The Navier–Stokes equations for steady flow, with zero viscosity and a body force corresponding to the Coriolis force, are
where  is the fluid velocity,  is the fluid density, and  the pressure. If we assume that  is a scalar potential and the advective term on the left may be neglected (reasonable if the Rossby number is much less than unity) and that the flow is incompressible (density is constant), the equations become:
where  is the angular velocity vector. If the curl of this equation is taken, the result is the Taylor–Proudman theorem:
To derive this, one needs the vector identities
and 
and 
(because the curl of the gradient is always equal to zero).
Note that  is also needed (angular velocity is divergence-free).
The vector form of the Taylor–Proudman theorem is perhaps better understood by expanding the dot product:
In coordinates for which , the equations reduce to
if .  Thus, all three components of the velocity vector are uniform along any line parallel to the z-axis.