Let us, initially, analysis only a two dimension situation. Assume that $F(x,y)$ is a force dependent on the particle position (give by $x,y$) is proposed that if
\begin{equation} \frac{\partial \frac{\partial F_x}{\partial x}}{\partial y} = \frac{\partial \frac{\partial F_y}{\partial y}}{\partial x} \end{equation}
where $F_x$ is the component of the force in $x$ axis and $F_y$ is the component of the force in $y$ axis.
The proposition is: If the equation above is true to the force $F$ then this force is conservative and we can write a potential function associated with this force.
Is this proposition true?
Remembering that the equation above is the same of say:
\begin{equation} \frac{\partial \frac{\partial (-U)}{\partial x}}{\partial y} = \frac{\partial \frac{\partial (-U)}{\partial y}}{\partial x} \end{equation}
implying for example that the work done by this force in any path does not depend on the path only in the initial and final positions..
in this lecture one can see better what is asked: http://youtu.be/R74EhX8O0Nw?t=1h1m49s
