# Is angular velocity in space frame same as that in the body frame?

The angular velocity of a rotating body can be expressed either in the space (fixed) frame, or in a frame fixed with the rotating body, as explained in this answer. Also, I understand how the angular velocity in the body frame is independent of the origin (that is fixed at some point on the body), as explained on the wikipedia page.

My question is whether the angular velocity in the space frame, and that in the body frame are equal.

The other school could be called the Angular Velocity Addition Theorem school, or Kane's school. In this school one talks about the angular velocity of rigid body B as measured in a frame attached to rigid body A. Since this school distinguishes which body is being measured and which frame it is being measured in, it uses the symbol $^A\omega^B$ to indicate the angular velocity of body B in frame A. The angular velocity can be expressed in terms of basis vectors that are fixed in A, fixed B, fixed in an auxiliary frame or combinations of frames. This school would say the angular velocity of body B in frame B is always zero, because to them the phrase "angular velocity in frame B" means as measured in frame B. The two schools would agree that "the angular velocity of body B as measured in body A is the same no matter what basis vectors are used."