Chapter 13.3, Problem 25E

Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then

$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}\frac{\partial P}{\partial z}=\frac{\partial R}{\partial x}\frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}$