Pressure

The pressure, P, within a fluid medium at equilibrium due to gravitational forces is defined as , where ρf is the mass density of the fluid, g is the gravitational acceleration, and z is directed in the downward positive direction. Because the formula assumes pressure calculated within a singular fluid medium, at the surface, or where z=0, pressure is equal to zero. As an object increases in depth, the pressure it is subjected to increases. Due to the difference in depth between each surface point on a submersed object, there will be a different magnitude of pressure acting on the top and bottom of the object, causing an upward force.

The Divergence Theorem

The vector field around the surface boundary of an object submerged in a fluid medium (having divergence associated with an outward facing normal vector) can be used to relate the surface flux of the forces acting on the object with the forces acting within the volume of the object. This relationship is defined using the divergence theorem (or Gauss’s theorem):