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Student[VectorCalculus]

 ScalarPotential
 compute the scalar potential of a vector field

 Calling Sequence ScalarPotential(v)

Parameters

 v - vector field or Vector-valued procedure; specify the components of the vector field

Description

 • The ScalarPotential(v) calling sequence computes the scalar potential of the vector field $v$.  This is a function $f$ such that $∇f=v$.  If a scalar potential does not exist, NULL is returned.
 • If $v$ is a vector field, an algebraic expression is returned. If $v$ is a Vector-valued procedure, a procedure is returned.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{VectorCalculus}]\right):$
 > $v≔\mathrm{VectorField}\left(⟨x,y,z⟩\right)$
 ${v}{≔}\left({x}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({y}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({z}\right){\stackrel{{_}}{{e}}}_{{z}}$ (1)
 > $\mathrm{ScalarPotential}\left(v\right)$
 $\frac{{1}}{{2}}{}{{x}}^{{2}}{+}\frac{{1}}{{2}}{}{{y}}^{{2}}{+}\frac{{1}}{{2}}{}{{z}}^{{2}}$ (2)
 > $\mathrm{Gradient}\left(\right)$
 $\left({x}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({y}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({z}\right){\stackrel{{_}}{{e}}}_{{z}}$ (3)
 > $v≔\mathrm{VectorField}\left(⟨y,-x,0⟩\right)$
 ${v}{≔}\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{-}{x}{\stackrel{{_}}{{e}}}_{{y}}$ (4)
 > $\mathrm{ScalarPotential}\left(v\right)$
 > $\mathrm{den}≔{x}^{2}+{y}^{2}+{z}^{2}$
 ${\mathrm{den}}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}$ (5)
 > $\mathrm{ScalarPotential}\left(\left(x,y,z\right)→\frac{⟨x,y,z⟩}{\mathrm{den}}\right)$
 $\left({x}{,}{y}{,}{z}\right){→}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}\right)$ (6)
 > $\mathrm{SetCoordinates}\left({\mathrm{spherical}}_{r,\mathrm{φ},\mathrm{θ}}\right)$
 ${{\mathrm{spherical}}}_{{r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}}$ (7)
 > $v≔\mathrm{VectorField}\left(⟨r,0,0⟩\right)$
 ${v}{≔}\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}$ (8)
 > $\mathrm{ScalarPotential}\left(v\right)$
 $\frac{{1}}{{2}}{}{{r}}^{{2}}$ (9)