Pressure due to fluid column at rest

The pressure a mountaineer feels DECREASE with altitude and the pressure a diver feels INCREASE with depth are both examples of HYDROSTATIC PRESSURE -- pressure due to a static fluid. To find how pressure changes with depth below a liquid surface, consider a cylindrical sample of the liquid, cross-sectional area A, with its top face at depth $h_1$ and its bottom face at the greater depth $h_2$ (both measured from the free surface). Let $F_1=P_1A$ be the downward force on the top face and $F_2=P_2A$ the upward force on the bottom face. For the sample to be in equilibrium, the net upward force must balance both the downward force from above AND the weight of the sample itself: $F_2=F_1+mg$. The mass of the sample is $m=\rho V=\rho A(h_2-h_1)$, so its weight (the gravitational force $F_G$) is $\rho A(h_2-h_1)g$. Substituting and cancelling the common factor A from every term gives $P_2=P_1+\rho(h_2-h_1)g$. Taking level 1 at the free surface itself (so $h_1=0$ and $P_1$ equals the atmospheric pressure $P_a$) and level 2 at depth h (so $P_2=P$, the pressure being sought) gives the standard result $P=P_a+\rho g h$: the pressure at depth h is always greater than the pressure at the surface. If the atmospheric contribution is ignored (or the pressure is measured relative to the atmosphere), this simplifies to $P=\rho g h$. Since $\rho$ and g are both fixed for a given liquid, this shows the pressure due to the fluid column depends ONLY on the vertical height h of fluid above the point in question -- never on the cross-sectional area, the total volume, or even the shape of the container. This surprising independence from shape is dramatized by the HYDROSTATIC PARADOX: three vessels of very different shapes -- one narrow and tall, one wide and squat, one irregularly bulging -- connected at the bottom by a common horizontal pipe and filled with the same liquid, all settle to exactly the SAME liquid level, because the liquid at the bottom of each vessel experiences exactly the same pressure regardless of how much liquid t …