Editing Vorticity (section)

==Mathematical definition and properties==
Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by <math>\boldsymbol{\omega}</math>, defined as the [[curl (mathematics)|curl]] of the velocity field <math>\mathbf v</math> describing the continuum motion. In [[Cartesian coordinates]]:

:<math>\begin{align}
  \boldsymbol{\omega} = \nabla \times \mathbf v = \left(
         \dfrac{\partial v_z}{\partial y} - \dfrac{\partial v_y}{\partial z},
         \dfrac{\partial v_x}{\partial z} - \dfrac{\partial v_z}{\partial x},
         \dfrac{\partial v_y}{\partial x} - \dfrac{\partial v_x}{\partial y}
       \right) \,.
\end{align}</math>

We may also express this in index notation as <math> \omega_i=\varepsilon_{ijk}\frac{\partial v_k}{\partial x_j}</math>.<ref>{{Cite book |last=Kundu |first=Pijush K. |title=Fluid mechanics |last2=Cohen |first2=Ira M. |last3=Dowling |first3=David R. |last4=Tryggvason |first4=Gretar |date=2016 |publisher=Elsevier, Academic Press |isbn=978-0-12-405935-1 |edition=Sixth |location=Amsterdam Boston Heidelberg London}}</ref> In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.

In a two-dimensional flow where the velocity is independent of the <math>z</math>-coordinate and has no <math>z</math>-component, the vorticity vector is always parallel to the <math>z</math>-axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector <math>\hat{z}</math>:
:<math>\begin{align}
  \boldsymbol{\omega} = \nabla \times \mathbf v    = \left(\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}\right)\mathbf e_z\,.
\end{align}</math>

The vorticity is also related to the flow's [[circulation (fluid dynamics)|circulation]] (line integral of the velocity) along a closed path by the (classical) [[Stokes' theorem]]. Namely, for any [[infinitesimal]] [[Differential (infinitesimal)|surface element]] {{math|''C''}} with [[normal (geometry)|normal direction]] <math>\mathbf n</math> and area <math>dA</math>, the circulation <math>d\Gamma</math> along the [[perimeter]] of <math>C</math> is the [[dot product]] <math>\boldsymbol{\omega} \cdot (\mathbf n \, dA)</math> where <math>\boldsymbol{\omega}</math> is the vorticity at the center of <math>C</math>.<ref name=Clancy7.11>Clancy, L.J., ''Aerodynamics'', Section 7.11</ref>

Since vorticity is an axial vector, it can be associated with a second-order antisymmetric tensor <math>\boldsymbol\Omega</math> (the so-called vorticity or rotation tensor), which is said to be the dual of <math>\boldsymbol\omega</math>. The relation between the two quantities, in index notation, are given by

:<math>\Omega_{ij}=\frac{1}{2}\varepsilon_{ijk}\omega_k, \qquad \omega_i = \varepsilon_{ijk}\Omega_{jk}</math>

where <math>\varepsilon_{ijk}</math> is the three-dimensional [[Levi-Civita symbol|Levi-Civita tensor]]. The vorticity tensor is simply the antisymmetric part of the tensor <math>\nabla\mathbf v</math>, i.e.,

:<math>\boldsymbol\Omega = \frac{1}{2}\left[ (\nabla\mathbf v)^T-\nabla\mathbf v\right] \quad \text{or} \quad \Omega_{ij} = \frac{1}{2}\left(\frac{\partial v_j}{\partial x_i}-\frac{\partial v_i}{\partial x_j}\right).</math>