t, 0, \cos t)$ and
$$\int_C F.dr =\int_0^{2\pi} \cos^2 t \ dt >0$$