Chapter 13.2, Problem 13.79P

Prove that a force F(x, y, z) is conservative if, and only if, the following relations are satisfied:

$\frac{\partial F_x}{\partial y}=\frac{\partial F_y}{\partial x}\frac{\partial F_y}{\partial z}=\frac{\partial F_z}{\partial y}\frac{\partial F_z}{\partial x}=\frac{\partial F_x}{\partial z}$