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The curl of a vector field is a vector function, with each point corresponding to the infinitesimal rotation of the original vector field at said point, with the direction of the vector being the axis of rotation and the magnitude being the magnitude of rotation. Curl can be calculated by taking the cross product of the vector field and the del operator.

\text{curl}(\vec{\mathbf v}) =\nabla\times\vec{\mathbf v}
=\left|\begin{matrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\[2pt]\dfrac{\part}{\part x}&\dfrac{\part}{\part y}&\dfrac{\part}{\part z}\\[2pt]v_x&v_y&v_z\end{matrix}\right|
=\left(\frac{\part}{\part y}v_z-\frac{\part}{\part z}v_y\right)\mathbf{\hat i}+\left(\frac{\part}{\part z}v_x-\frac{\part}{\part x}v_z\right)\mathbf{\hat j}+\left(\frac{\part}{\part x}v_y-\frac{\part}{\part y}v_x\right)\mathbf{\hat k}

When looking at a two dimensional vector field, we can consider the \mathbf{\hat k} component to be 0, giving us

\nabla\times\vec{\mathbf v}=\left|\begin{matrix}\mathbf{\hat i}&\mathbf{\hat j}&\mathbf{\hat k}\\[2pt]\dfrac{\part}{\part x}&\dfrac{\part}{\part y}&\dfrac{\part}{\part z}\\[2pt]v_x&v_y&0\end{matrix}\right|=\left(\frac{\part}{\part x}v_y-\frac{\part}{\part y}v_x\right)\mathbf{\hat k}

(as all partial derivatives with respect to z will be 0).

Properties of curl

If the curl of a vector field is zero everywhere it is said to be irrotational. Any irrotational vector field will be conservative if the function is simply connected (it has no "holes" or point of discontinuity). If any scalar function has continuous second order partial derivatives, the gradient of the function will be irrotational.

Curl is very useful for computing line integrals over vector fields, as it is used in Green's and Stokes' theorems. It is also heavily used in physics, especially in electromagnetism.

See also

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