Archimedes' principle can be justified via arguments using some
elementary classical mechanics. We use a Cartesian coordinate system oriented such that the -axis is normal to the surface of the
fluid.
Let be The Gravitational Field (taken to be a constant)
and let denote the submerged region of the body. To obtain
the net force of buoyancy acting on the object, we
integrate the pressure over the boundary of this region
Where is the outward pointing normal to the boundary of
. The negative sign is there because pressure points in the
direction of the inward normal. It is a consequence of
Stokes' theorem that for a differentiable scalarfield and for
any a compact three-manifold with
boundary, we have
therefore we can write
Now, it turns out that where
is the volume density of the fluid. Here is why. Imagine a cubical
element of fluid whose height is , whose top and bottom
surface area is (in the plane), and whose mass is
. Let us consider the forces acting on the bottom surface
of this fluid element. Let the z-coordinate of its bottom surface
be . Then, there is an upward force equal to </math>p(z)\Delta
A\mathbf{e}_z-p(z +
\Delta z)\Delta A\mathbf{e}_z + \Delta m\mathbf{g}a simple manipulation of this equation along with dividing by
gives
taking the limit gives
Similar arguments for the and directions yield
putting this all together we obtain as
desired. Substituting this into the integral expression for the
buoyant force obtained above using Stokes' theorem, we have
where we can pull and outside of the integral
since they are assumed to be constant. But notice that
is equal to , the mass of the
displaced fluid so that
But by Newton's second law, the buoyant force must balance the
weight of the object which is given by . It follows
from the above expression for the buoyant force that
which is precisely the statement of Archimedes' Principle.