The Curl(F) command computes the curl of the vector field F in R^3.  This is a synonym for $\mathrm{Del}\&xF$ or CrossProduct(Del, F).

If F is a Vector valued procedure, the default coordinate system is used, and it must be indexed by the coordinate names.  Otherwise, F must be a vector field.

If F is a procedure, the result is a procedure.  Otherwise, the result is a vector field.

The Curl() command returns the differential form of the curl operator in the current coordinate system.  For more information, see SetCoordinates.

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y,z\right]\right)$

${\mathrm{cartesian}}_{x,y,z}$

$F≔\mathrm{VectorField}\left(⟨y\,-x\,0⟩\right)$

$F≔\left(y\right){\stackrel{\_}{e}}_x\+\left(-x\right){\stackrel{\_}{e}}_y\+\left(0\right){\stackrel{\_}{e}}_z$

$\mathrm{Curl}\left(F\right)$

$\left(0\right){\stackrel{\_}{e}}_x\+\left(0\right){\stackrel{\_}{e}}_y\+\left(\mathrm{-2}\right){\stackrel{\_}{e}}_z$

$\mathrm{Curl}\left(\right)$

$\left(\frac{\partial }{\partial y}{\mathrm{VF}}_3\left(x\,y\,z\right)-\frac{\partial }{\partial z}{\mathrm{VF}}_2\left(x\,y\,z\right)\right){\stackrel{\_}{e}}_x\+\left(\frac{\partial }{\partial z}{\mathrm{VF}}_1\left(x\,y\,z\right)-\frac{\partial }{\partial x}{\mathrm{VF}}_3\left(x\,y\,z\right)\right){\stackrel{\_}{e}}_y\+\left(\frac{\partial }{\partial x}{\mathrm{VF}}_2\left(x\,y\,z\right)-\frac{\partial }{\partial y}{\mathrm{VF}}_1\left(x\,y\,z\right)\right){\stackrel{\_}{e}}_z$

$\mathrm{Del}\&xF$

$\left(0\right){\stackrel{\_}{e}}_x\+\left(0\right){\stackrel{\_}{e}}_y\+\left(\mathrm{-2}\right){\stackrel{\_}{e}}_z$

$\nabla \&xF$

$\left(0\right){\stackrel{\_}{e}}_x\+\left(0\right){\stackrel{\_}{e}}_y\+\left(\mathrm{-2}\right){\stackrel{\_}{e}}_z$

$\mathrm{CrossProduct}\left(\mathrm{Del}\,F\right)$

$\left(0\right){\stackrel{\_}{e}}_x\+\left(0\right){\stackrel{\_}{e}}_y\+\left(\mathrm{-2}\right){\stackrel{\_}{e}}_z$

$\mathrm{Curl}\left(\left(x\,y\,z\right)\mapsto ⟨x^2\,y^2\,z^2⟩\right)$

$\left(x\,y\,z\right)\mapsto \mathrm{Vector}\left(\left[0\,0\,0\right]\,\mathrm{attributes}=\left[\mathrm{vectorfield}\,\mathrm{coords}={\mathrm{cartesian}}_{x,y,z}\right]\right)$

$\mathrm{SetCoordinates}\left('\mathrm{cylindrical}'\left[r,\mathrm{\theta },z\right]\right)$

${\mathrm{cylindrical}}_{r,\mathrm{\theta },z}$

$\mathrm{Curl}\left(\left(r\,\mathrm{\theta }\,z\right)\mapsto ⟨f\left(r\,\mathrm{\theta }\,z\right)\,g\left(r\,\mathrm{\theta }\,z\right)\,h\left(r\,\mathrm{\theta }\,z\right)⟩\right)$

$\left(r\,\mathrm{\θ}\,z\right)\→\mathrm{VectorCalculus}:-\mathrm{Vector}\left(\left[\frac{\frac{\partial }{\partial \mathrm{\θ}}h\left(r\,\mathrm{\θ}\,z\right)-r\left(\frac{\partial }{\partial z}g\left(r\,\mathrm{\θ}\,z\right)\right)}{r}\,\frac{\partial }{\partial z}f\left(r\,\mathrm{\θ}\,z\right)-\left(\frac{\partial }{\partial r}h\left(r\,\mathrm{\θ}\,z\right)\right)\,\frac{g\left(r\,\mathrm{\θ}\,z\right)\+r\left(\frac{\partial }{\partial r}g\left(r\,\mathrm{\θ}\,z\right)\right)-\left(\frac{\partial }{\partial \mathrm{\θ}}f\left(r\,\mathrm{\θ}\,z\right)\right)}{r}\right]\,\mathrm{attributes}\=\left[\mathrm{vectorfield}\,\mathrm{coords}\={\mathrm{cylindrical}}_{r\,\mathrm{\θ}\,z}\right]\right)$