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.

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

$R\left[x\right]≔\mathrm{VectorField}\left(-z\mathrm{D}\left[y\right]+y\mathrm{D}\left[z\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$R_x≔-z\left(\frac{\ⅆ}{\ⅆy}\right)+y\left(\frac{\ⅆ}{\ⅆz}\right)$

$R\left[y\right]≔\mathrm{VectorField}\left(-x\mathrm{D}\left[z\right]+z\mathrm{D}\left[x\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$R_y≔z\left(\frac{\ⅆ}{\ⅆx}\right)-x\left(\frac{\ⅆ}{\ⅆz}\right)$

$R\left[z\right]≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\,z\right]\right)$

$R_z≔-y\left(\frac{\ⅆ}{\ⅆx}\right)+x\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{\Sigma }≔\mathrm{Distribution}\left(R\left[x\right]\,R\left[y\right]\,R\left[z\right]\right)$

$\mathrm{\Sigma }≔\left\{-\frac{y\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆy}\,-\frac{z\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆz}\right\}$

$\mathrm{GetVectorFields}\left(\mathrm{\Sigma }\right)$

$\left[-\frac{y\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆy}\,-\frac{z\left(\frac{\ⅆ}{\ⅆx}\right)}{x}+\frac{\ⅆ}{\ⅆz}\right]$

$\mathrm{GetAnnihilator}\left(\mathrm{\Sigma }\right)$

$\left[\mathrm{dx}+\frac{y\mathrm{dy}}{x}+\frac{z\mathrm{dz}}{x}\right]$

The GetVectorFields and GetAnnihilator commands were introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.