IsInvolutive - Maple Programming Help

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IsInvolutive

check if a Distribution object is in involution

IsIntegrable

a synonym for IsInvolutive

Integrals

calculate the integrals of an involutive Distribution object

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsInvolutive( dist)

IsIntegrable( dist)

Integrals( dist)

Parameters

dist

-

a Distribution object.

Description

• 

The IsInvolutive (or IsIntegrable) method returns true if the distribution specified by dist is in involution.

• 

A distribution is involutive (also known as integrable, or completely integrable) if the Lie bracket of any two vector fields lying in dist also lies in dist.

• 

The Integrals method  returns a list of the functionally independent integrals of an involutive distribution, or the string "not known" if Maple was unable to find all the integrals.

• 

A function fx1,,xn is an integral of distribution dist on a space with coordinates x1,,xn if every vector field X lying in dist satisfies X_(f(x[1], ..., x[n]))= 0.

• 

Because successful integration of PDE by Maple cannot be guaranteed (see pdsolve), it is possible that Integrals is unable to return an answer.

• 

These methods are associated with the Distribution object. For more detail see Overview of the Distribution object.

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)

IsInvolutiveΣ

true

(5)

IsIntegrableΣ

true

(6)

INIntegralsΣ

INx2+y2+z2

(7)

ρopIN

ρx2+y2+z2

(8)

Since rho is an integral of distribution Sigma, it should be annihilated by every vector field lying in Sigma...

vfsGetVectorFieldsΣ

vfsyxx+y,zxx+z

(9)

mapXXρ,vfs

0,0

(10)

Compatibility

• 

The IsInvolutive, IsIntegrable and Integrals commands were introduced in Maple 2020.

• 

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

See Also

Distribution (Object overview)

LieAlgebrasOfVectorFields[Distribution]

VectorField (Object overview)

LieAlgebrasOfVectorFields[VectorField]

LieAlgebrasOfVectorFields (Package overview)