The Curl(F) calling sequence computes the curl of the vector field F in R^3.  This is equivalent to $\mathrm{Del}\&xF$ and 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(c) calling sequence returns the differential form of the curl operator in the coordinate system specified by c, which can be given as:

* an indexed name, e.g., ${\mathrm{spherical}}_{r,\mathrm{\phi },\mathrm{\theta }}$

* a name, e.g., spherical; default coordinate names will be used

* a list of names, e.g., $\left[r\,\mathrm{\phi }\,\mathrm{\theta }\right]$; the current coordinate system will be used, with these as the coordinate names

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

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

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

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

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

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

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

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

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

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

$\left(\frac{\frac{\partial }{\partial \mathrm{\theta }}{\mathrm{VF}}_3\left(r\,\mathrm{\theta }\,z\right)-\frac{\partial }{\partial z}\left(r{\mathrm{VF}}_2\left(r\,\mathrm{\theta }\,z\right)\right)}{r}\right){\stackrel{\_}{e}}_r\+\left(\frac{\partial }{\partial z}{\mathrm{VF}}_1\left(r\,\mathrm{\theta }\,z\right)-\frac{\partial }{\partial r}{\mathrm{VF}}_3\left(r\,\mathrm{\theta }\,z\right)\right){\stackrel{\_}{e}}_{\mathrm{\θ}}\+\left(\frac{\frac{\partial }{\partial r}\left(r{\mathrm{VF}}_2\left(r\,\mathrm{\theta }\,z\right)\right)-\frac{\partial }{\partial \mathrm{\theta }}{\mathrm{VF}}_1\left(r\,\mathrm{\theta }\,z\right)}{r}\right){\stackrel{\_}{e}}_z$

$\mathrm{Curl}\left(\left[s\,\mathrm{\phi }\,w\right]\right)$

$\left(\frac{\frac{\partial }{\partial \mathrm{\phi }}{\mathrm{VF}}_3\left(s\,\mathrm{\phi }\,w\right)-\frac{\partial }{\partial w}\left(s{\mathrm{VF}}_2\left(s\,\mathrm{\phi }\,w\right)\right)}{s}\right){\stackrel{\_}{e}}_s\+\left(\frac{\partial }{\partial w}{\mathrm{VF}}_1\left(s\,\mathrm{\phi }\,w\right)-\frac{\partial }{\partial s}{\mathrm{VF}}_3\left(s\,\mathrm{\phi }\,w\right)\right){\stackrel{\_}{e}}_{\mathrm{\φ}}\+\left(\frac{\frac{\partial }{\partial s}\left(s{\mathrm{VF}}_2\left(s\,\mathrm{\phi }\,w\right)\right)-\frac{\partial }{\partial \mathrm{\phi }}{\mathrm{VF}}_1\left(s\,\mathrm{\phi }\,w\right)}{s}\right){\stackrel{\_}{e}}_w$

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

$\left(\frac{\frac{\partial }{\partial \mathrm{\phi }}\left(r\sin \left(\mathrm{\phi }\right){\mathrm{VF}}_3\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)-\frac{\partial }{\partial \mathrm{\theta }}\left(r{\mathrm{VF}}_2\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)}{r^2\sin \left(\mathrm{\phi }\right)}\right){\stackrel{\_}{e}}_r\+\left(\frac{\frac{\partial }{\partial \mathrm{\theta }}{\mathrm{VF}}_1\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)-\frac{\partial }{\partial r}\left(r\sin \left(\mathrm{\phi }\right){\mathrm{VF}}_3\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)}{r\sin \left(\mathrm{\phi }\right)}\right){\stackrel{\_}{e}}_{\mathrm{\φ}}\+\left(\frac{\frac{\partial }{\partial r}\left(r{\mathrm{VF}}_2\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)\right)-\frac{\partial }{\partial \mathrm{\phi }}{\mathrm{VF}}_1\left(r\,\mathrm{\phi }\,\mathrm{\theta }\right)}{r}\right){\stackrel{\_}{e}}_{\mathrm{\θ}}$

$\mathrm{Curl}\left(\mathrm{spherical}\left[\mathrm{\alpha },\mathrm{\psi },\mathrm{\gamma }\right]\right)$

$\left(\frac{\frac{\partial }{\partial \mathrm{\psi }}\left(\mathrm{\alpha }\sin \left(\mathrm{\psi }\right){\mathrm{VF}}_3\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)-\frac{\partial }{\partial \mathrm{\gamma }}\left(\mathrm{\alpha }{\mathrm{VF}}_2\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)}{{\mathrm{\alpha }}^2\sin \left(\mathrm{\psi }\right)}\right){\stackrel{\_}{e}}_{\mathrm{\α}}\+\left(\frac{\frac{\partial }{\partial \mathrm{\gamma }}{\mathrm{VF}}_1\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)-\frac{\partial }{\partial \mathrm{\alpha }}\left(\mathrm{\alpha }\sin \left(\mathrm{\psi }\right){\mathrm{VF}}_3\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)}{\mathrm{\alpha }\sin \left(\mathrm{\psi }\right)}\right){\stackrel{\_}{e}}_{\mathrm{\ψ}}\+\left(\frac{\frac{\partial }{\partial \mathrm{\alpha }}\left(\mathrm{\alpha }{\mathrm{VF}}_2\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)\right)-\frac{\partial }{\partial \mathrm{\psi }}{\mathrm{VF}}_1\left(\mathrm{\alpha }\,\mathrm{\psi }\,\mathrm{\gamma }\right)}{\mathrm{\alpha }}\right){\stackrel{\_}{e}}_{\mathrm{\γ}}$

$\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)$

$\mathrm{cartesian}$

$\mathrm{Del}\&xF$

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

$\nabla \&xF$

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

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

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

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

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

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

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

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

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