  An extension of the adiabatic factorization of the time evolution operator is
studied for spin in a general time varying magnetic field $B(t)$. When $B(t)$
changes adiabatically, such a factorization reduces to the product of the
geometric operator which embodies the Berry phase phenomenon and a usual
dynamical operator. For a general time variation of $B(t)$, there should be
another operator $N(t)$ in the factorization that is related to non-adiabatic
transitions. A simple and explicit expression for the instantaneous angular
velocity of this operator is derived. This is done in a way that is independent
of any specific representation of spin. Two classes of simple conditions are
given under which the operator $N(t)$ can be made explicit. As a special case,
a generalization of the traditional magnetic resonance condition is pointed
out.
