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JetCalculus[HorizontalExteriorDerivative] - calculate the horizontal exterior derivative of a bi-form on a jet space

Calling Sequences



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



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. Let (xi, uα, uiα, uijα, ..., uij  kα, ....) be a local system of jet coordinates. Every differential form on JE can be expressed locally in terms of a sum of wedge products of 1-forms dxi on M and contact 1-forms,

 Θα = duαuℓαdxℓ,     Θiα = duiαuiℓαdxℓ ,  ....  ,  Θijkα = duijkαuijkℓα dxℓ , .... .

Note that exterior derivatives of the contact 1-forms are

dΘα = dxℓ  Θℓ α,     dΘiα = dx  Θiℓ α,  .... ,  dΘijkα = dx  Θijkℓ α.

A differential pform ω  ΩpJ∞ is called a bi-form of degree r,s if  it is a sum of wedge products of  r 1-forms on M  and s contact 1-forms, that is,

ω = Ai1i2ir a1 as               dxi1dxi2   dxir   Ca1Ca2  Cas,   where each Cak is a contact 1-form.

The space of all p-forms then decomposes as a direct sum of bi-forms

ΩpJ∞  = r+s =p Ωr,sJ∞E

The above formulas for the exterior derivative of the contact forms shows that d:Ωr,sJE Ωr+1,sJE Ωr,s+1JEand therefore d  = dH  + dV,   where

  dH :Ωr,sJE Ωr+1,sJE  and   dV :Ωr,sJE Ωr,s+1JE.

The differential operator dH is called the horizontal exterior derivative and the differential operator dV is called the vertical exterior derivative. One has that

dHdH =0,  dHdV + dVdH =0,  and  dVdV =0.

The coordinate formulas for the horizontal exterior derivative are

dHxi = dxi ,      dHuij  kα = uij  kℓα dxℓ,       dHdxi = 0,       dHΘijkα = dxℓ   Θijkℓα.

The coordinate formulas for the vertical exterior derivative are

dVxi =0,    dVuij  kα = Θij  kα ,      dVdxi = 0,     dVΘijkα = 0.


The command HorizontalExteriorDerivative(ω) returns the horizontal exterior derivative dHω. The horizontal degree of ω must be less than the dimension of the base manifold M. The vertical exterior derivative is computed with the command VerticalExteriorDerivative.


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




Example 1.

Create the jet space J2E for the bundle E with coordinates x, y, u, v  x, y.



Calculate the horizontal exterior derivative of a function.

E > 


E > 


E > 





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

E > 



E > 





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

E > 

ω2:=Cu2 &wedge Cv2


E > 




See Also

DifferentialGeometry, JetCalculus, ExteriorDerivative, VerticalExteriorDerivative, HorizontalHomotopy, VerticalHomotopy

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