easy question about conservative vector fields Suppose I have a vector field $f: A \subset R^2 \to R^2 $.  Write $f = (M,N) $. Does it follow this?
$$ \frac{ \partial M}{\partial y} = \frac{ \partial N}{ \partial x } \quad \iff \quad f \text{ is conservative}$$

No. It does not. If $A$ is the punctured plane then the vector field with $M = -y/(x^2+y^2)$ and $N = x/(x^2+y^2)$ is not conservative. This can be seen by integrating around any closed contour which contains the origin. It will give a nontrivial result depending on how many times the curve winds around the origin. On the other hand, if $A$ is simply connected then there is an equivalence between satisfying the criteria $M_y=N_x$ and $(M,N)$ being conservative.
