Curl - Maple Help

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 $\mathrm{Del}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$ 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

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

To display the differential form of the curl operator:

 > $\mathrm{Curl}\left(\right)$
 $\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{3}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{2}}{}\left({x}{,}{y}{,}{z}\right)\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{3}}{}\left({x}{,}{y}{,}{z}\right)\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{2}}{}\left({x}{,}{y}{,}{z}\right){-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({x}{,}{y}{,}{z}\right)\right){\stackrel{{_}}{{e}}}_{{z}}$ (3)
 > $\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,\mathrm{\theta },z\right]\right):$
 > $\mathrm{Curl}\left(\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{3}}{}\left({r}{,}{\mathrm{\theta }}{,}{z}\right){-}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({r}{}{{\mathrm{VF}}}_{{2}}{}\left({r}{,}{\mathrm{\theta }}{,}{z}\right)\right)}{{r}}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({r}{,}{\mathrm{\theta }}{,}{z}\right){-}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{3}}{}\left({r}{,}{\mathrm{\theta }}{,}{z}\right)\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({r}{}{{\mathrm{VF}}}_{{2}}{}\left({r}{,}{\mathrm{\theta }}{,}{z}\right)\right){-}\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({r}{,}{\mathrm{\theta }}{,}{z}\right)}{{r}}\right){\stackrel{{_}}{{e}}}_{{z}}$ (4)
 > $\mathrm{Curl}\left(\left[s,\mathrm{\phi },w\right]\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{3}}{}\left({s}{,}{\mathrm{\phi }}{,}{w}\right){-}\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({s}{}{{\mathrm{VF}}}_{{2}}{}\left({s}{,}{\mathrm{\phi }}{,}{w}\right)\right)}{{s}}\right){\stackrel{{_}}{{e}}}_{{s}}{+}\left(\frac{{\partial }}{{\partial }{w}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({s}{,}{\mathrm{\phi }}{,}{w}\right){-}\frac{{\partial }}{{\partial }{s}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{3}}{}\left({s}{,}{\mathrm{\phi }}{,}{w}\right)\right){\stackrel{{_}}{{e}}}_{{\mathrm{φ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{s}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({s}{}{{\mathrm{VF}}}_{{2}}{}\left({s}{,}{\mathrm{\phi }}{,}{w}\right)\right){-}\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({s}{,}{\mathrm{\phi }}{,}{w}\right)}{{s}}\right){\stackrel{{_}}{{e}}}_{{w}}$ (5)
 > $\mathrm{Curl}\left(\mathrm{spherical}\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({r}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{VF}}}_{{3}}{}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right)\right){-}\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({r}{}{{\mathrm{VF}}}_{{2}}{}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right)\right)}{{{r}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\theta }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right){-}\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({r}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}{{\mathrm{VF}}}_{{3}}{}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right)\right)}{{r}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right)}\right){\stackrel{{_}}{{e}}}_{{\mathrm{φ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{r}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({r}{}{{\mathrm{VF}}}_{{2}}{}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right)\right){-}\frac{{\partial }}{{\partial }{\mathrm{\phi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({r}{,}{\mathrm{\phi }}{,}{\mathrm{\theta }}\right)}{{r}}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}$ (6)
 > $\mathrm{Curl}\left(\mathrm{spherical}\left[\mathrm{\alpha },\mathrm{\psi },\mathrm{\gamma }\right]\right)$
 $\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\psi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{\alpha }}{}{\mathrm{sin}}{}\left({\mathrm{\psi }}\right){}{{\mathrm{VF}}}_{{3}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\psi }}{,}{\mathrm{\gamma }}\right)\right){-}\frac{{\partial }}{{\partial }{\mathrm{\gamma }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{\alpha }}{}{{\mathrm{VF}}}_{{2}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\psi }}{,}{\mathrm{\gamma }}\right)\right)}{{{\mathrm{\alpha }}}^{{2}}{}{\mathrm{sin}}{}\left({\mathrm{\psi }}\right)}\right){\stackrel{{_}}{{e}}}_{{\mathrm{α}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\gamma }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\psi }}{,}{\mathrm{\gamma }}\right){-}\frac{{\partial }}{{\partial }{\mathrm{\alpha }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{\alpha }}{}{\mathrm{sin}}{}\left({\mathrm{\psi }}\right){}{{\mathrm{VF}}}_{{3}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\psi }}{,}{\mathrm{\gamma }}\right)\right)}{{\mathrm{\alpha }}{}{\mathrm{sin}}{}\left({\mathrm{\psi }}\right)}\right){\stackrel{{_}}{{e}}}_{{\mathrm{ψ}}}{+}\left(\frac{\frac{{\partial }}{{\partial }{\mathrm{\alpha }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{\alpha }}{}{{\mathrm{VF}}}_{{2}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\psi }}{,}{\mathrm{\gamma }}\right)\right){-}\frac{{\partial }}{{\partial }{\mathrm{\psi }}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{VF}}}_{{1}}{}\left({\mathrm{\alpha }}{,}{\mathrm{\psi }}{,}{\mathrm{\gamma }}\right)}{{\mathrm{\alpha }}}\right){\stackrel{{_}}{{e}}}_{{\mathrm{γ}}}$ (7)

Nabla is a synonym for Del.

 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\right)$
 ${\mathrm{cartesian}}$ (8)
 > $\mathrm{Del}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$
 $\left({0}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({-2}\right){\stackrel{{_}}{{e}}}_{{z}}$ (9)
 > $\nabla \phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$
 $\left({0}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({-2}\right){\stackrel{{_}}{{e}}}_{{z}}$ (10)
 > $\mathrm{CrossProduct}\left(\mathrm{Del},F\right)$
 $\left({0}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({-2}\right){\stackrel{{_}}{{e}}}_{{z}}$ (11)
 > $\mathrm{Curl}\left(\left(x,y,z\right)↦⟨{x}^{2},{y}^{2},{z}^{2}⟩\right)$
 $\left({x}{,}{y}{,}{z}\right){↦}{\mathrm{VectorCalculus}}{:-}{\mathrm{Vector}}{}\left(\left[{0}{,}{0}{,}{0}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{vectorfield}}{,}{\mathrm{coords}}{=}{{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}\right]\right)$ (12)
 > $\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,\mathrm{\theta },z\right]\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{\theta }}{,}{z}}$ (13)
 > $\mathrm{Curl}\left(\left(r,\mathrm{\theta },z\right)↦⟨f\left(r,\mathrm{\theta },z\right),g\left(r,\mathrm{\theta },z\right),h\left(r,\mathrm{\theta },z\right)⟩\right)$
 $\left({r}{,}{\mathrm{θ}}{,}{z}\right){→}{\mathrm{VectorCalculus}}{:-}{\mathrm{Vector}}{}\left(\left[\frac{\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{h}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}{r}{}\left(\frac{{\partial }}{{\partial }{z}}{}{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}{,}\frac{{\partial }}{{\partial }{z}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){-}\left(\frac{{\partial }}{{\partial }{r}}{}{h}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){,}\frac{{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right){+}{r}{}\left(\frac{{\partial }}{{\partial }{r}}{}{g}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right){-}\left(\frac{{\partial }}{{\partial }{\mathrm{θ}}}{}{f}{}\left({r}{,}{\mathrm{θ}}{,}{z}\right)\right)}{{r}}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{vectorfield}}{,}{\mathrm{coords}}{=}{{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{θ}}{,}{z}}\right]\right)$ (14)