A solid occupies a region E with surface S and is immersed in a liquid with constant density p. We set up a coordinate system so that the xy-plane coincides with the surface of the liquid, and positive values of z are measured downward into the liquid. Then the pressure at depth z is p = pgz, where t is the acceleration due to gravity (see Section 8.3). The total buoyant force on the solid due to the pressure distribution is given by the surface integralwhere n is the outer unit normal. Use the result of Exercise 31 to show that F = -Wk, where W is the weight of the liquid displaced by the solid. (That F is directed upward because z is directed downward.) The result is Archimedes’ Principle:
The buoyant force on an object equals the weight of the displaced liquid.

F= -Jf pn.