Say I have an arbitrary body $\Omega$ and surface $\partial \Omega$ submerged in a hydrostatic fluid of density $\rho$ under the influence of gravity. How does one show Archimedes Principle? i.e.
$$\int_{\partial \Omega}(P_0 + \rho g h)d\vec{S}=\text{vol}(\Omega)\rho g\vec{e_2}.$$
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