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VectorCalculus

 ScalarPotential

 Calling Sequence ScalarPotential(v)

Parameters

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

Description

 • The ScalarPotential(v) command computes the scalar potential of the vector field v.  This is a function f such that $\mathrm{Gradient}\left(f\right)=v$.  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

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