# Vorticity

Vorticity is a vector field variable which is derived from the velocity vector. Mathematically, it is defined as the curl of the velocity vector

$\omega \equiv \textrm{curl}(u) \equiv \nabla \times u$

In tensor notation, vorticity is given by:

$\omega_i = \epsilon_{ijk} \frac{\partial u_k}{\partial x_j}$

where $\epsilon_{ijk}$ is the alternating tensor. The components of vorticity in Cartesian coordinates are;:

$\omega_1 = \frac{\partial u_3}{\partial x_2} - \frac{\partial u_2}{\partial x_3}$
$\omega_2 = \frac{\partial u_1}{\partial x_3} - \frac{\partial u_3}{\partial x_1}$
$\omega_3 = \frac{\partial u_2}{\partial x_1} - \frac{\partial u_1}{\partial x_2}$

This can be obtained by using determinants

$\omega = \begin{vmatrix} \hat{e}_1 & \hat{e}_2 & \hat{e}_3 \\ \frac{\partial}{\partial x_1} & \frac{\partial}{\partial x_2} & \frac{\partial}{\partial x_3} \\ u_1 & u_2 & u_3 \end{vmatrix}$

where $\hat{e}_1, \hat{e}_2, \hat{e}_3$ are the unit vectors for the Cartesian coordinate system.

## Physical Significance

The vorticity can be seen as a vector having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point.