Editing Hydrostatic equilibrium (section)

=== Derivation from force summation ===
{{further|Mechanical equilibrium}}
[[Newton's laws of motion]] state that a volume of a fluid that is not in motion or that is in a state of constant velocity must have zero net force on it. This means the sum of the forces in a given direction must be opposed by an equal sum of forces in the opposite direction. This force balance is called a hydrostatic equilibrium.

The fluid can be split into a large number of [[cuboid]] volume elements; by considering a single element, the action of the fluid can be derived.

There are three forces: the force downwards onto the top of the cuboid from the pressure, ''P'', of the fluid above it is, from the definition of [[pressure]],
<math display="block">F_\text{top} = - P_\text{top} A</math>
Similarly, the force on the volume element from the pressure of the fluid below pushing upwards is
<math display="block">F_\text{bottom} = P_\text{bottom} A</math>

Finally, the [[weight]] of the volume element causes a force downwards.  If the [[density]] is ''ρ'', the volume is ''V'' and ''g'' the [[standard gravity]], then:
<math display="block">F_\text{weight} = -\rho g V</math>
The volume of this cuboid is equal to the area of the top or bottom, times the height&nbsp;– the formula for finding the volume of a cube.
<math display="block">F_\text{weight} = -\rho g A h</math>

By balancing these forces, the total force on the fluid is
<math display="block">\sum F = F_\text{bottom} + F_\text{top} + F_\text{weight} = P_\text{bottom} A - P_\text{top} A - \rho g A h</math>
This sum equals zero if the fluid's velocity is constant.  Dividing by A,
<math display="block">0 = P_\text{bottom} - P_\text{top} - \rho g h</math>
Or,
<math display="block">P_\text{top} - P_\text{bottom} = - \rho g h</math>
''P''<sub>top</sub> − ''P''<sub>bottom</sub> is a change in pressure, and ''h'' is the height of the volume element—a change in the distance above the ground.  By saying these changes are [[infinitesimal]]ly small, the equation can be written in [[differential equation|differential]] form.
<math display="block">dP = - \rho g \, dh</math>
Density changes with pressure, and gravity changes with height, so the equation would be:
<math display="block">dP = - \rho(P) \, g(h) \, dh</math>