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Hydrostatic pressure

Let us consider an incompressible fluid at rest in the presence of a homogeneous gravitational field so it is supported by walls of some, possibly open, vessel. Inside a volume of the fluid there is the well known hydrostatic pressure, let us denote it by $ p$. Assume the fluid has one free8.6 part of its boundary, i.e. one part of its surface forms a level, i.e. plane or its part. Let us denote by $ \vec{r}_0$ a position vector8.7 of an arbitrary point of this plane. Hence for $ p$ holds img206 Notice the relation ([*]) is valid inside the volume of the fluid only. If we need to extend its domain, it is necessary to assure zero values for $ p$ outside of the volume of the fluid. For instance, this is satisfied naturally, if we consider $ \varrho$ a constant scalar field but vanishing outside the volume of the fluid.