Assume that the pressure in a volume of air can be expressed to first order in a local coordinate system (x, y, z) as p = p₀ + f x + O(x², y², xy, . . .). Show from the balance of forces on a small volume element dV that the force per unit volume is f in the −x-direction. Using the fact that the coordinate system can be chosen arbitrarily, show that more generally f _p = −∇P.