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A mathematical derivation of the force exerted by an \emph{inhomogeneous}(i.e., compressible) fluid on the surface of an \emph{arbitrarily-shaped} bodyimmersed in it is not found in literature, which may be attributed to our truston Archimedes' law of buoyancy. However, this law, also known as Archimedes'principle (AP), does not yield the force observed when the body is in contactto the container walls, as is more evident in the case of a block immersed in aliquid and in contact to the bottom, in which a \emph{downward} force that\emph{increases with depth} is observed. In this work, by taking into accountthe surface integral of the pressure force exerted by a fluid over the surfaceof a body, the general validity of AP is checked. For a body fully surroundedby a fluid, homogeneous or not, a gradient version of the divergence theoremapplies, yielding a volume integral that simplifies to an upward force whichagrees to the force predicted by AP, as long as the fluid density is a\emph{continuous function of depth}. For the bottom case, this approach yieldsa downward force that increases with depth, which contrasts to AP but is inagreement to experiments. It also yields a formula for this force which showsthat it increases with the area of contact.

Using surface integrals for checking the Archimedes' law of buoyancy