  We establish a complete set of hydrodynamic equations for a spin-1
Bose-Einstein condensate (BEC), which are equivalent to the multi-component
Gross-Pitaevskii equations and expressed in terms of only observable physical
quantities: the spin density and the nematic (or quadrupolar) tensor in
addition to the density and the mass current that appear in the hydrodynamic
description of a scalar BEC. The obtained hydrodynamic equations involve a
generalized Mermin-Ho relation that is valid regardless of the spatiotemporal
dependence of the spin polarization. Low-lying collec- tive modes for phonons
and magnons are reproduced by linearizing the hydrodynamic equations. We also
apply the single-mode approximation to the hydrodynamic equations and find a
complete set of analytic solutions.
