Answer
$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
$\dfrac{\partial P}{\partial z}=\dfrac{\partial R}{\partial x}$
$\dfrac{\partial Q}{\partial z}=\dfrac{\partial R}{\partial y}$
Work Step by Step
$f_x=P=\dfrac{\partial f}{\partial x} $ and $Q=f_y=\dfrac{\partial f}{\partial y} ; R=f_z=\dfrac{\partial f}{\partial z}$
1) $\dfrac{\partial P}{\partial y}=\dfrac{\partial}{\partial y}(\dfrac{\partial f}{\partial x})$
and $\dfrac{\partial^2 f}{\partial x \partial y}=\dfrac{\partial Q}{\partial x}$
2) $\dfrac{\partial P}{\partial z}=\dfrac{\partial}{\partial z}(\dfrac{\partial f}{\partial y})$
and $\dfrac{\partial^2 f}{\partial x \partial z}=\dfrac{\partial R}{\partial x}$
3) $\dfrac{\partial Q}{\partial z}=\dfrac{\partial}{\partial z}(\dfrac{\partial f}{\partial y})$
and $\dfrac{\partial^2 f}{\partial z \partial y}=\dfrac{\partial R}{\partial y}$
Hence, we get $\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
$\dfrac{\partial P}{\partial z}=\dfrac{\partial R}{\partial x}$
$\dfrac{\partial Q}{\partial z}=\dfrac{\partial R}{\partial y}$
