= Rotating-axes theorem
{title2=$(d\mathbf a/dt)_I=(d\mathbf a/dt)_R+\boldsymbol\Omega\times\mathbf a$}
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 $\dot{\mathbf e}_j=\boldsymbol\Omega\times\mathbf e_j$, proving the displayed identity by differentiating $\mathbf a=\sum_ja_j\mathbf e_j$. Applying it twice to position produces the <Coriolis acceleration> $2\boldsymbol\Omega\times\mathbf u$ and the centripetal term $\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf x)$. A time-dependent rotation adds $\dot{\boldsymbol\Omega}\times\mathbf x$; constant angular velocity removes that last term.
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