VectorPotential - Maple Help

VectorCalculus

 VectorPotential
 compute a vector potential of a vector field in R^3

 Calling Sequence VectorPotential(v)

Parameters

 v - vector field or Vector valued procedure; specify the components of the vector field in R^3

Description

 • The VectorPotential(v) command computes a vector potential of the vector field v.  The result is a vector field F such that $\mathrm{Curl}\left(F\right)=v$.  If a vector potential does not exist, NULL is returned.
 • If v is a vector field, a vector field is returned.  If v is a procedure, a procedure is returned.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y,z\right]\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (1)
 > $v≔\mathrm{VectorField}\left(⟨y,-x,0⟩\right)$
 ${v}{≔}\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{x}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (2)
 > $\mathrm{VectorPotential}\left(v\right)$
 $\left({-}{x}{}{z}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{y}{}{z}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (3)
 > 
 $\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{x}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (4)
 > $\mathrm{SetCoordinates}\left('\mathrm{cylindrical}'\left[r,\mathrm{\theta },z\right]\right)$
 ${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{\theta }}{,}{z}}$ (5)
 > $v≔\mathrm{VectorField}\left(⟨r,0,-2z⟩\right)$
 ${v}{≔}\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left({-}{2}{}{z}\right){\stackrel{{_}}{{e}}}_{{z}}$ (6)
 > $\mathrm{VectorPotential}\left(v\right)$
 $\left({0}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({-}{r}{}{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{z}{-}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{z}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (7)
 > 
 $\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left({-}{2}{}{z}\right){\stackrel{{_}}{{e}}}_{{z}}$ (8)