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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

with(DifferentialGeometry): with(JetCalculus):

 

Example 1.

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

DGsetup([x, y], [u, v], E, 2):

 

Calculate the vertical exterior derivative of a function.

E > 

F := f(x, y, u[], v[], u[1], u[2], v[1], v[2]):

E > 

PDEtools[declare](F, quiet):

E > 

VerticalExteriorDerivative(F);

_DGbiform,E,0,1,3,fu,4,fv,5,fu1,6,fu2,7,fv1,8,fv2

(2.1)

 

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

E > 

omega1 := evalDG(u[2, 2]*Dx + v[1, 1, 1]*Dy);

ω1_DGbiform,E,1,0,1,u2,2,2,v1,1,1

(2.2)
E > 

VerticalExteriorDerivative(omega1);

_DGbiform,E,1,1,1,11,−1,2,19,−1

(2.3)

 

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

E > 

omega2 := evalDG(v[1, 1]*u[2, 2 ,2]*Cu[2] &w Cv[2]);

ω2_DGbiform,E,0,2,6,8,v1,1u2,2,2

(2.4)
E > 

VerticalExteriorDerivative(omega2);

_DGbiform,E,0,3,6,8,12,u2,2,2,6,8,18,v1,1

(2.5)

See Also

DifferentialGeometry

JetCalculus

ExteriorDerivative

HorizontalExteriorDerivative

HorizontalHomotopy

VerticalHomotopy