AssignTransformationType - Maple Help
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JetCalculus[AssignTransformationType] - assign a type (one of projectable, point, contact, differential substitution, generalized differential substitution, generic) to a transformation

Calling Sequences

     AssignTransformationType()

Parameters

            - a transformation

 

Description

Examples

Description

• 

Let and be two fiber bundles, and let ,  be the associated bundles of jets.

[i] A map which sends the fibers of to fibers of (and hence covers a map  is called a projectable transformation.

[ii] A map is called a point transformation.

[iii] A transformation  is called a contact transformation if the fiber dimensions of E and  are 1 and  pulls back the contact form on  to a multiple of the contact form on .

[iv] If  and covers the identity map then is called a differential substitution.

[v] A map is called a generalized differential substitution.

[vi] A transformation not of one the types [i]--[v] is called generic.

Explicit coordinate formulas for these various types of maps are given in Example 1.

• 

 The command AssignTransformationType( ) returns the transformation , but with internal representation  of  changed to encode its transformation type. The type of a transformation and its prolongation order can be determined by the command DGinfo with the keyword "TransformationType".

• 

Any transformation of type [i]--[v] can be prolonged to higher order jet spaces. See Prolong for further information.

• 

The command AssignTransformationType is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form AssignTransformationType(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-AssignTransformationType(...).

Examples

 

Example 1.

First initialize various jet spaces of one or two independent variables and one dependent variable and prolong them to order 4.

 

Case 1. Projectable transformations from  to .

K > 

(2.1)

 

When a transformation is first defined, it is not given a type.

E > 

(2.2)

 

Now assign the transformation  a type.

E > 

(2.3)
E > 

(2.4)

 

This indicates that the transformation is a projectable transformation, the 0 indicates that the transformation has not been prolonged to a jet space.

 

Case 2. Point transformations:

E > 

(2.5)
E > 

(2.6)
E > 

(2.7)

 

Case 3. Contact transformations:

E > 

(2.8)
E > 

(2.9)
E > 

(2.10)

 

By the conventions adopted here, a contact transformation need not be a local diffeomorphism so that, in particular, the dimensions of the bundles  and  need not coincide.

E > 

(2.11)
F > 

(2.12)
F > 

(2.13)

 

Case 4. Differential Substitutions:

F > 

(2.14)
E > 

(2.15)
E > 

E > 

(2.16)

 

Case 5. Generalized Differential Substitutions:

E > 

(2.17)
E > 

E > 

(2.18)

 

Case 6. Generic:

E > 

(2.19)
E > 

F > 

(2.20)

See Also

DifferentialGeometry

JetCalculus

AssignVectorType

DGinfo

Prolong

Transformation

 


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