JetCalculus[IntegrationByParts] - apply the integration by parts operator to a differential bi-form
ω - a differential bi-form on a jet space
Let π:E→M 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 = duα−uℓαdxℓ. Let Ωn,sJ∞E be the space of all differential bi-forms of horizontal degree n and vertical degree s. Let ω ∈Ωn,sJ∞E and let Eαω ∈ Ωn−1,sJ∞E be the components of the Euler-Lagrange operator applied to ω. Then the integration by parts operator I: Ωn,sJ∞E→Ωn,sJ∞E 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 n−1, 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 n−1, s such that ω = dH η.
[iii] I is a projection operator in the sense that I∘I = 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(...).
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.
PDEtools[declare](a(x), b(x), c(x), quiet):
omega1 := Dx &wedge evalDG(a(x)*Cu + b(x)*Cu + c(x)*Cu[1, 1] + d(x)*Cu[1, 1, 1]);
Apply the integration by parts operator to a bi-form ω2 of vertical degree 2.
omega2 := Dx &wedge evalDG(a(x)*Cu&w Cu + b(x)*Cu &w Cu[1,1] + c(x)*Cu &w Cu[1,1]);
omega3 := IntegrationByParts(omega2);
Verify that the integration by parts operator is a projection operator by applying it to ω3 – the result is ω3 again.
Create the jet space J3E for the bundle E with coordinates x,y, u, v→ x,y.
DGsetup([x, y], [u, v], E, 3):
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.
omega4 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu + b(x, y)*Cv + c(x, y)*Cu + d(x, y)*Cu + e(x, y)*Cv + f(x, y)*Cv);
Apply the integration by parts operator to a type (2, 2) bi-form ω5.
omega5 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu &w Cv);
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.
eta := evalDG(u*Dx &w Cu &w Cv &w Cu[1, 1]);
omega6 := HorizontalExteriorDerivative(eta);
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