Editing Taylor–Proudman theorem (section)

==Derivation==
The [[Navier–Stokes equations]] for steady flow, with zero [[viscosity]] and a body force corresponding to the Coriolis force, are
:<math>
\rho({\mathbf u}\cdot\nabla){\mathbf u}={\mathbf F}-\nabla p,</math>
where <math>{\mathbf u}</math> is the fluid velocity, <math>\rho</math> is the fluid density, and <math>p</math> the pressure. If we assume that <math>F=\nabla\Phi=-2\rho\mathbf\Omega\times{\mathbf u}</math> is a [[scalar potential]] and the [[Advection|advective]] term on the left may be neglected (reasonable if the [[Rossby number]] is much less than unity) and that the [[Incompressible flow|flow is incompressible]] (density is constant), the equations become:

:<math>
2\rho\mathbf\Omega\times{\mathbf u}=-\nabla p,</math>
where <math>\Omega</math> is the [[angular velocity]] vector. If the [[Curl (mathematics)|curl]] of this equation is taken, the result is the Taylor–Proudman theorem:
:<math>
({\mathbf\Omega}\cdot\nabla){\mathbf u}={\mathbf  0}.
</math>

To derive this, one needs the [[Vector calculus identities|vector identities]]

:<math>\nabla\times(A\times B)=A(\nabla\cdot B)-(A\cdot\nabla)B+(B\cdot\nabla)A-B(\nabla\cdot A)</math> 

and 

:<math>\nabla\times(\nabla p)=0\ </math>

and 
:<math>\nabla\times(\nabla \Phi)=0\ </math>

(because the [[Curl (mathematics)|curl]] of the gradient is always equal to zero).
Note that <math>\nabla\cdot{\mathbf\Omega}=0</math> 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]]:

:<math>
\Omega_x\frac{\partial {\mathbf u}}{\partial x} + \Omega_y\frac{\partial {\mathbf u}}{\partial y} + \Omega_z\frac{\partial {\mathbf u}}{\partial z}=0.
</math>

In coordinates for which <math>\Omega_x=\Omega_y=0</math>, the equations reduce to
:<math>
\frac{\partial{\mathbf u}}{\partial z}=0,</math>
if <math>\Omega_z\neq 0</math>.  Thus, ''all three'' components of the velocity vector are uniform along any line parallel to the z-axis.