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Mathematics-Online lexicon:

Curl


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

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}
.$

()

Example:


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  automatically generated 8/ 1/2011