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VectorCalculus

 Curl
 compute the curl of a vector field in R^3

 Calling Sequence Curl(F)

Parameters

 F - (optional) vector field or a Vector valued procedure; specify the components of the vector field

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{SetCoordinates}\left({'\mathrm{cartesian}'}_{x,y,z}\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (1)
 > $F≔\mathrm{VectorField}\left(⟨y,-x,0⟩\right)$
 ${F}{≔}\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{-}{x}{\stackrel{{_}}{{e}}}_{{y}}$ (2)
 > $\mathrm{Curl}\left(F\right)$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (3)
 > $\mathrm{Curl}\left(\right)$
 $\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{VF}}{[}{3}{]}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{VF}}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{VF}}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{VF}}{[}{3}{]}{}\left({x}{,}{y}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{VF}}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{VF}}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{z}}$ (4)
 > $\mathrm{Del}&xF$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (5)
 > $\mathrm{Nabla}&xF$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (6)
 > $\mathrm{CrossProduct}\left(\mathrm{Del},F\right)$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (7)
 > $\mathrm{Curl}\left(\left(x,y,z\right)→⟨{x}^{2},{y}^{2},{z}^{2}⟩\right)$
 $\left({x}{,}{y}{,}{z}\right){→}{\mathrm{VectorCalculus:-Vector}}{}\left(\left[{0}{,}{0}{,}{0}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{vectorfield}}{,}{\mathrm{coords}}{=}{{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}\right]\right)$ (8)
 > $\mathrm{SetCoordinates}\left({'\mathrm{cylindrical}'}_{r,\mathrm{θ},z}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}$ (9)
 > $\mathrm{Curl}\left(\left(r,\mathrm{θ},z\right)→⟨f\left(r,\mathrm{θ},z\right),g\left(r,\mathrm{θ},z\right),h\left(r,\mathrm{θ},z\right)⟩\right)$
 $\left({r}{,}{\mathrm{θ}}{,}{z}\right){→}{\mathrm{VectorCalculus:-Vector}}{}\left(\left[\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{h}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}{r}{}\left(\frac{{\partial }}{{\partial }{z}}{}{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}{,}\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{r}}{}{h}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){,}\frac{{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){+}{r}{}\left(\frac{{\partial }}{{\partial }{r}}{}{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{vectorfield}}{,}{\mathrm{coords}}{=}{{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}\right]\right)$ (10)