An infinitesimal transformation of a Vector is given by

(1) 
where the Matrix
is infinitesimal and I 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) 

(3) 
Now let

(4) 
The inverse
is then
, since

(5) 
Since we are defining our infinitesimal transformation to be a rotation, Orthogonality
of Rotation Matrices
requires that

(6) 
but

(7) 

(8) 
so
and the infinitesimal rotation is Antisymmetric. It must therefore have
a Matrix of the form

(9) 
The differential change in a vector upon application of the Rotation Matrix is then

(10) 
Writing in Matrix form,
Therefore,

(13) 
where

(14) 
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

(15) 
This can be written as an operator equation, known as the Rotation Operator, defined as

(16) 
See also Acceleration, Euler Angles, Rotation, Rotation Matrix, Rotation Operator
© 19969 Eric W. Weisstein
19990526