LieAlgebrasOfVectorFields - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for PDEs (and ODEs) : LieAlgebrasOfVectorFields : Distribution : LieAlgebrasOfVectorFields/Distribution/constructor

LieAlgebrasOfVectorFields

  

Distribution

  

Construct a Distribution object

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Distribution(vf1,..., 'space'=vars)

Distribution(str, 'space'=vars)

Distribution(L)

Parameters

vf1, ...,

-

VectorField object, or sequence of VectorField objects

vars

-

(optional) list of names of coordinates of the space

str

-

a string: either "trivial" or "universal"

L

-

an LAVF object.

Description

• 

The Distribution command constructs and returns a Distribution object (see Overview of the Distribution object).

• 

In the first calling sequence, a sequence of VectorField objects must be provided which span the distribution at each point. The input vector fields are further processed by the constructor, so that the Distribution may be stored internally and displayed with a different basis.  The space= option is not necessary, since the space can be inferred from the vector fields.  For completeness, an empty sequence of vector fields is permitted: in this case, the space= option is required, and a trivial (0-dimensional) distribution is constructed on the specified space.

• 

For convenience the second calling sequence construct the trivial (0-dimensional) distribution and 'universal' distribution (which spans the whole of tangent space at each point). In these sequences, the space= option must be specified.

• 

In the final calling sequence, L is a Lie algebra of vector fields LAVF object whose determining system is of order 0. An exception is raised if the determining system of L is not 0th order.

• 

These commands are part of the LieAlgebrasOfVectorFields package, for more detail see Overview of the LieAlgebrasOfVectorFields package.

Examples

withLieAlgebrasOfVectorFields:

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

RxVectorFieldzDy+yDz,space=x,y,z

Rxzy+yz

(1)

RyVectorFieldxDz+zDx,space=x,y,z

Ryzxxz

(2)

RzVectorFieldyDx+xDy,space=x,y,z

Rzyx+xy

(3)

Construct the associated distribution...

ΣDistributionRx,Ry,Rz

Σyxx+y,zxx+z

(4)

A trivial Distribution has empty basis...

Distributiontrivial,space=x,y,z

(5)

The universal Distribution is spanned by the basis vectors of tangent space at each point.

Distributionuniversal,space=x,y,z

z,x,y

(6)

Compatibility

• 

The LieAlgebrasOfVectorFields[Distribution] command was introduced in Maple 2020.

• 

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

See Also

Distribution (Object overview)

VectorField (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LAVF (Object overview)

LieAlgebrasOfVectorFields (Package overview)