 Solution

Take $z$ positive downwards, and let the surface pressure be $p_0$. Static force balance gives the hydrostatic pressure equation $dp/dz=\rho g$. HenceOn a submerged body's surface, pressure traction is $-p\mathbf{n}$, where $\mathbf{n}$ points out of the body. The divergence theorem gives the resultantThus the buoyancy is upward with magnitude $\rho gV$, proving Archimedes' principle. The constant surface pressure makes no contribution because ${\int }_{\partial V}\mathbf{n}dS=0$.

For a floating iceberg of total volume $V$, the displaced seawater volume is $V-V_I$. Neglecting air buoyancy, equality of weight and buoyancy gives ${\rho }_{I}gV={\rho }_{w}g(V-V_I)$. Consequentlywith $0<{\rho }_I<{\rho }_w$, as required for partial flotation.