The velocity \( \mathbf{u} \) of the fluid particle is obtained by time differentiation:
\[ \mathbf{u} = \frac{d\mathbf{x}}{dt} = \frac{d\mathbf{X}}{dt} + \frac{d\mathbf{x'}}{dt} = \mathbf{U} + \frac{d}{dt} \left( x'_1 \mathbf{e}'_1 + x'_2 \mathbf{e}'_2 + x'_3 \mathbf{e}'_3 \right) \] \[ = \mathbf{U} + {\frac{dx'_1}{dt} \mathbf{e}'_1 + \frac{dx'_2}{dt} \mathbf{e}'_2 + \frac{dx'_3}{dt} \mathbf{e}'_3} + {x'_1 \frac{d\mathbf{e}'_1}{dt} + x'_2 \frac{d\mathbf{e}'_2}{dt} + x'_3 \frac{d\mathbf{e}'_3}{dt}} \] \[ = \mathbf{U} + \mathbf{u'} + \mathbf{\Omega} \times \mathbf{x'} \]To find the acceleration \( \mathbf{a} \) of a fluid particle at P, take the time derivative to find:
\[ \mathbf{a} = \frac{d \mathbf{u}}{dt} = \frac{d}{dt} \left( \mathbf{U} + \mathbf{u}' + \boldsymbol{\Omega} \times \mathbf{x}' \right) = \frac{d \mathbf{U}}{dt} + \mathbf{a}' + 2 \boldsymbol{\Omega} \times \mathbf{u}' + \frac{d \boldsymbol{\Omega}}{dt} \times \mathbf{x}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}') \]\[ \mathbf{a} = \frac{d \mathbf{u}}{dt} = \frac{d}{dt} \left( \mathbf{U} + \mathbf{u}' + \boldsymbol{\Omega} \times \mathbf{x}' \right) \] \[ = \underbrace{\frac{d \mathbf{U}}{dt}}_{\text{acceleration of O' with respect to O}} + \underbrace{\mathbf{a}'}_{\text{fluid particle acceleration viewed in the noninertial frame}} + \underbrace{2 \boldsymbol{\Omega} \times \mathbf{u}'}_{\text{Coriolis acceleration}} \] \[ + \underbrace{\frac{d \boldsymbol{\Omega}}{dt} \times \mathbf{x}'}_{\text{acceleration caused by angular acceleration of the noninertial frame}} + \underbrace{\boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{x}')}_{\text{centripetal acceleration}} \]