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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 $∇×F=v$.  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

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