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Student[VectorCalculus]

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

 Calling Sequence Curl(F) Curl(c)

Parameters

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

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{VectorCalculus}]\right):$
 > $F≔\mathrm{VectorField}\left(⟨y,-x,0⟩\right)$
 ${F}{≔}\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{-}{x}{\stackrel{{_}}{{e}}}_{{y}}$ (1)
 > $\mathrm{Curl}\left(F\right)$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (2)

To display the differential form of the curl operator:

 > $\mathrm{Curl}\left(\right)$
 $\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{VF}}{[}{3}{]}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{VF}}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{VF}}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{VF}}{[}{3}{]}{}\left({x}{,}{y}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{VF}}{[}{2}{]}{}\left({x}{,}{y}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{y}}{}{\mathrm{VF}}{[}{1}{]}{}\left({x}{,}{y}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{z}}$ (3)
 > $\mathrm{SetCoordinates}\left({\mathrm{cylindrical}}_{r,\mathrm{θ},z}\right):$
 > $\mathrm{Curl}\left(\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{\mathrm{VF}}{[}{3}{]}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{z}}{}\left({r}{}{\mathrm{VF}}{[}{2}{]}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)\right)}{{r}}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left(\frac{{\partial }}{{\partial }{z}}{}{\mathrm{VF}}{[}{1}{]}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{r}}{}{\mathrm{VF}}{[}{3}{]}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{r}}{}\left({r}{}{\mathrm{VF}}{[}{2}{]}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{\mathrm{VF}}{[}{1}{]}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}\right){\stackrel{{_}}{{e}}}_{{z}}$ (4)
 > $\mathrm{Curl}\left(\left[s,\mathrm{φ},w\right]\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{φ}}}{}{\mathrm{VF}}{[}{3}{]}{}\left({s}{,}{\mathrm{φ}}{,}{w}\right){-}\left(\frac{{\partial }}{{\partial }{w}}{}\left({s}{}{\mathrm{VF}}{[}{2}{]}{}\left({s}{,}{\mathrm{φ}}{,}{w}\right)\right)\right)}{{s}}\right){\stackrel{{_}}{{e}}}_{{s}}{+}\left(\frac{{\partial }}{{\partial }{w}}{}{\mathrm{VF}}{[}{1}{]}{}\left({s}{,}{\mathrm{φ}}{,}{w}\right){-}\left(\frac{{\partial }}{{\partial }{s}}{}{\mathrm{VF}}{[}{3}{]}{}\left({s}{,}{\mathrm{φ}}{,}{w}\right)\right)\right){\stackrel{{_}}{{e}}}_{{\mathrm{φ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{s}}{}\left({s}{}{\mathrm{VF}}{[}{2}{]}{}\left({s}{,}{\mathrm{φ}}{,}{w}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{φ}}}{}{\mathrm{VF}}{[}{1}{]}{}\left({s}{,}{\mathrm{φ}}{,}{w}\right)\right)}{{s}}\right){\stackrel{{_}}{{e}}}_{{w}}$ (5)
 > $\mathrm{Curl}\left(\mathrm{spherical}\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{φ}}}{}\left({r}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{VF}}{[}{3}{]}{}\left({r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}\left({r}{}{\mathrm{VF}}{[}{2}{]}{}\left({r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}\right)\right)\right)}{{{r}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{\mathrm{VF}}{[}{1}{]}{}\left({r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}\right){-}\left(\frac{{\partial }}{{\partial }{r}}{}\left({r}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right){}{\mathrm{VF}}{[}{3}{]}{}\left({r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}\right)\right)\right)}{{r}{}{\mathrm{sin}}{}\left({\mathrm{φ}}\right)}\right){\stackrel{{_}}{{e}}}_{{\mathrm{φ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{r}}{}\left({r}{}{\mathrm{VF}}{[}{2}{]}{}\left({r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{φ}}}{}{\mathrm{VF}}{[}{1}{]}{}\left({r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}\right)\right)}{{r}}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}$ (6)
 > $\mathrm{Curl}\left({\mathrm{spherical}}_{\mathrm{α},\mathrm{ψ},\mathrm{γ}}\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{ψ}}}{}\left({\mathrm{α}}{}{\mathrm{sin}}{}\left({\mathrm{ψ}}\right){}{\mathrm{VF}}{[}{3}{]}{}\left({\mathrm{α}}{,}{\mathrm{ψ}}{,}{\mathrm{γ}}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{γ}}}{}\left({\mathrm{α}}{}{\mathrm{VF}}{[}{2}{]}{}\left({\mathrm{α}}{,}{\mathrm{ψ}}{,}{\mathrm{γ}}\right)\right)\right)}{{{\mathrm{α}}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{ψ}}\right)}\right){\stackrel{{_}}{{e}}}_{{\mathrm{α}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{γ}}}{}{\mathrm{VF}}{[}{1}{]}{}\left({\mathrm{α}}{,}{\mathrm{ψ}}{,}{\mathrm{γ}}\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{α}}}{}\left({\mathrm{α}}{}{\mathrm{sin}}{}\left({\mathrm{ψ}}\right){}{\mathrm{VF}}{[}{3}{]}{}\left({\mathrm{α}}{,}{\mathrm{ψ}}{,}{\mathrm{γ}}\right)\right)\right)}{{\mathrm{α}}{}{\mathrm{sin}}{}\left({\mathrm{ψ}}\right)}\right){\stackrel{{_}}{{e}}}_{{\mathrm{ψ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{α}}}{}\left({\mathrm{α}}{}{\mathrm{VF}}{[}{2}{]}{}\left({\mathrm{α}}{,}{\mathrm{ψ}}{,}{\mathrm{γ}}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{ψ}}}{}{\mathrm{VF}}{[}{1}{]}{}\left({\mathrm{α}}{,}{\mathrm{ψ}}{,}{\mathrm{γ}}\right)\right)}{{\mathrm{α}}}\right){\stackrel{{_}}{{e}}}_{{\mathrm{γ}}}$ (7)

