<P> where R = R 1 x ^ + R 2 y ^ + R 3 z ^ (\ displaystyle \ mathbf (R) = R_ (1) \ mathbf (\ hat (x)) + R_ (2) \ mathbf (\ hat (y)) + R_ (3) \ mathbf (\ hat (z)) \!) is the displacement vector from the centre of mass to the new point, and δ is the Kronecker delta . </P> <P> For diagonal elements (when i = j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem . </P> <P> The generalized version of the parallel axis theorem can be expressed in the form of coordinate - free notation as </P> <Dl> <Dd> J = I + m ((R ⋅ R) E 3 − R ⊗ R), (\ displaystyle \ mathbf (J) = \ mathbf (I) + m \ left (\ left (\ mathbf (R) \ cdot \ mathbf (R) \ right) \ mathbf (E) _ (3) - \ mathbf (R) \ otimes \ mathbf (R) \ right),) </Dd> </Dl>

Theorem of parallel axis of moment of inertia