Revision as of 12:12, 15 April 2009 by imported>Paul Wormer
Given a 3-dimensional vector field F(r), the curl (also known as rotation) of F(r) is the differential vector operator nabla (symbol ∇) applied to F. The application of ∇ is in the form of a cross product:
![{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )\;{\stackrel {\mathrm {def} }{=}}\;\mathbf {e} _{x}\left({\frac {\partial F_{y}}{\partial z}}-{\frac {\partial F_{z}}{\partial y}}\right)+\mathbf {e} _{y}\left({\frac {\partial F_{z}}{\partial x}}-{\frac {\partial F_{x}}{\partial z}}\right)+\mathbf {e} _{z}\left({\frac {\partial F_{x}}{\partial y}}-{\frac {\partial F_{y}}{\partial x}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18f91847fb6a64e6ccd50771b2c3280e46c35269)
where ex, ey, and ez are unit vectors along the axes of a Cartesian coordinate system of axes.
As any cross product the curl may be written in a few alternative ways.
As a determinant (evaluate along the first row):
![{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )={\begin{vmatrix}\mathbf {e} _{x}&\mathbf {e} _{y}&\mathbf {e} _{z}\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fe5579b777351c2e061a025ef7a2a55e0ff72a5)
As a vector-matrix-vector product
![{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} )=\left(\mathbf {e} _{x},\;\mathbf {e} _{y},\;\mathbf {e} _{z}\right)\;{\begin{pmatrix}0&{\frac {\partial }{\partial z}}&-{\frac {\partial }{\partial y}}\\-{\frac {\partial }{\partial z}}&0&{\frac {\partial }{\partial x}}\\{\frac {\partial }{\partial y}}&-{\frac {\partial }{\partial x}}&0\\\end{pmatrix}}{\begin{pmatrix}F_{x}\\F_{y}\\F_{z}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cadafd58683c6a25a342c531363598be7b1c7eba)
In terms of the antisymmetric Levi-Civita symbol
![{\displaystyle {\Big (}{\boldsymbol {\nabla }}\times \mathbf {F} (\mathbf {r} ){\Big )}_{\alpha }=\sum _{\beta ,\gamma =x,y,z}\epsilon _{\alpha \beta \gamma }{\frac {\partial F_{\gamma }}{\partial \beta }},\qquad \alpha =x,y,z,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae1c5c1c7e59ed2f7cbe7da3d1f4dc6085847c4)
(the component of the curl along the Cartesian α-axis).