An infinitesimal transformation of a vector is given by
(1)
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where the matrix is infinitesimal and is the identity matrix. (Note that the infinitesimal transformation may not correspond to an inversion, since inversion is a discontinuous process.) The commutativity of infinitesimal transformations and is established by the equivalence of
(2)
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(3)
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(4)
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(5)
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Now let
(6)
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The inverse is then , since
(7)
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(8)
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(9)
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Since we are defining our infinitesimal transformation to be a rotation, orthogonality of rotation matrices requires that
(10)
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but
(11)
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(12)
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(13)
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so and the infinitesimal rotation is antisymmetric. It must therefore have a matrix of the form
(14)
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The differential change in a vector upon application of the rotation matrix is then
(15)
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Writing in matrix form,
(16)
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(17)
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(18)
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(19)
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Therefore,
(20)
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where
(21)
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The total rotation observed in the stationary frame will be a sum of the rotational velocity and the velocity in the rotating frame. However, note that an observer in the stationary frame will see a velocity opposite in direction to that of the observer in the frame of the rotating body, so
(22)
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This can be written as an operator equation, known as the rotation operator, defined as
(23)
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