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\end{aligned}</math>
which holds even if ''A''(''t'') does not rotate uniformly. Therefore, the angular velocity tensor is:
: <math>\Omega = \frac {dA} {dt} A^{-1} = \frac {dA} {dt} A^{\mathsf{T}},</math>
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its [[Hodge dual]] is a vector, which is precisely the previous angular velocity vector <math>\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]</math>.
=== Exponential of
If we know an initial frame ''A''(0) and we are given a ''constant'' angular velocity tensor '''Ω''', we can obtain ''A''(''t'') for any given ''t''. Recall the matrix differential equation:
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which shows a connection with the [[Lie group]] of rotations.
===
We prove that angular velocity tensor is [[Skew-symmetric matrix|skew symmetric]], i.e. <math>\Omega = \frac {dA(t)}{dt} \cdot A^\text{T} </math> satisfies <math>\Omega^\text{T} = -\Omega</math>.
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