Order the steps to derive the hydrostatic equilibrium equation from the concept of pressure variation in a fluid due to gravity.

To derive hydrostatic equilibrium, start by picking a tiny fluid slice of height \(dz\) and cross‑sectional area \(A\). The pressure at the bottom of the slice is higher than at the top, so the net upward force from pressure is \(P(z)A - P(z+dz)A\). This upward force must balance the slice’s weight, \(\rho g A\,dz\), giving \(P(z)-P(z+dz)=\rho g\,dz\). Rearranging yields the differential relation \(dP/dz=-\rho g\). Integrating this from the surface downward, with the surface pressure as a boundary condition, produces the hydrostatic pressure formula \(P(z)=P_0+\rho g z\).