Archimedes principle states that when a solid body is immersed in a fluid, the net force acting on the body by the fluid is directly up and equals the total weight of the fluid displaced.
Denote the set of points in three dimensions occupied by the body as V . Then for dA an increment of area on the surface of this body, the force acting on this increment of area would equal −pdAn where n is the exterior unit normal. Therefore, since the fluid does not move,
Which equals the total weight of the displaced fluid and you note the force is directed upward as claimed. Here ρ is the density and (17.5) is being used. There is an interesting point in the above explanation. Why does the second equation hold? Imagine that V were filled with fluid. Then the equation follows from (17.5) because in this equation g = −gk.