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JetCalculus[IntegrationByParts] - apply the integration by parts operator to a differential bi-form
Calling Sequences
IntegrationByParts(omega)
Parameters
omega - a differential bi-form on the jet space of a fiber bundle E -> M
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Description
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Let E -> M be a fiber bundle over an n dimensional base manifold M and Omega^(n, s) be the space of all differential bi-forms of horizontal degree n and vertical degree s. The integration by parts operator is a total differential operator I: Omega^(n, s) -> Omega^(n, s) which is intrinsically characterized by the following properties. [i] For any differential bi-form eta of type (n - 1, s), I(dH(eta)) = 0, where dH(eta) is the horizontal exterior derivative of eta. [ii] If omega is a type (n, s) bi-form and I(omega) = 0, then there exists a bi-form eta of type (n - 1, s) such that omega = dH(eta). [iii] I is a projection operator in the sense that I^2 = I.
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The coordinate formula for I is I(omega) = 1/s*Cu^a &w EL_a(omega) (sum on a) where EL_a(omega) are the components of the Euler-Lagrange operator applied to omega.
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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(...).
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Examples
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with(DifferentialGeometry): with(JetCalculus):
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Example 1.
Create the jet space J^3(E) for the bundle E = R x R with coordinates (x, u) -> (x).
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DGsetup([x], [u], E, 3):
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Apply the integration by parts operator to a bi-form omega1 of vertical degree 1.
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PDEtools[declare](a(x), b(x), c(x), quiet):
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omega1 := Dx &wedge evalDG(a(x)*Cu[] + b(x)*Cu[1] + c(x)*Cu[1, 1] + d(x)*Cu[1, 1, 1]);
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![omega1 := a*Dx*`^`*Cu[]+b*Dx*`^`*Cu[1]+c*Dx*`^`*Cu[1, 1]+d(x)*Dx*`^`*Cu[1, 1, 1]](/support/helpjp/helpview.aspx?si=6561/file05761/math121.png)
| (2.1) |
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IntegrationByParts(omega1);
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![(-d[x, x, x]+c[x, x]-b[x]+a)*Dx*`^`*Cu[]](/support/helpjp/helpview.aspx?si=6561/file05761/math128.png)
| (2.2) |
Apply the integration by parts operator to a bi-form omega2 of vertical degree 2.
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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]);
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![omega2 := a*Dx*`^`*Cu[]*`^`*Cu[1]+b*Dx*`^`*Cu[]*`^`*Cu[1, 1]+c*Dx*`^`*Cu[1]*`^`*Cu[1, 1]](/support/helpjp/helpview.aspx?si=6561/file05761/math141.png)
| (2.3) |
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omega3 := IntegrationByParts(omega2);
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![omega3 := (-(1/2)*c[x, x]-b[x]+a)*Dx*`^`*Cu[]*`^`*Cu[1]-(3/2)*(c[x]*Dx*`^`*Cu[]*`^`*Cu[1, 1])-c*Dx*`^`*Cu[]*`^`*Cu[1, 1, 1]](/support/helpjp/helpview.aspx?si=6561/file05761/math148.png)
| (2.4) |
Verify that the integration by parts operator is a projection operator by applying it to omega3--the result is omega3 again!
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IntegrationByParts(omega3);
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![(-(1/2)*c[x, x]-b[x]+a)*Dx*`^`*Cu[]*`^`*Cu[1]-(3/2)*(c[x]*Dx*`^`*Cu[]*`^`*Cu[1, 1])-c*Dx*`^`*Cu[]*`^`*Cu[1, 1, 1]](/support/helpjp/helpview.aspx?si=6561/file05761/math163.png)
| (2.5) |
Example 3.
Create the jet space J^3(E) for the bundle E = R^2 x R^2 with coordinates (x, y, u, v) -> (x, y).
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DGsetup([x, y], [u, v], E, 3):
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PDEtools[declare](a(x, y), b(x, y), c(x, y), d(x, y), e(x, y), f(x, y), quiet):
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Apply the integration by parts operator to a type (2, 1) bi-form.
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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]);
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![omega4 := a*Dx*`^`*Dy*`^`*Cu[]+b*Dx*`^`*Dy*`^`*Cv[]+c*Dx*`^`*Dy*`^`*Cu[1]+d*Dx*`^`*Dy*`^`*Cu[2]+e*Dx*`^`*Dy*`^`*Cv[1]+f*Dx*`^`*Dy*`^`*Cv[2]](/support/helpjp/helpview.aspx?si=6561/file05761/math194.png)
| (2.6) |
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IntegrationByParts(omega4);
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![(-d[y]-c[x]+a)*Dx*`^`*Dy*`^`*Cu[]+(-f[y]-e[x]+b)*Dx*`^`*Dy*`^`*Cv[]](/support/helpjp/helpview.aspx?si=6561/file05761/math201.png)
| (2.7) |
Apply the integration by parts operator to a type (2, 2) bi-form.
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omega5 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu[1] &w Cv[1]);
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![omega5 := a*Dx*`^`*Dy*`^`*Cu[1]*`^`*Cv[1]](/support/helpjp/helpview.aspx?si=6561/file05761/math212.png)
| (2.8) |
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IntegrationByParts(omega5);
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![-(1/2)*(a[x]*Dx*`^`*Dy*`^`*Cu[]*`^`*Cv[1])-(1/2)*(a*Dx*`^`*Dy*`^`*Cu[]*`^`*Cv[1, 1])+(1/2)*(a[x]*Dx*`^`*Dy*`^`*Cv[]*`^`*Cu[1])+(1/2)*(a*Dx*`^`*Dy*`^`*Cv[]*`^`*Cu[1, 1])](/support/helpjp/helpview.aspx?si=6561/file05761/math219.png)
| (2.9) |
Apply the integration by parts operator to a (2, 3) bi-form omega6 which is the horizontal exterior derivative of a type (1, 3) bi-form eta.
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eta := evalDG(u[1]*Dx &w Cu[2] &w Cv[1] &w Cu[1, 1]);
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![eta := u[1]*Dx*`^`*Cu[2]*`^`*Cv[1]*`^`*Cu[1, 1]](/support/helpjp/helpview.aspx?si=6561/file05761/math234.png)
| (2.10) |
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omega6 := HorizontalExteriorDerivative(eta);
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![omega6 := -u[1, 2]*Dx*`^`*Dy*`^`*Cu[2]*`^`*Cv[1]*`^`*Cu[1, 1]-u[1]*Dx*`^`*Dy*`^`*Cu[2]*`^`*Cv[1]*`^`*Cu[1, 1, 2]+u[1]*Dx*`^`*Dy*`^`*Cu[2]*`^`*Cu[1, 1]*`^`*Cv[1, 2]-u[1]*Dx*`^`*Dy*`^`*Cv[1]*`^`*Cu[1, 1]*`^`*Cu[2, 2]](/support/helpjp/helpview.aspx?si=6561/file05761/math241.png)
| (2.11) |
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IntegrationByParts(omega6);
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![`0`*Dx*`^`*Dy*`^`*Cu[]*`^`*Cv[]*`^`*Cu[1]](/support/helpjp/helpview.aspx?si=6561/file05761/math248.png)
| (2.12) |
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