The VectorPotential(v) calling sequence computes the vector potential of the vector field v.  The result is a vector field F such that $\mathrm{Curl}\left(F\right)=v$ (that is, the curl of $F$ is the field $v$).  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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$v≔\mathrm{VectorField}\left(⟨y\,-x\,0⟩\right)$

$v≔\left(y\right){\stackrel{\_}{e}}_x\+\left(-x\right){\stackrel{\_}{e}}_y\+\left(0\right){\stackrel{\_}{e}}_z$

$\mathrm{VectorPotential}\left(v\right)$

$\left(-xz\right){\stackrel{\_}{e}}_x\+\left(-yz\right){\stackrel{\_}{e}}_y\+\left(0\right){\stackrel{\_}{e}}_z$

$\left(y\right){\stackrel{\_}{e}}_x\+\left(-x\right){\stackrel{\_}{e}}_y\+\left(0\right){\stackrel{\_}{e}}_z$

$\mathrm{SetCoordinates}\left(\mathrm{cylindrical}\left[r,\mathrm{\theta },z\right]\right)$

${\mathrm{cylindrical}}_{r,\mathrm{\theta },z}$

$v≔\mathrm{VectorField}\left(⟨r\,0\,-2z⟩\right)$

$v≔\left(r\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\θ}}\+\left(-2z\right){\stackrel{\_}{e}}_z$

$\mathrm{VectorPotential}\left(v\right)$

$\left(0\right){\stackrel{\_}{e}}_r\+\left(-r{\cos \left(\mathrm{\theta }\right)}^{2}z-r{\sin \left(\mathrm{\theta }\right)}^{2}z\right){\stackrel{\_}{e}}_{\mathrm{\θ}}\+\left(0\right){\stackrel{\_}{e}}_z$

$\left(r\right){\stackrel{\_}{e}}_r\+\left(0\right){\stackrel{\_}{e}}_{\mathrm{\θ}}\+\left(-2z\right){\stackrel{\_}{e}}_z$