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]);
 ${\mathrm{ω1}}{≔}{a}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{+}{b}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{c}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{+}{d}{}\left({x}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}{,}{1}}$ (2.1)
 E > IntegrationByParts(omega1);
 $\left({-}{{d}}_{{x}{,}{x}{,}{x}}{+}{{c}}_{{x}{,}{x}}{-}{{b}}_{{x}}{+}{a}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}$ (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]);
 ${\mathrm{ω2}}{≔}{a}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{b}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{+}{c}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}$ (2.3)
 E > omega3 := IntegrationByParts(omega2);
 ${\mathrm{ω3}}{≔}\left({-}\frac{{{c}}_{{x}{,}{x}}}{{2}}{-}{{b}}_{{x}}{+}{a}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{-}\frac{{3}{}{{c}}_{{x}}}{{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{-}{c}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}{,}{1}}$ (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);
 $\left({-}\frac{{{c}}_{{x}{,}{x}}}{{2}}{-}{{b}}_{{x}}{+}{a}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{-}\frac{{3}{}{{c}}_{{x}}}{{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{-}{c}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}{,}{1}}$ (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]);
 ${\mathrm{ω4}}{≔}{a}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{+}{b}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{\left[\right]}{+}{c}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}{d}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{+}{e}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}{+}{f}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}$ (2.6)
 E > IntegrationByParts(omega4);
 $\left({-}{{d}}_{{y}}{-}{{c}}_{{x}}{+}{a}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{+}\left({-}{{f}}_{{y}}{-}{{e}}_{{x}}{+}{b}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{\left[\right]}$ (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]);
 ${\mathrm{ω5}}{≔}{a}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}$ (2.8)
 E > IntegrationByParts(omega5);
 ${-}\frac{{{a}}_{{x}}}{{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}{-}\frac{{a}}{{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}{,}{1}}{+}\frac{{{a}}_{{x}}}{{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{+}\frac{{a}}{{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}$ (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]);
 ${\mathrm{\eta }}{≔}{{u}}_{{1}}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}$ (2.10)
 E > omega6 := HorizontalExteriorDerivative(eta);
 ${\mathrm{ω6}}{≔}{-}{{u}}_{{1}{,}{2}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{-}{{u}}_{{1}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}{,}{2}}{+}{{u}}_{{1}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}{,}{2}}{-}{{u}}_{{1}}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}}$ (2.11)
 E > IntegrationByParts(omega6);
 ${0}{}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cv}}}_{\left[\right]}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}$ (2.12)