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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 =
\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

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.

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