- Angular velocity tensor
In

physics , the**angular velocity tensor**is defined as a matrix T such that::$oldsymbolomega(t)\; imes\; mathbf\{r\}(t)\; =\; T(t)\; mathbf\{r\}(t)$

It allows us to express the

cross product :$oldsymbolomega(t)\; imes\; mathbf\{r\}(t)$as a matrix multiplication. It is, by definition, askew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements::$T(t)\; =egin\{pmatrix\}0\; -omega\_z(t)\; omega\_y(t)\; \backslash omega\_z(t)\; 0\; -omega\_x(t)\; \backslash -omega\_y(t)\; omega\_x(t)\; 0\; \backslash end\{pmatrix\}$**Coordinate-free description**At any instant, $t$, the angular velocity tensor is a linear map between the position vectors $mathbf\{r\}(t)$and their velocity vectors $mathbf\{v\}(t)$ of a rigid body rotating around the origin:

:$mathbf\{v\}\; =\; Tmathbf\{r\}$

where we omitted the $t$ parameter, and regard $mathbf\{v\}$ and $mathbf\{r\}$ as elements of the same 3-dimensional

Euclidean vector space $V$.The relation between this linear map and the angular velocity

pseudovector $omega$ is the following.Because of "T" is the derivative of an

orthogonal transformation , the:$B(mathbf\{r\},mathbf\{s\})\; =\; (Tmathbf\{r\})\; cdot\; mathbf\{s\}$

bilinear form isskew-symmetric . (Here $cdot$ stands for thescalar product ). So we can apply the fact ofexterior algebra that there is a uniquelinear form $L$ on $Lambda^2\; V$ that:$L(mathbf\{r\}wedge\; mathbf\{s\})\; =\; B(mathbf\{r\},mathbf\{s\})$ ,

where $mathbf\{r\}wedge\; mathbf\{s\}\; in\; Lambda^2\; V$ is the

wedge product of $mathbf\{r\}$ and $mathbf\{s\}$.Taking the

dual vector "L"* of "L" we get:$(Tmathbf\{r\})cdot\; mathbf\{s\}\; =\; L^*\; cdot\; (mathbf\{r\}wedge\; mathbf\{s\})$

Introducing $omega\; :=\; *L^*$, as the

Hodge dual of "L"* , and apply further Hodge dual identities we arrive at:$(Tmathbf\{r\})\; cdot\; mathbf\{s\}\; =\; *\; (\; *L^*\; wedge\; mathbf\{r\}\; wedge\; mathbf\{s\})\; =\; *\; (omega\; wedge\; mathbf\{r\}\; wedge\; mathbf\{s\})\; =\; *(omega\; wedge\; mathbf\{r\})\; cdot\; mathbf\{s\}\; =\; (omega\; imes\; mathbf\{r\})\; cdot\; mathbf\{s\}$

where :$omega\; imes\; mathbf\{r\}\; :=\; *(omega\; wedge\; mathbf\{r\})$

by definition.

Because $mathbf\{s\}$ is an arbitrary vector, from nondegeneracy of

scalar product follows:$Tmathbf\{r\}\; =\; omega\; imes\; mathbf\{r\}$

**See also***

Angular velocity

*Rigid body dynamics

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