From this and the infinitesimal rotation operator and its Hermitian conjugate, and ignoring second order term in ( \ delta \ theta ) ^ 2, one can derive the commutation relation with the rotation generator:
12.
Although spinors can be defined purely as elements of a representation space of the spin group ( or its Lie algebra of infinitesimal rotations ), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra.
13.
A single discrete Thomas rotation ( as opposed to the series of infinitesimal rotations that add up to the Thomas precession ) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.
14.
The generalized force for a rotation is the torque "'N "', since the work done for an infinitesimal rotation \ delta \ boldsymbol \ phi is dW = \ mathbf { N } \ cdot \ delta \ boldsymbol \ phi.
15.
In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7 ( and, trivially, in dimension 0 or 1 ) is the curl of a vector field a vector field, and only in 3 or Curl : Generalizations; in brief, the curl of a vector field is a bivector field, which may be interpreted as the special orthogonal Lie algebra of infinitesimal rotations; however, this cannot be identified with a vector field because the dimensions differ there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions ( and more generally \ textstyle { \ binom { n } { 2 } = \ frac { 1 } { 2 } n ( n-1 ) } dimensions of rotations in " n " dimensions ).
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