In Goldstein (2ed) sec 4.9 - Rate of change of a vector, why does he say that the instantaneous angular velocity $\omega$ is not a derivative of any vector?
$$ (d\textbf{G})_{space} = (d\textbf{G})_{body} + d\pmb\Omega \times \textbf{G} $$
The time rate of change of the vector $\textbf{G}$ as seen by the two observers is then obtained by dividing the terms by the differential time element $dt$ under consideration;
$$ \bigg ( \frac{d\textbf{G}} {dt} \bigg)_{space} = \bigg ( \frac{d\textbf{G}} {dt} \bigg)_{body} + \pmb\omega \times \textbf{G} $$
Here $\pmb\omega$ is the instantaneous angular velocity of the body defined by the relation*
$$\pmb\omega dt = d\pmb\Omega $$
*As $\pmb\omega$ is not the derivative of any vector, it is sometimes described as a nonholonomic vector, in analogy to the nonintegrable differential constraints.
