Hereafter, hydrostatic eqilibrium will be understood to imply hydrostatic equilibrium of a fluid in the presence of a field of gravity of the earth (i.e., geo-hydrostatic equilibrium).
Consider a vertical column of fluid of height z - h and cross sectional area A. We will consider the density of the fluid as defined in the previous article is endowed with the additional property that the integral
represents the weight of the vertical column of height z - h. Under hydrostatic conditions, the net surface
force 
must balance the weight W, thus
If this equation is differentiated with respect to z we obtain
which is the scalar form of the differential equation for hydrostatic equilibrium.
The complete vector equation can be obtained by a more general approach. The property (13) can be generalized in the form of the following vector equation
where 
is the total force exerted upon an
arbitrary volume of fluid V by the field of gravity.
Equation (16) has the advantage that is is independent
of the coordinate system chosen.
THe force acting on a small element of surface area
dA of the volume V is 
where 
is the unit outward normal to the surface at the
point in question and p is the fluid pressure
at that point. The total surface force is therefore given by the surface integral
where A is the total surface area of the arbitrary volume V.
The condition of hydrostatic balance can therefore be expressed by
.
In order to put this in differential form, we will make
use of a mathematical integral tranformation known as the
theorem of Gauss.
This theorem states that for any continuous position function
f
where is the unit outward normal at any point on the
surface A. Actually, it can be shown that the divergence
theorem (Eq. 154, Art. 1.071, p. 65) is a special case
of the theorem of Gauss.
in general is that the integrand vanish. Therefore
This is the complete statement of the hydrostatic equation in vector form.
If the above theorem is applied to the pressure function p, Eq. (17) takes the form
However, since V is arbitrary, it follows that the only way that Eq. 19 can be satisfied in general is that the integrand vanish. Therefore
This is the complete statement of the hydrostatic equation in vector form.
Equation (20) implies several things:
, hence
for a given interval of p the spacing of isobaric surfaces
is inversely proportial to 
and g;
under perfect hydrostatic conditions, therefore the isobaric
surfaces are perpendicular to 
and are therefore
level surfaces; and
can be expressed as the gradient of the
geopotential 
. Consequently, the hydrostatic equation can
be written in the form
If we let 
represent any differential change of position,
then the scalar products 
and
are equivalent to the total space
differentials dp and 
.
Consequently, the scalar product of on Eq. (21) leads to
However, since the isobaric surfaces must be level under hydrostatic
conditions, as implied by Eq. (20) or (21), it therefore follows
that p is a function of only, since the latter function defines
the level surfaces. Consequently, the left hand side of
Eq. (22) is a function only of and hence must
also be a function only of . Therefore the isopycnal surfaces
are level surfaces under perfectly hydrostatic conditions.
In summary, the orientation of the surfaces of p, , and
are identical under hydrostatic conditions and would form
approximately spheroidal surfaces on the earth under this
condition.