>
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with(DifferentialGeometry): with(JetCalculus):
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Example 1.
Create the jet space for the bundle with coordinates
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DGsetup([x], [u], E, 3):
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Apply the integration by parts operator to a bi-form of vertical degree 1.
E >
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PDEtools[declare](a(x), b(x), c(x), quiet):
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E >
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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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E >
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IntegrationByParts(omega1);
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Apply the integration by parts operator to a bi-form of vertical degree 2.
E >
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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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E >
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omega3 := IntegrationByParts(omega2);
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Verify that the integration by parts operator is a projection operator by applying it to – the result is again.
E >
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IntegrationByParts(omega3);
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Example 3.
Create the jet space for the bundle with coordinates .
E >
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DGsetup([x, y], [u, v], E, 3):
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E >
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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
E >
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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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E >
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IntegrationByParts(omega4);
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Apply the integration by parts operator to a type (2, 2) bi-form
E >
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omega5 := Dx &wedge Dy &wedge evalDG(a(x, y)*Cu[1] &w Cv[1]);
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E >
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IntegrationByParts(omega5);
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Apply the integration by parts operator to a (2, 3) bi-form which is the horizontal exterior derivative of a type (1, 3) bi-form
E >
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eta := evalDG(u[1]*Dx &w Cu[2] &w Cv[1] &w Cu[1, 1]);
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E >
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omega6 := HorizontalExteriorDerivative(eta);
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E >
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IntegrationByParts(omega6);
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