Green, Stokes & divergence

For a plane vector field $\mathbf{F}=M\mathbf{i}+N\mathbf{j}$, a stream function $g(x,y)$ satisfies $g_y=M$ and $g_x=-N$. Such a $g$ exists exactly when $\partial M/\partial x+\partial N/\partial y=0$, since then ${\partial }_x(g_y)={\partial }_y(g_x)$. Integrate $g_y=M$ with respect to $y$ to get $g$ up to a function of $x$, then differentiate with respect to $x$ and match $-N$ to find that function. If the divergence is not zero, no stream function exists.