Hydrostatic equilibrium represents that state of a fluid for which all of it sparts are macroscopically at rest relative to an externally applied, invariant field of force. In such a state, the absolute rate of deformation is zero, and its velocity relative to the field of force is zero. The sea at rest in the presence of the field of gravity would represent a state of geohydrostatic equilibrium.
In a fluid which is in a state of hydrostatic equilibrium
the pressure is isotropic, that is, the normal
stress
acting on any prescribed surface within the
fluid is independent of the orientation of that surface for
a given point in the fluid. Another way of saying the same
thing is that the hydrostatic pressure is a scalar quantity.
This property of isotropy of the hydrostatic pressure can be proven from the basin definition of a fluid. Proof: Consider a small tetrahedron of fluid of arbitrary dimensions which is contained within a fluid region (Figure 3.02-1).
The forces to which the tetrahedron of fluid is subjected under
hydrostatic conditions are the surface forces
and pA, and the
volume force components 
where
p is pressure, 
is force per unit volume, A is area,
and V is volume. No shearing stresses can exist at the
surfaces 
or A if the fluid is in a
state of hydrostatic equilibrium. The balance of forces under
static conditions is therefore
where the angles 
are connected by
the relation
and, from geometrical considerations,
where d is the length of the line which passes through the point 0 and is normal to the surface ABC. Furthermore, the surface areas of the right tetrahedron are connected by the relation
It follows from relations (2) and (3) that
but since 
, 
and
it follows that
or
using relation (4). Therefore
or
The volume V is 
, consequently Eqs. (9)
can be expressed as
If we now take the limit as 
approach zero,
considering that 
are finite, then we find that
at the point 0. That is, the hydrostatic pressure at a given point is independent of direction, and therefore isotropic.