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

Calling Sequences

     IntegrationByParts(ω )

Parameters

     ω     - a differential bi-form on a jet space

 

Description

Examples

Description

• 

Let π:EM be a fiber bundle, with base dimension n and fiber dimension m and let π∞:J∞E  M be the infinite jet bundle of E. Let (xi, uα, uiα, uijα, ..., uij  kα, ....) be a local system of jet coordinates and let Θα = duαuℓαdxℓ. Let Ωn,sJE be the space of all differential bi-forms of horizontal degree n and vertical degree s. Let ω Ωn,sJE and let Eαω  Ωn1,sJE be the components of the Euler-Lagrange operator applied to ω. Then the integration by parts operator I: Ωn,sJEΩn,sJE is defined by

Iω = 1sΘα Eαω.

The operator I is intrinsically characterized by the following properties.

[i] For any differential bi-form η of type n1, s,  IdHη = 0 where dH η is the horizontal exterior derivative of η.

[ii]  If ω is a type n,s bi-form and Iω =0, then there exists a bi-form of type n1, s such that ω = dH η.

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

• 

The command IntegrationByParts(ω) returns the typen, s bi-form Iω.

• 

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 J3E for the bundle E with coordinates x,u x.

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

 

Apply the integration by parts operator to a bi-form ω1 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]);

ω1aDxCu+bDxCu1+cDxCu1,1+dxDxCu1,1,1

(2.1)
E > 

IntegrationByParts(omega1);

dx,x,x+cx,xbx+aDxCu

(2.2)

 

Apply the integration by parts operator to a bi-form ω2 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]);

ω2aDxCuCu1+bDxCuCu1,1+cDxCu1Cu1,1

(2.3)
E > 

omega3 := IntegrationByParts(omega2);

ω3cx,x2bx+aDxCuCu13cx2DxCuCu1,1cDxCuCu1,1,1

(2.4)

 

Verify that the integration by parts operator is a projection operator by applying it to ω3 – the result is ω3 again.

E > 

IntegrationByParts(omega3);

cx,x2bx+aDxCuCu13cx2DxCuCu1,1cDxCuCu1,1,1

(2.5)

 

Example 3.

Create the jet space J3E for the bundle E with coordinates x,y, u, v x,y.

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 ω4.

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]);

ω4aDxDyCu+bDxDyCv+cDxDyCu1+dDxDyCu2+eDxDyCv1+fDxDyCv2

(2.6)
E > 

IntegrationByParts(omega4);

dycx+aDxDyCu+fyex+bDxDyCv

(2.7)

 

Apply the integration by parts operator to a type (2, 2) bi-form ω5.

E > 

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

ω5aDxDyCu1Cv1

(2.8)
E > 

IntegrationByParts(omega5);

ax2DxDyCuCv1a2DxDyCuCv1,1+ax2DxDyCvCu1+a2DxDyCvCu1,1

(2.9)

 

Apply the integration by parts operator to a (2, 3) bi-form ω6which is the horizontal exterior derivative of a type (1, 3) bi-form η.

E > 

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

ηu1DxCu2Cv1Cu1,1

(2.10)
E > 

omega6 := HorizontalExteriorDerivative(eta);

ω6u1,2DxDyCu2Cv1Cu1,1u1DxDyCu2Cv1Cu1,1,2+u1DxDyCu2Cu1,1Cv1,2u1DxDyCv1Cu1,1Cu2,2

(2.11)
E > 

IntegrationByParts(omega6);

0DxDyCuCvCu1

(2.12)

See Also

DifferentialGeometry

JetCalculus

HorizontalExteriorDerivative

HorizontalHomotopy

 


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