*#redirect [[Angular velocity]] ▼{{Unreferenced|date=February 2007}}
In [[physics]], the '''angular velocity tensor''' is defined as a matrix T such that:
:<math>
\boldsymbol\omega(t) \times \mathbf{r}(t) = T(t) \mathbf{r}(t) </math>
It allows us to express the [[cross product]]
:<math>\boldsymbol\omega(t) \times \mathbf{r}(t) </math>
as a matrix multiplication. It is, by definition, a [[skew-symmetric matrix]] with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements:
:<math>
T(t) =
\begin{pmatrix}
0 & -\omega_z(t) & \omega_y(t) \\
\omega_z(t) & 0 & -\omega_x(t) \\
-\omega_y(t) & \omega_x(t) & 0 \\
\end{pmatrix}</math>
==Coordinate-free description==
At any instant, <math>t</math>, the angular velocity tensor is a linear map between the position vectors <math> \mathbf{r}(t) </math>
and their velocity vectors <math> \mathbf{v}(t) </math> of a rigid body rotating around the origin:
:<math> \mathbf{v} = T\mathbf{r} </math>
where we omitted the <math>t</math> parameter, and regard <math> \mathbf{v} </math> and <math> \mathbf{r} </math> as elements of the same 3-dimensional [[Euclidean vector space]] <math>V</math>.
The relation between this linear map and the angular velocity [[pseudovector]] <math>\omega</math> is the following.
Because of ''T'' is the derivative of an [[orthogonal transformation]], the
:<math>B(\mathbf{r},\mathbf{s}) = (T\mathbf{r}) \cdot \mathbf{s} </math>
[[bilinear form]] is [[skew-symmetric]]. (Here <math>\cdot</math> stands for the [[scalar product]]). So we can apply the fact of [[exterior algebra]] that there is a unique [[linear form]] <math>L</math> on <math>\Lambda^2 V </math> that
:<math>L(\mathbf{r}\wedge \mathbf{s}) = B(\mathbf{r},\mathbf{s})</math> ,
where <math>\mathbf{r}\wedge \mathbf{s} \in \Lambda^2 V </math> is the [[wedge product]] of <math>\mathbf{r}</math> and <math>\mathbf{s}</math>.
Taking the [[dual vector]] ''L''* of ''L'' we get
:<math> (T\mathbf{r})\cdot \mathbf{s} = L^* \cdot (\mathbf{r}\wedge \mathbf{s}) </math>
Introducing <math> \omega := *L^* </math>, as the [[Hodge dual]] of ''L''* , and apply further Hodge dual identities we arrive at
:<math> (T\mathbf{r}) \cdot \mathbf{s} = * ( *L^* \wedge \mathbf{r} \wedge \mathbf{s}) = * (\omega \wedge \mathbf{r} \wedge \mathbf{s}) = *(\omega \wedge \mathbf{r}) \cdot \mathbf{s} = (\omega \times \mathbf{r}) \cdot \mathbf{s} </math>
where
:<math>\omega \times \mathbf{r} := *(\omega \wedge \mathbf{r}) </math>
by definition.
Because <math>\mathbf{s}</math> is an arbitrary vector, from nondegeneracy of [[scalar product]] follows
:<math> T\mathbf{r} = \omega \times \mathbf{r}</math>
==Viewing as a vector field==
For angular velocity tensor maps velocities to positions, it is a [[vector field]]. In particular, this vector field is a [[Killing vector field]] belonging to an element of the [[Lie algebra]] so(3) of the 3-dimensional [[rotation group]], SO(3).
==See also==
* [[Rigid body dynamics]]
{{DEFAULTSORT:Angular Velocity Tensor}}
[[Category:Physical quantities]]
[[Category:Tensors]]
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