The gradient of the pressure field Vp(r, t) is very important in meteorology. Consider a parcel of air contained in the rectangular parallelepiped dx dy dz, and whose center is located at the point r. This volume element is embedded in a surrounding pressure field p(r) = p(x,y, z). Let us compute the x component of the net force on the volume element. As indicated in Figure 9.8, the left-most face experiences a force due to the external field to the right:

the right-most face experiences a force to the left fright face = —p(x + idx, y, z) dy dz. (9.28)

The net force on the volume element is dFxnet = — (p(x + 2 dx, y, z) — p(x — 1 dx, y, z)) dy dz d p d p = —- dxdydz = —- dV (9.29)

dx dx where dV is the volume of the infinitesimal material element. Newton's Second Law (force is mass times acceleration) tells us that

where ax is the x component of acceleration and dM is the mass contained in the parcel. We can divide each side by dV and obtain d p

d x where p is the density of the air in the parcel. Put in more conventional form we have:

If we evaluate the y and z components in a similar fashion we can summarize the result in vector form a = —Vp P