GetVectorFields - Maple Help

GetVectorFields

get vector fields that form a basis for a Distribution object

GetAnnihilator

get one-forms that form a co-basis for a Distribution object

 Calling Sequence GetVectorFields( dist) GetAnnihilator( dist)

Parameters

 dist - a Distribution object

Description

 • The GetVectorFields method returns a list of VectorField objects which at each point form a basis of the subspace of tangent space associated with a distribution.  In the case where Distribution dist is in involution, the basis is chosen to commute.
 • Similarly the GetAnnihilator method returns a list of OneForm objects which form a co-basis for the forms that annihilate the distribution.
 • Both VectorField and OneForm objects are part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.
 • These methods are associated with the Distribution object. For more detail see Overview of the Distribution object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$

Build vector fields associated with 3-d spatial rotations...

 > ${R}_{x}≔\mathrm{VectorField}\left(-z{\mathrm{D}}_{y}+y{\mathrm{D}}_{z},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right){+}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)$ (1)
 > ${R}_{y}≔\mathrm{VectorField}\left(-x{\mathrm{D}}_{z}+z{\mathrm{D}}_{x},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{y}}{≔}{z}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{z}}\right)$ (2)
 > ${R}_{z}≔\mathrm{VectorField}\left(-y{\mathrm{D}}_{x}+x{\mathrm{D}}_{y},\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{z}}{≔}{-}{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (3)

Construct the associated distribution...

 > $\mathrm{Σ}≔\mathrm{Distribution}\left({R}_{x},{R}_{y},{R}_{z}\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{y}}{,}{-}\frac{{z}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{z}}\right\}$ (4)
 > $\mathrm{GetVectorFields}\left(\mathrm{Σ}\right)$
 $\left[{-}\frac{{y}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{y}}{,}{-}\frac{{z}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right)}{{x}}{+}\frac{{ⅆ}}{{ⅆ}{z}}\right]$ (5)
 > $\mathrm{GetAnnihilator}\left(\mathrm{Σ}\right)$
 $\left[{\mathrm{dx}}{+}\frac{{y}{}{\mathrm{dy}}}{{x}}{+}\frac{{z}{}{\mathrm{dz}}}{{x}}\right]$ (6)

Compatibility

 • The GetVectorFields and GetAnnihilator commands were introduced in Maple 2020.