Rotating-axes theorem ($(d\mathbf{a}/dt{)}_I=(d\mathbf{a}/dt{)}_R+\mathbf{\Omega }\times \mathbf{a}$)

The time derivative of a vector in inertial axes equals its component derivative in rotating axes plus the change of the rotating basis itself. Each basis vector obeys ${\dot{\mathbf{e}}}_j=\mathbf{\Omega }\times {\mathbf{e}}_j$, proving the displayed identity by differentiating $\mathbf{a}={\sum }_j{a}_j{\mathbf{e}}_j$. Applying it twice to position produces the Coriolis acceleration $2\mathbf{\Omega }\times \mathbf{u}$ and the centripetal term $\mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{x})$. A time-dependent rotation adds $\dot{\mathbf{\Omega }}\times \mathbf{x}$; constant angular velocity removes that last term.