KF Object as Operator - Maple Programming Help

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KF Object as Operator

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

K(X,Y)

Parameters

X,Y

-

VectorField objects (see LieAlgebrasOfVectorFields[VectorField] for how to construct one)

Description

• 

Let K be the Killing form of a LAVF object L. The K can act as a symmetric bilinear operator on vector fields.

• 

This method is associated with the local KF object. For more detail, see Overview of the KF object for more detail.

Examples

withLieAlgebrasOfVectorFields:

Typesetting:-Settingsuserep=true:

Typesetting:-Suppressαx,y,βx,y,ξx,y,ηx,y:

VVectorFieldξx,yDx+ηx,yDy,space=x,y

Vξx+ηy

(1)

E2LHPDEdiffξx,y,y,y=0,diffηx,y,x=diffξx,y,y,diffηx,y,y=0,diffξx,y,x=0,indep=x,y,dep=ξ,η

E2ξy,y=0,ηx=ξy,ηy=0,ξx=0,indep=x,y,dep=ξ,η

(2)

We first construct a LAVF object for E(2).

LLAVFV,E2

Lξx+ηy&whereξy,y=0,ξx=0,ηx=ξy,ηy=0

(3)

IsLieAlgebraL

true

(4)

Get its KillingForm as a local KF object.

KKillingFormL

KX,Y→2yXxyYx

(5)

Now we create a second vector field on the same space

XV

Xξx+ηy

(6)

Ysubsξ=α,η=β,V

Yαx+βy

(7)

KX,Y

2ξyαy

(8)

Compatibility

• 

The KF Object as Operator command was introduced in Maple 2020.

• 

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

See Also

LieAlgebrasOfVectorFields (Package overview)

LAVF (Object overview)

KF (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields[LHPDE]

LieAlgebrasOfVectorFields[LAVF]

IsLieAlgebra

KillingForm