Question on the 3D Kinematics equation.

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kwnmanu94
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How exactly did the terms in parenthesis of the equation (14) become (15)?

I don't even know what the Transport Theorem is about?
 
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What textbook is this? Since the text directly references Transport Theorem I would expect it to be described earlier. Perhaps look in the index or try compare with transport theorem where you can see it relates the coordinate-wise time derivative of a point coordinated by two different coordinate system that rotate relative to each other.