Overview of the Distribution Object - Maple Programming Help

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Overview of the Distribution Object

 

Description

List of Distribution object Methods

Examples

Description

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The Distribution object provides a general toolkit for dealing with distributions. A user can query properties of a Distribution object, compute associated quantities, and combine distributions in various ways.  Some existing Maple utility functions such as indets, type, has,...etc are overloaded for use with the Distribution object.

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A distribution in the differential-geometric sense is a specification of a subspace of tangent space at each point of a manifold M.

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A Distribution object can be constructed via the Distribution constructor. To construct a Distribution object, see LieAlgebrasOfVectorFields[Distribution].

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The Distribution object is an exported item in the LieAlgebrasOfVectorFields package. To construct and access a Distribution object, the LieAlgebrasOfVectorFields package must be loaded (i.e. with(LieAlgebrasOfVectorFields);). For more information, see Overview of the LieAlgebrasOfVectorFields package.

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Once a Distribution object S has been constructed, each method in the Distribution object S can be accessed by either the short form command(S, otherArguments) or the long form S:-command(S, otherArguments).

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A Distribution object is displayed (via its ModulePrint method) as a set of VectorField objects which form a basis for it at each point.

List of Distribution object Methods

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The following is a list of available commands in a Distribution object.

AreSameSpace

CauchyDistribution

Codimension

DChange

DerivedDistribution

Dimension

GetAnnihilator

GetSpace

GetVectorFields

Integrals

Intersection

IsIntegrable

IsInvariant

IsInvolutive

IsSubspace

IsTrivial

VectorSpaceSum

 

 

 

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The following Maple builtins functions are extended so that they work for a Distribution object: type, has, hastype, indets, convert. See Distribution Object Overloaded Builtins for more detail.

Examples

withLieAlgebrasOfVectorFields:

We first build vector fields associated with 3-d cylinder (2-dim x-y rotation, and z translation and uniform scaling)

X1VectorFieldyDx+xDy,space=x,y,z

X1yx+xy

(1)

X2VectorFieldDz,space=x,y,z

X2z

(2)

X3VectorFieldzDz,space=x,y,z

X3zz

(3)

ΣDistributionX1,X2,X3

Σz,yxx+y

(4)

ΩDistributionVectorFieldDx,space=x,y,z

Ωx

(5)

We can request the dimension of this distribution

DimensionΣ

2

(6)

IsInvolutiveΣ

true

(7)

IsIntegrableΣ

true

(8)

We can check if x-translation is subspace of Sigma

IsSubspaceΩ,Σ

false

(9)

Sum of these two distributions covers all (x,y,z) space.

VectorSpaceSumΣ,Ω

z,x,y

(10)

These two distribution don't intersect

IntersectionΣ,Ω

(11)

The invariant of Sigma

IntegralsΣ

x2+y2

(12)

Finding other distributions..

CauchyDistributionΣ

z,yxx+y

(13)

DerivedDistributionΣ

z,yxx+y

(14)

typeΣ,Distribution

true

(15)

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

LieAlgebrasOfVectorFields[Distribution]

LieAlgebrasOfVectorFields[VectorField]