## Calculus 8th Edition

$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$ $\dfrac{\partial P}{\partial z}=\dfrac{\partial R}{\partial x}$ $\dfrac{\partial Q}{\partial z}=\dfrac{\partial R}{\partial y}$
$f_x=P=\dfrac{\partial f}{\partial x}$ 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 $\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$ $\dfrac{\partial P}{\partial z}=\dfrac{\partial R}{\partial x}$ $\dfrac{\partial Q}{\partial z}=\dfrac{\partial R}{\partial y}$