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...

 > $R\left[x\right]≔\mathrm{VectorField}\left(-z\mathrm{D}\left[y\right]+y\mathrm{D}\left[z\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{x}}{≔}{-}{z}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $R\left[y\right]≔\mathrm{VectorField}\left(-x\mathrm{D}\left[z\right]+z\mathrm{D}\left[x\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{y}}{≔}{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{-}{x}{}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (2)
 > $R\left[z\right]≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y,z\right]\right)$
 ${{R}}_{{z}}{≔}{-}{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{x}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (3)

Construct the associated distribution...

 > $\mathrm{\Sigma }≔\mathrm{Distribution}\left(R\left[x\right],R\left[y\right],R\left[z\right]\right)$
 ${\mathrm{\Sigma }}{≔}\left\{{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right\}$ (4)
 > $\mathrm{GetVectorFields}\left(\mathrm{\Sigma }\right)$
 $\left[{-}\frac{{y}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{,}{-}\frac{{z}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}}{{x}}{+}\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]$ (5)
 > $\mathrm{GetAnnihilator}\left(\mathrm{\Sigma }\right)$
 $\left[{\mathrm{dx}}{+}\frac{{y}{}{\mathrm{dy}}}{{x}}{+}\frac{{z}{}{\mathrm{dz}}}{{x}}\right]$ (6)

Compatibility

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