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Archimedes’ Principle

States that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
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The statement of the theorem

Let Ω⊂R3\Omega \subset \mathbb{R}^3 be the continuous domain occupied by the fluid, and let V⊂ΩV \subset \Omega be the volume of the object submerged in the fluid. Assume the fluid is in hydrostatic equilibrium, characterized by a pressure field p(r⃗)p(\vec{r}) such that ∇p=−ρ(r⃗)g⃗\nabla p = -\rho(\vec{r}) \vec{g}, where ρ(r⃗)\rho(\vec{r}) is the fluid density and g⃗\vec{g} is the gravitational acceleration vector. The buoyant force F⃗B\vec{F}_B acting on the object is the net force exerted by the pressure differential over the object's surface ∂V\partial V. Mathematically, this force is given by the surface integral:\n\nF⃗B=−∯∂Vpn⃗dS\vec{F}_B = -\oiint_{\partial V} p \vec{n} dS \n\nBy applying the Divergence Theorem and the hydrostatic pressure gradient relation, the buoyant force can be expressed as the volume integral of the pressure force density over the submerged volume VV:\n\nF⃗B=∭V(−∇p)dV=∭Vρg⃗dV\vec{F}_B = \iiint_{V} (-\nabla p) dV = \iiint_{V} \rho \vec{g} dV \n\nSince ρ\rho and g⃗\vec{g} are assumed constant throughout the fluid domain, we obtain:\n\nF⃗B=ρVg⃗\vec{F}_B = \rho V \vec{g} \n\nThis result is equated to the weight of the displaced fluid, WdispW_{disp}. The mass of the displaced fluid is mdisp=ρVm_{disp} = \rho V, and its weight is Wdisp=mdispg=ρVgW_{disp} = m_{disp} g = \rho V g. Thus, the principle states the vector equality:\n\nF⃗B=Wdisp  ⟹  ρVg⃗=ρVg⃗\vec{F}_B = W_{disp} \implies \rho V \vec{g} = \rho V \vec{g}