For a incompressible fluid at rest there are no shear or deforming forces and
Any stresses are normal to any surface drawn in the fluid and the pressure inside the fluid is the same in all directions.
Since
though the fluid is incompressible the density
need not be the same everywhere. We can writ
Draw a cube in the fluid.
The pressure on the face A is
and the force is
The total force in the x direction is then
Similarly
and
The total force is
As
Dividing by
gives
This force must be balanced by an external force. If the external forces are due to a potential field
the force per unit mass is
The total force is then
since the fluid is in equilibrium.
If
is constant then