For a nonrotating observer, both the components of \( \mathbf{B} \) and the unit vectors change over time. The scalar components \( B_1, B_2, B_3 \) are the same for both rotating and nonrotating observers \[ \left( \frac{d\mathbf{B}}{dt} \right)_I = \frac{dB_1}{dt} \mathbf{i}_1 + \frac{dB_2}{dt} \mathbf{i}_2 + \frac{dB_3}{dt} \mathbf{i}_3 + B_1 \frac{d\mathbf{i}_1}{dt} + B_2 \frac{d\mathbf{i}_2}{dt} + B_3 \frac{d\mathbf{i}_3}{dt} \]

Time derivatives of the unit vectors \[ B_1 \frac{d\mathbf{i}_1}{dt} + B_2 \frac{d\mathbf{i}_2}{dt} + B_3 \frac{d\mathbf{i}_3}{dt} = B_1 (\boldsymbol{\Omega} \times \mathbf{i}_1) + B_2 (\boldsymbol{\Omega} \times \mathbf{i}_2) + B_3 (\boldsymbol{\Omega} \times \mathbf{i}_3) \\ = \boldsymbol{\Omega} \times (B_1 \mathbf{i}_1 + B_2 \mathbf{i}_2 + B_3 \mathbf{i}_3) = \boldsymbol{\Omega} \times \mathbf{B} \Rightarrow \boxed{\left( \frac{d\mathbf{B}}{dt} \right)_I = \left( \frac{d\mathbf{B}}{dt} \right)_R + \boldsymbol{\Omega} \times \mathbf{B}} \]Observers in both frames agree on the rate of change of \( \boldsymbol{\Omega} \) because \( \boldsymbol{\Omega} \times \boldsymbol{\Omega} = 0 \), \(\boxed{\left( \frac{d\mathbf{B}}{dt} \right)_I = \left( \frac{d\mathbf{B}}{dt} \right)_R + \boldsymbol{\Omega} \times \mathbf{B}}\) remains valid even if \( \boldsymbol{\Omega} \) varies in time or direction

Let \( \mathbf{r} \) be the position vector of an arbitrary fluid element, \(\boxed{\left( \frac{d\mathbf{B}}{dt} \right)_I = \left( \frac{d\mathbf{B}}{dt} \right)_R + \boldsymbol{\Omega} \times \mathbf{B}}\) \(\Rightarrow \left( \frac{d\mathbf{r}}{dt} \right)_I = \left( \frac{d\mathbf{r}}{dt} \right)_R + \boldsymbol{\Omega} \times \mathbf{r} \)
This means the velocity in the inertial frame \( \mathbf{u}_I \) is the sum of the velocity in the rotating frame and the velocity due to rotation \( \mathbf{u}_I = \mathbf{u}_R + \boldsymbol{\Omega} \times \mathbf{r} \), \( \mathbf{u}_R \) is called the relative velocity
According to Newton’s second law, the acceleration in the inertial frame determines the force per unit mass. Applying \(\boxed{\left( \frac{d\mathbf{B}}{dt} \right)_I = \left( \frac{d\mathbf{B}}{dt} \right)_R + \boldsymbol{\Omega} \times \mathbf{B}}\) to \( \mathbf{u}_I \) gives \( \left( \frac{d\mathbf{u}_I}{dt} \right)_I = \left( \frac{d\mathbf{u}_I}{dt} \right)_R + \boldsymbol{\Omega} \times \mathbf{u}_I \)

¹ Pedlosky, J. (1982). Geophysical Fluid Dynamics. Springer study edition. Springer, Berlin, Heidelberg.