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Mathematics-Online lexicon:

Curl

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Let $ \Phi: D \longrightarrow \mathbb{R}^3 $ be a differentiable vector field defined on a region $ D $ and given by

$\displaystyle \par \left(\begin{array}{c} x \\ y \\ z \end{array}\right) \par... ...eft(\begin{array}{c} p(x,y,z) \\ q(x,y,z) \\ r(x,y,z) \end{array}\right) \ . $

Then its curl is defined as

$\displaystyle \operatorname{curl} \Phi = \left(\begin{array}{c} r_y - q_z \\ p_z - r_x \\ q_x - p_y \end{array}\right) \ . $

Formally with $ \nabla = \left(\begin{array}{c} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} \end{array}\right) $ the curl may be calculated as follows

$\displaystyle \nabla \times \Phi = \left\vert\begin{array}{ccc} e_1 & e_2 & e_... ...al y} & \frac{\partial}{\partial z} \\ p & q & r \end{array}\right\vert \ . $

For $ 2$ - dimensional vector fields $ \Phi$ given by

$\displaystyle \left(\begin{array}{c} x \\ y \end{array}\right) \ \mapsto \ \left(\begin{array}{c} p(x,y) \\ q(x,y) \end{array}\right) \ . $

a scalar curl is defined as

$\displaystyle \operatorname{curl} \Phi = q_x - p_y \ .$

If one extends $ \Phi$ to a $ 3$ - dimensional vector field $ \hat{\Phi} $ putting $ r = 0$ and computes the curl of $ \hat{\Phi} $, then the scalar curl is just the third component of $ \operatorname{curl} \hat{\Phi} .$

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  automatically generated 8/ 1/2011