Archimedes's Principle states that the upward buoyant force on an object immersed in liquid is equal to the weight of the volume of liquid displaced by the object. Here, we rify Archimedes' principle as follows: we use the dot product along with the divergence theorem to compute the forces on a boat hull.

 

 

Suppose a boat hull is labeled as the surface S (see figure below) with the water line for the hull at z=0. The pressure, p, is depth-dependent: p=-pg: where p is the density of water, g is the gravitational constant, and z s0 is the depth. The force from the pressure, denoted by the vector F, on a small patch dA on S. is the pressure pointing into the hull surface F= -pn = pgzn. Assume that the shape of the hull is a triangular trough with cross section specified by the figure below.