with the rotation vector pointed along the z-axis of rotation namely,

where is either or

Time derivatives in the two frames

Introduce unit vectors , now representing standard unit basis vectors in the general rotating frame. As they rotate they will remain normalized and perpendicular to each other. If they rotate at the speed of about an axis along the rotation vector then each unit vector of the rotating coordinate system (such as or ) abides by the following equation:

So if denotes the transformation

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