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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, y], [u], E, 3):
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Define a type (1, 0) form and show that it is -closed.
E >
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omega1 := evalDG((u[1, 2]*u[1, 1, 1] + u[1 ,1]*u[1, 1, 2])*Dx + (u[1, 2]*u[1 ,1, 2] + u[1, 1]*u[1, 2, 2])*Dy);
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E >
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HorizontalExteriorDerivative(omega1);
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Apply the ZigZag command to to obtain a form .
E >
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theta1 := ZigZag(omega1);
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Check that is -closed and that its [1, 0] component matches .
E >
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ExteriorDerivative(theta1);
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E >
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convert(theta1, DGbiform, [1, 0]);
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Example 2.
Define a type (2, 0) form and show that its Euler-Lagrange form is 0.
E >
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omega2 := evalDG((- u[2, 2]*u[1, 2] - u[2]*u[1, 2, 2] - u[1]*u[1, 2, 2] - u[1, 2]^2)*Dx &w Dy);
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E >
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EulerLagrange(omega2);
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Apply the ZigZag command to to obtain a 2-form .
E >
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theta2 := ZigZag(omega2);
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Check that is closed and that its [2, 0] component matches .
E >
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ExteriorDerivative(theta2);
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E >
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convert(theta2, DGbiform, [2, 0]);
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Example 3.
Define a type (2, 1) form and show that .
E >
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omega3 := EulerLagrange(u[1]*u[2]^2*Dx &w Dy);
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E >
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IntegrationByParts(VerticalExteriorDerivative(omega3));
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Apply the ZigZag command to to obtain a form .
E >
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theta3 := ZigZag(omega3);
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Check that is closed and that its [2, 1] component matches
E >
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ExteriorDerivative(theta3);
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E >
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convert(theta3, DGbiform, [2, 1]);
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