PX285 - J2 - hydrostatic limit and archimedes' principle

P + ρ g z = constant

P (z) = P_{a t m} + ρ g (z_{1} - z)

$z_{1}$ can be taken to be zero, and $P_{a t m}$ is constant
the total upward force on the object, which is the projection of the force on the z-axis

\vec{F} \cdot \hat{k} = - \subset \supset \iint_{S} P (z) \hat{k} \cdot d \vec{s} = - ∭_{V} \vec{\nabla} \cdot (P (z) \hat{k}) d V

where, $S$ is the surface of the object, and $d \vec{s}$ is pointed outwards; $V$ is the volume of the object

∭_{V} \vec{\nabla} \cdot (ρ g z \hat{k}) d V = ∭_{V} ρ g d V = g ∭_{V} ρ d V = M g

where, $\vec{\nabla} \cdot z \hat{k} = 1$

archimedes' principle

the vertical force on a submerged object is equal to the weight of the fluid displaced by the object

\vec{F} \cdot \hat{k} = ∭_{V} ρ d V g = M g

where, $ρ$ and $M$ are the density and the mass of the fluid