Write out the components of the vorticity tensor, \(\tilde{W}\) in Cartesian coordinates. Show that the tensor is antisymmetric, i.e., \(W_{i j}=-W_{j i}\), and has only three distinct components.

Show that the components of the vorticity tensor can be represented in terms of the components of the vorticity vector as

\[\left(\begin{array}{ccc}0 & \omega_{z} & -\omega_{y} \\-\omega_{z} & 0 &\omega_{x}\\\omega_{y} & -\omega_{x} & 0\end{array}\right)\]

where \(\omega_{x}\) etc. are the components of the vorticity vector \(\boldsymbol{\omega}=abla \times \boldsymbol{v}\).

Now show by direct matrix expansion that

\[\tilde{W} \cdot \delta \boldsymbol{r}=(1 / 2) \boldsymbol{\omega} \times \delta \boldsymbol{r}\]

The RHS is the angular velocity of motion and hence this verifies that \(\tilde{W} \cdot \delta \boldsymbol{r}\) is a purely rotational motion of the line element \(\delta \boldsymbol{r}\).