I want to know why $\vec F=\hat\imath $ is a conservative vector field, and $\vec F =\hat\jmath$ is an non conservative vector field. As my professor told me I can draw some closed paths on the vector field, such as a circle. However, I don't see $\vec F= \hat\jmath $ is non conservative after I drew a circle and a rectangle, but I see $\vec F=\hat\imath $ is conservative.
Can someone tell me where I didn't wrong?
The unit vectors $\hat{i}$ and $\hat{j}$ are conservative because they are the gradients of scalar fields $f(x,y,z) = x$ and $g(x,y,z) = y$, respectively.