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Potential of a homogeneous gravitational field

A homogeneous gravitational field is characterized by a constant vector field , the well known gravitational acceleration. Let us consider some volume or vessel filled with an incompressible fluid of density img196 1. The gravitational field exerts a force on the fluid. Its force density img197 is given by the following well known formula img198 It is a simple task to find a potential to (crossref). Let us remind, a potential (if it exists) of some given force field img199 is defined as a certain function img91 satisfying the following equation img200 We can see that to the given by ([*]) there exists a potential, let us denote it by $ \varphi$, stating img202 where $ C \in \mathbb{R}$ is a constant of integration8.4 and $ \vec{r}$ is a position vector8.5.