IntegrationByParts - Maple Help

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

Calling Sequences

IntegrationByParts()

Parameters

$\mathrm{ω}$     - a differential bi-form on a jet space

Description

 • Let be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let be the infinite jet bundle of $E$. Let , ..., be a local system of jet coordinates and let . Let ${\mathrm{\Omega }}^{\left(n,s\right)}\left({J}^{\infty }\left(E\right)\right)$ 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 $\mathrm{ω}$. Then the integration by parts operator  is defined by

The operator is intrinsically characterized by the following properties.

[i] For any differential bi-form $\mathrm{η}$ of type where is the horizontal exterior derivative of $\mathrm{η}$.

[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(${\mathrm{\omega }}$) returns the typebi-form $I\left(\mathrm{ω}\right)$.
 • 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 ${\mathrm{ω}}_{1}$ 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]);
 ${\mathrm{ω1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{a}\right]{,}\left[\left[{1}{,}{3}\right]{,}{b}\right]{,}\left[\left[{1}{,}{4}\right]{,}{c}\right]{,}\left[\left[{1}{,}{5}\right]{,}{d}{}\left({x}\right)\right]\right]\right]\right)$ (2.1)
 E > IntegrationByParts(omega1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{{d}}_{{x}{,}{x}{,}{x}}{+}{{c}}_{{x}{,}{x}}{-}{{b}}_{{x}}{+}{a}\right]\right]\right]\right)$ (2.2)

Apply the integration by parts operator to a bi-form ${\mathrm{ω}}_{2}$ 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]);
 ${\mathrm{ω2}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{2}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{a}\right]{,}\left[\left[{1}{,}{2}{,}{4}\right]{,}{b}\right]{,}\left[\left[{1}{,}{3}{,}{4}\right]{,}{c}\right]\right]\right]\right)$ (2.3)
 E > omega3 := IntegrationByParts(omega2);
 ${\mathrm{ω3}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{2}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}\frac{{{c}}_{{x}{,}{x}}}{{2}}{-}{{b}}_{{x}}{+}{a}\right]{,}\left[\left[{1}{,}{2}{,}{4}\right]{,}{-}\frac{{3}{}{{c}}_{{x}}}{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{5}\right]{,}{-}{c}\right]\right]\right]\right)$ (2.4)

Verify that the integration by parts operator is a projection operator by applying it to ${\mathrm{ω}}_{3}$ – the result is ${\mathrm{ω}}_{3}$ again.

 E > IntegrationByParts(omega3);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{2}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}\frac{{{c}}_{{x}{,}{x}}}{{2}}{-}{{b}}_{{x}}{+}{a}\right]{,}\left[\left[{1}{,}{2}{,}{4}\right]{,}{-}\frac{{3}{}{{c}}_{{x}}}{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{5}\right]{,}{-}{c}\right]\right]\right]\right)$ (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 ${\mathrm{ω}}_{4}.$

 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]);
 ${\mathrm{ω4}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{a}\right]{,}\left[\left[{1}{,}{2}{,}{4}\right]{,}{b}\right]{,}\left[\left[{1}{,}{2}{,}{5}\right]{,}{c}\right]{,}\left[\left[{1}{,}{2}{,}{6}\right]{,}{d}\right]{,}\left[\left[{1}{,}{2}{,}{7}\right]{,}{e}\right]{,}\left[\left[{1}{,}{2}{,}{8}\right]{,}{f}\right]\right]\right]\right)$ (2.6)
 E > IntegrationByParts(omega4);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{{d}}_{{y}}{-}{{c}}_{{x}}{+}{a}\right]{,}\left[\left[{1}{,}{2}{,}{4}\right]{,}{-}{{f}}_{{y}}{-}{{e}}_{{x}}{+}{b}\right]\right]\right]\right)$ (2.7)

Apply the integration by parts operator to a type (2, 2) bi-form ${\mathrm{ω}}_{5}.$

 E > omega5 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu[1] &w Cv[1]);
 ${\mathrm{ω5}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{2}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{5}{,}{7}\right]{,}{a}\right]\right]\right]\right)$ (2.8)
 E > IntegrationByParts(omega5);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{2}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{7}\right]{,}{-}\frac{{{a}}_{{x}}}{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{3}{,}{12}\right]{,}{-}\frac{{a}}{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{4}{,}{5}\right]{,}\frac{{{a}}_{{x}}}{{2}}\right]{,}\left[\left[{1}{,}{2}{,}{4}{,}{9}\right]{,}\frac{{a}}{{2}}\right]\right]\right]\right)$ (2.9)

Apply the integration by parts operator to a (2, 3) bi-form ${\mathrm{ω}}_{6}$which is the horizontal exterior derivative of a type (1, 3) bi-form $\mathrm{η}.$

 E > eta := evalDG(u[1]*Dx &w Cu[2] &w Cv[1] &w Cu[1, 1]);
 ${\mathrm{\eta }}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{3}\right]\right]{,}\left[\left[\left[{1}{,}{6}{,}{7}{,}{9}\right]{,}{{u}}_{{1}}\right]\right]\right]\right)$ (2.10)
 E > omega6 := HorizontalExteriorDerivative(eta);
 ${\mathrm{ω6}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{3}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{6}{,}{7}{,}{9}\right]{,}{-}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{7}{,}{16}\right]{,}{-}{{u}}_{{1}}\right]{,}\left[\left[{1}{,}{2}{,}{6}{,}{9}{,}{13}\right]{,}{{u}}_{{1}}\right]{,}\left[\left[{1}{,}{2}{,}{7}{,}{9}{,}{11}\right]{,}{-}{{u}}_{{1}}\right]\right]\right]\right)$ (2.11)
 E > IntegrationByParts(omega6);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{3}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (2.12)