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</math>
Such that its associated rotation matrix is <math>A = I + d\Phi(t)</math>. When it is divided by the time, this will yield the [[angular velocity]] vector.
==Order of rotations==
These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals .<ref>{{Harv|Goldstein|Poole|Safko|2002|loc=§4.8}}</ref> To understand what this means, consider
:<math> dA_{\mathbf{x}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -d\theta \\ 0 & d\theta & 1 \end{bmatrix}.</math>
First, test the orthogonality condition, {{math|1=''Q''<sup>T</sup>''Q'' = ''I''}}. The product is
:<math> dA_\mathbf{x}^\textsf{T} \, dA_\mathbf{x} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 + d\theta^2 & 0 \\ 0 & 0 & 1 + d\theta^2 \end{bmatrix},</math>
differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.
Next, examine the square of the matrix,
:<math> dA_{\mathbf{x}}^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 - d\theta^2 & -2d\theta \\ 0 & 2\,d\theta & 1 - d\theta^2 \end{bmatrix}.</math>
Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,
:<math>dA_\mathbf{y} = \begin{bmatrix} 1 & 0 & d\phi \\ 0 & 1 & 0 \\ -d\phi & 0 & 1 \end{bmatrix}.</math>
Compare the products {{math|''dA''<sub>'''x'''</sub> ''dA''<sub>'''y'''</sub>}} to {{math|''dA''<sub>'''y'''</sub>''dA''<sub>'''x'''</sub>}},
:<math>\begin{align}
dA_{\mathbf{x}}\,dA_{\mathbf{y}} &= \begin{bmatrix} 1 & 0 & d\phi \\ d\theta\,d\phi & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix} \\
dA_{\mathbf{y}}\,dA_{\mathbf{x}} &= \begin{bmatrix} 1 & d\theta\,d\phi & d\phi \\ 0 & 1 & -d\theta \\ -d\phi & d\theta & 1 \end{bmatrix}. \\
\end{align}</math>
Since <math>d\theta \, d\phi</math> is second-order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is ''commutative''. In fact,
:<math> dA_{\mathbf{x}}\,dA_{\mathbf{y}} = dA_{\mathbf{y}}\,dA_{\mathbf{x}},</math>
again to first order. In other words, {{em|the order in which infinitesimal rotations are applied is irrelevant}}.
This useful fact makes, for example, derivation of rigid body rotation relatively simple. 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. 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. Technically, this dismissal of any second order terms amounts to [[Group contraction]].
== Generators of rotations ==
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