  Using a generalized Madelung transformation, we derive the hydrodynamic
representation of the Dirac equation in arbitrary curved space-times coupled to
an electromagnetic field. We obtain Dirac-Euler equations for fermions
involving a continuity equation and a first integral of the Bernoulli equation.
Comparing between the Dirac and Klein-Gordon equations we obtain the balance
equation for fermion particles. We also use the correspondence between fermions
and bosons to derive the hydrodynamic representation of the Weyl equation which
is a chiral form of the Dirac equation.
