holds. If it is an exact differential so A = ∂ ∂ Q ∂ ∂ x and B = ∂ ∂ Q ∂ ∂ y , then Q is a differentiable (smoothly continuous) function along x and y , so ( ∂ ∂ A ∂ ∂ y ) x = ∂ ∂ 2 Q ∂ ∂ y ∂ ∂ x = ∂ ∂ 2 Q ∂ ∂ x ∂ ∂ y = ( ∂ ∂ B ∂ ∂ x ) y . If ( ∂ ∂ A ∂ ∂ y ) x = ( ∂ ∂ B ∂ ∂ x ) y holds, then A and B are differentiable (again, smoothly continuous) functions along y and x respectively, and ( ∂ ∂ A ∂ ∂ y ) x = ∂ ∂ 2 Q ∂ ∂ y ∂ ∂ x = ∂ ∂ 2 Q ∂ ∂ x ∂ ∂ y = ( ∂ ∂ B ∂ ∂ x ) y is only the case.