To see this exemplified, consult infinitesimal rotations SO ( 3 ).
2.
The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.
3.
Taylor expanding to first order in " ? " gives the infinitesimal rotation operator:
4.
It turns out that " the order in which infinitesimal rotations are applied is irrelevant ".
5.
For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space.
6.
Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded and in this case commutativity appears.
7.
Therefore, R = e ^ { Wt } is a rotation matrix and in a time dt is an infinitesimal rotation matrix.
8.
But one must always be careful to distinguish ( the first order treatment of ) these infinitesimal rotation matrices from both finite rotation matrices and from Lie algebra elements.
9.
Over real numbers, this characterization is used in interpreting the curl of a vector field ( naturally a 2-vector ) as an infinitesimal rotation or " curl ", hence the name.
10.
When contrasting the behavior of finite rotation matrices in the BCH formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals one finds a bona fide vector space.
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