Let $\mathrm{\π}\:E\to M$be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let ${\mathrm{\π}}^{\mathrm{\∞}}\:J^{\mathrm{\∞}}\left(E\right) \to M$be the infinite jet bundle of $E$. Let $\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\α}}\, ....\)$be a local system of jet coordinates and let ${\mathrm{\Θ}}^{\mathrm{\α}} \= {\mathrm{du}}^{\mathrm{\α}}-u_{\mathrm{\ℓ}}^{\mathrm{\α}}{\mathrm{dx}}^{\mathrm{\ℓ}}$. Let ${\mathrm{\Omega }}^{\left(n\,s\right)}\left(J^{\infty }\left(E\right)\right)$ be the space of all differential bi-forms of horizontal degree$n$and vertical degree $s\.$Let $\mathrm{\ω} \in {\mathrm{\Omega }}^{\left(n\,s\right)}\left(J^{\infty }\left(E\right)\right)$and let $E_{\mathrm{\α}}\left(\mathrm{\ω}\right) \in {\mathrm{\Ω}}^{\left(n-1\,s\right)}\left(J^{\infty }\left(E\right)\right)$ be the components of the Euler-Lagrange operator applied to $\mathrm{\ω}$. Then the integration by parts operator $I\: {\mathrm{\Omega }}^{\left(n\,s\right)}\left(J^{\infty }\left(E\right)\right)\to {\mathrm{\Omega }}^{\left(n\,s\right)}\left(J^{\infty }\left(E\right)\right)$ is defined by

The command IntegrationByParts($\mathrm{\omega }$) returns the type$\left(n\, s\right)$bi-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(...).

with(DifferentialGeometry): with(JetCalculus):

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

PDEtools[declare](a(x), b(x), c(x), quiet):

omega1 := Dx &wedge evalDG(a(x)*Cu[] + b(x)*Cu[1] + c(x)*Cu[1, 1] + d(x)*Cu[1, 1, 1]);

$\mathrm{\ω1}≔a\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}+b\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1+c\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{1,1}+d\left(x\right)\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{1,1,1}$

IntegrationByParts(omega1);

$\left(-d_{x,x,x}+c_{x,x}-b_x+a\right)\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

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]);

$\mathrm{\ω2}≔a\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1+b\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_{1,1}+c\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_{1,1}$

omega3 := IntegrationByParts(omega2);

$\mathrm{\ω3}≔\left(-\frac{c_{x,x}}{2}-b_x+a\right)\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1-\frac{3c_x}{2}\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_{1,1}-c\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_{1,1,1}$

IntegrationByParts(omega3);

$\left(-\frac{c_{x,x}}{2}-b_x+a\right)\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1-\frac{3c_x}{2}\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_{1,1}-c\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_{1,1,1}$

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):

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]);

$\mathrm{\ω4}≔a\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}+b\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_{\left[\right]}+c\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_1+d\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_2+e\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_1+f\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_2$

IntegrationByParts(omega4);

$\left(-d_y-c_x+a\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}+\left(-f_y-e_x+b\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_{\left[\right]}$

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

$\mathrm{\ω5}≔a\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_1\bigwedge {\mathrm{Cv}}_1$

IntegrationByParts(omega5);

$-\frac{a_x}{2}\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cv}}_1-\frac{a}{2}\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cv}}_{1,1}+\frac{a_x}{2}\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1+\frac{a}{2}\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_{1,1}$

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

$\mathrm{\eta }≔u_1\mathrm{Dx}\bigwedge {\mathrm{Cu}}_2\bigwedge {\mathrm{Cv}}_1\bigwedge {\mathrm{Cu}}_{1,1}$

omega6 := HorizontalExteriorDerivative(eta);

$\mathrm{\ω6}≔-u_{1,2}\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_2\bigwedge {\mathrm{Cv}}_1\bigwedge {\mathrm{Cu}}_{1,1}-u_1\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_2\bigwedge {\mathrm{Cv}}_1\bigwedge {\mathrm{Cu}}_{1,1,2}+u_1\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_2\bigwedge {\mathrm{Cu}}_{1,1}\bigwedge {\mathrm{Cv}}_{1,2}-u_1\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cv}}_1\bigwedge {\mathrm{Cu}}_{1,1}\bigwedge {\mathrm{Cu}}_{2,2}$

IntegrationByParts(omega6);

$0\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cv}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1$