We can now compute the angular velocity of the particle about the \( z \) axis, \( \omega_z \), by combining all these results \[ \omega_z = \lim_{\Delta t \to 0} \frac{1}{2} (\Delta \alpha - \Delta \beta) / \Delta t = \lim_{\Delta t \to 0} \frac{1}{2} \left( \frac{\Delta \eta}{\Delta x} - \frac{\Delta \xi}{\Delta y} \right) / \Delta t = \frac{1}{2} \lim_{\Delta t \to 0} \left( \frac{\partial v_y}{\partial x} \Delta t - \frac{\partial v_x}{\partial y} \Delta t \right) / \Delta t \]
\( \omega_z = \frac{1}{2} \left( \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \right) \)
By considering the rotation of pairs of perpendicular line segments in the \( yz \) and \( xz \) planes, one can show similarly that
\(
\omega_x = \frac{1}{2} \left( \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \right), \quad
\omega_y = \frac{1}{2} \left( \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \right)
\)
Then \( \vec{\omega} = \omega_x \hat{i} + \omega_y \hat{j} + \omega_z \hat{k} \quad \Rightarrow \)
\(
\vec{\omega} = \frac{1}{2} \left[
\hat{i} \left( \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z} \right) +
\hat{j} \left( \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x} \right) +
\hat{k} \left( \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y} \right)
\right]
\)
We recognize the term in the square brackets as
\(\boxed{
\text{curl } \vec{V} = \nabla \times \vec{V}}
\)
Then, in vector notation, we can write \(\quad \vec{\omega} = \frac{1}{2} \nabla \times \vec{V} \)
It is worth noting here that we should not confuse rotation of a fluid particle with flow consisting of circular streamlines, or vortex flow. As we saw in the Example in class, in such a flow the particles could rotate as they move in a circular motion, but they do not have to!