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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]);

ω1_DGbiform,E,1,1,1,2,a,1,3,b,1,4,c,1,5,dx

(2.1)
E > 

IntegrationByParts(omega1);

_DGbiform,E,1,1,1,2,dx,x,x+cx,xbx+a

(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]);

ω2_DGbiform,E,1,2,1,2,3,a,1,2,4,b,1,3,4,c

(2.3)
E > 

omega3 := IntegrationByParts(omega2);

ω3_DGbiform,E,1,2,1,2,3,cx,x2bx+a,1,2,4,3cx2,1,2,5,c

(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);

_DGbiform,E,1,2,1,2,3,cx,x2bx+a,1,2,4,3cx2,1,2,5,c

(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]);

ω4_DGbiform,E,2,1,1,2,3,a,1,2,4,b,1,2,5,c,1,2,6,d,1,2,7,e,1,2,8,f

(2.6)
E > 

IntegrationByParts(omega4);

_DGbiform,E,2,1,1,2,3,dycx+a,1,2,4,fyex+b

(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]);

ω5_DGbiform,E,2,2,1,2,5,7,a

(2.8)
E > 

IntegrationByParts(omega5);

_DGbiform,E,2,2,1,2,3,7,ax2,1,2,3,12,a2,1,2,4,5,ax2,1,2,4,9,a2

(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]);

η_DGbiform,E,1,3,1,6,7,9,u1

(2.10)
E > 

omega6 := HorizontalExteriorDerivative(eta);

ω6_DGbiform,E,2,3,1,2,6,7,9,u1,2,1,2,6,7,16,u1,1,2,6,9,13,u1,1,2,7,9,11,u1

(2.11)
E > 

IntegrationByParts(omega6);

_DGbiform,E,2,3,1,2,3,4,5,0

(2.12)

See Also

DifferentialGeometry

JetCalculus

HorizontalExteriorDerivative

HorizontalHomotopy