IntegrationByParts - Maple Help
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JetCalculus[IntegrationByParts] - apply the integration by parts operator to a differential bi-form

Calling Sequences

     IntegrationByParts()

Parameters

          - a differential bi-form on a jet space

 

Description

Examples

Description

• 

Let be a fiber bundle, with base dimension  and fiber dimension  and let be the infinite jet bundle of . Let , ..., be a local system of jet coordinates and let . Let  be the space of all differential bi-forms of horizontal degreeand vertical degree Let and let  be the components of the Euler-Lagrange operator applied to . Then the integration by parts operator  is defined by

The operator is intrinsically characterized by the following properties.

[i] For any differential bi-form  of type where is the horizontal exterior derivative of .

[ii]  If is a type bi-form and then there exists a bi-form of type such that .

[iii] is a projection operator in the sense that .

• 

The command IntegrationByParts() returns the typebi-form .

• 

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

Examples

with(DifferentialGeometry): with(JetCalculus):

 

Example 1.

Create the jet space for the bundle with coordinates

DGsetup([x], [u], E, 3):

 

Apply the integration by parts operator to a bi-form  of vertical degree 1.

E > 

PDEtools[declare](a(x), b(x), c(x), quiet):

E > 

omega1 := Dx &wedge evalDG(a(x)*Cu[] + b(x)*Cu[1] + c(x)*Cu[1, 1] + d(x)*Cu[1, 1, 1]);

(2.1)
E > 

IntegrationByParts(omega1);

(2.2)

 

Apply the integration by parts operator to a bi-form  of vertical degree 2.

E > 

omega2 := Dx &wedge evalDG(a(x)*Cu[]&w Cu[1] + b(x)*Cu[] &w Cu[1,1] + c(x)*Cu[1] &w Cu[1,1]);

(2.3)
E > 

omega3 := IntegrationByParts(omega2);

(2.4)

 

Verify that the integration by parts operator is a projection operator by applying it to  – the result is  again.

E > 

IntegrationByParts(omega3);

(2.5)

 

Example 3.

Create the jet space for the bundle with coordinates .

E > 

DGsetup([x, y], [u, v], E, 3):

E > 

PDEtools[declare](a(x, y), b(x, y), c(x, y), d(x, y), e(x, y), f(x, y), quiet):

 

Apply the integration by parts operator to a type (2, 1) bi-form

E > 

omega4 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu[] + b(x, y)*Cv[] + c(x, y)*Cu[1] + d(x, y)*Cu[2] + e(x, y)*Cv[1] + f(x, y)*Cv[2]);

(2.6)
E > 

IntegrationByParts(omega4);

(2.7)

 

Apply the integration by parts operator to a type (2, 2) bi-form

E > 

omega5 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu[1] &w Cv[1]);

(2.8)
E > 

IntegrationByParts(omega5);

(2.9)

 

Apply the integration by parts operator to a (2, 3) bi-form which is the horizontal exterior derivative of a type (1, 3) bi-form

E > 

eta := evalDG(u[1]*Dx &w Cu[2] &w Cv[1] &w Cu[1, 1]);

(2.10)
E > 

omega6 := HorizontalExteriorDerivative(eta);

(2.11)
E > 

IntegrationByParts(omega6);

(2.12)

See Also

DifferentialGeometry

JetCalculus

HorizontalExteriorDerivative

HorizontalHomotopy

 


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