calculate the vertical exterior derivative of a bi-form on a jet space - Maple Help

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JetCalculus[VerticalExteriorDerivative] - calculate the vertical exterior derivative of a bi-form on a jet space

Calling Sequences

     VerticalExteriorDerivative(ω)

Parameters

      ω     - a differential bi-form on the jet space of a fiber bundle 

 

Description

Examples

Description

• 

Let π:EM be a fiber bundle, with base dimension n and fiber dimension m and let π∞ :J∞E M  be the infinite jet bundle of E. The  p -forms ΩpJ∞E can be graded by horizontal and vertical (or contact) degree and, with respect to this grading, the exterior derivative operator can be decomposed as d = dH  + dV. The horizontal exterior derivative dH  raises the horizontal degree by 1 and the vertical exterior derivative dV  raises the vertical degree by 1. For details, see HorizontalExteriorDerivative.

• 

The command VerticalExteriorDerivative(ω) returns the vertical exterior derivative dVω.

• 

The command VerticalExteriorDerivative is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form VerticalExteriorDerivative(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-VerticalExteriorDerivative(...).

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

Create the jet space J2Ewith coordinates x, y, u, v  x,y.

DGsetupx,y,u,v,E,2:

 

Calculate the vertical exterior derivative of a function.

E > 

Ffx,y,u[],v[],u1,u2,v1,v2:

E > 

PDEtools[declare]F,quiet:

E > 

VerticalExteriorDerivativeF

fuCu+fvCv+fu1Cu1+fu2Cu2+fv1Cv1+fv2Cv2

(2.1)

 

Calculate the vertical exterior derivative of a type (1, 0) bi-form.

E > 

ω1evalDGu2,2Dx+v1,1,1Dy

ω1u2,2Dx+v1,1,1Dy

(2.2)
E > 

VerticalExteriorDerivativeω1

DxCu2,2DyCv1,1,1

(2.3)

 

Calculate the vertical exterior derivative of a type (0, 2) bi-form.

E > 

ω2evalDGv1,1u2,2,2Cu2 &w Cv2

ω2v1,1u2,2,2Cu2Cv2

(2.4)
E > 

VerticalExteriorDerivativeω2

u2,2,2Cu2Cv2Cv1,1+v1,1Cu2Cv2Cu2,2,2

(2.5)

See Also

DifferentialGeometry

JetCalculus

ExteriorDerivative

HorizontalExteriorDerivative

HorizontalHomotopy

VerticalHomotopy

 


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