Nabla is a synonym for Del.

 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)$
 ${\mathrm{cartesian}}$ (8)
 > $\mathrm{Del}&xF$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (9)
 > $\mathrm{Nabla}&xF$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (10)
 > $\mathrm{CrossProduct}\left(\mathrm{Del},F\right)$
 ${-}{2}{\stackrel{{_}}{{e}}}_{{z}}$ (11)
 > $\mathrm{Curl}\left(\left(x,y,z\right)→⟨{x}^{2},{y}^{2},{z}^{2}⟩\right)$
 $\left({x}{,}{y}{,}{z}\right){→}{\mathrm{VectorCalculus:-Vector}}{}\left(\left[{0}{,}{0}{,}{0}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{vectorfield}}{,}{\mathrm{coords}}{=}{{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}\right]\right)$ (12)
 > $\mathrm{SetCoordinates}\left({\mathrm{cylindrical}}_{r,\mathrm{θ},z}\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}$ (13)
 > $\mathrm{Curl}\left(\left(r,\mathrm{θ},z\right)→⟨f\left(r,\mathrm{θ},z\right),g\left(r,\mathrm{θ},z\right),h\left(r,\mathrm{θ},z\right)⟩\right)$
 $\left({r}{,}{\mathrm{θ}}{,}{z}\right){→}{\mathrm{VectorCalculus:-Vector}}{}\left(\left[\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{h}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}{r}{}\left(\frac{{\partial }}{{\partial }{z}}{}{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}{,}\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{r}}{}{h}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){,}\frac{{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){+}{r}{}\left(\frac{{\partial }}{{\partial }{r}}{}{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{vectorfield}}{,}{\mathrm{coords}}{=}{{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}\right]\right)$ (14)