Editing PGE/RotatingFrame (section)

===Traditional Presentation===
At times, it can be useful to view the motion of a fluid from a frame of reference that is rotating with a uniform (''i.e.,'' time-independent) angular velocity <math>~\Omega_f</math>.  In order to transform any one of the [[PGE#Principal_Governing_Equations|principal governing equations]] from the inertial reference frame to such a rotating reference frame, we must specify the orientation as well as the magnitude of the angular velocity vector about which the frame is spinning, <math>{\vec\Omega}_f</math>; and the <math>~d/dt</math> operator, which denotes Lagrangian time-differentiation in the inertial frame, must everywhere be replaced as follows:
<div align="center">
<math>
\biggl[\frac{d}{dt} \biggr]_{inertial} \rightarrow \biggl[\frac{d}{dt} \biggr]_{rot} + {\vec{\Omega}}_f \times .
</math>

[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (11)

</div>

Operating on the fluid element's position vector, <math>\vec{x}</math>, we obtain the transformation,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\frac{d\vec{x}}{dt}\biggr|_\mathrm{inertial}</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="right"><math>\frac{d\vec{x}}{dt}\biggr|_\mathrm{rotating} + \vec\Omega \times \vec{x} \, ,</math></td>
</tr>
</table>
that is,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right"><math>\vec{v}_\mathrm{inertial}</math></td>
  <td align="center"><math>\rightarrow</math></td>
  <td align="right"><math>\vec{v}_\mathrm{rot} + \vec\Omega \times \vec{x} \, .</math></td>
</tr>
<tr>
  <td align="center" colspan="3" align="center">
[<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>], Chap. 4, &sect;25, Eq. (15)
  </td>
</tr>
</table>

Performing this transformation implies, for example, that
<div align="center">
<math>
\vec{v}_{inertial} = \vec{v}_{rot} + {\vec{\Omega}}_f \times \vec{x} ,
</math>
</div>
and,
<div align="center">
<math>
\biggl[ \frac{d\vec{v}}{dt}\biggr]_{inertial} = \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} + {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) 
</math>

<math>
= \biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} + 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - \frac{1}{2} \nabla \biggl[ |{\vec{\Omega}}_f \times \vec{x}|^2 \biggr] 
</math>
</div>
(If we were to allow <math>{\vec\Omega}_f</math> to be a function of time, an additional term involving the time-derivative of <math>{\vec\Omega}_f</math> also would appear on the right-hand-side of these last expressions; see, for example, Eq.~1D-42 in [[Appendix/References|BT87]].)  Note as well that the relationship between the fluid [[PGE/RotatingFrame#WikiVorticity|vorticity]] in the two frames is,
<div align="center">
<math>
[\vec\zeta]_{inertial} = [\vec\zeta]_{rot} + 2{\vec\Omega}_f .
</math>
</div>