push forward a total vector field by a transformation - Maple Help

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JetCalculus[PushforwardTotalVector] - push forward a total vector field by a transformation

Calling Sequences

     PushforwardTotalVector(φ)

Parameters

     φ       - a transformation between two jet spaces

 

Description

Examples

Description

• 

Let EM and FN be two fiber bundles with associated jet spaces JkE M and JℓF N and with jet coordinates (xi, uα, uiα, uijα, ..., uij  kα) and (ya, vρ, viρ, vij ρ, ..., vij  ℓρ) respectively. Let φ:JkE JF be a transformation and let φa= φa(xi, uα, uiα, uijα, ..., uij  kα) be the ya components of φ . Then the total Jacobian of φ is the m ×n matrix Diφa, where Di denotes the total derivative with respect to xi. The push forward of the total vector field X = Xi Di on JkE is the total vector X = Ya Da, where Ya = DiφaXi.

• 

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

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

First initialize two different jet spaces over bundles E1M1, E2M2. The dimension of the base spaces are dimM1 =2, dimM2 =1.

DGsetupx,y,u,E1,2:DGsetupt,v,E2,2:DGsetupp,q,r,w,E3,2:

 

Define a transformation φ:J2 E1  E2 and compute its total Jacobian (a 1 × 2 matrix).

E3 > 

φTransformationE1,E2,t=u2,0,v[]=xy

φt=u2,0,v=xy

(2.1)
E1 > 

J1TotalJacobianφ

J1u0,1,2u0,2,2

(2.2)

 

Define a vector field on M1 and its total part on J4E1.

E1 > 

XaD_x+bD_y

XD_xa+D_yb

(2.3)
E1 > 

totXProlongTotalVectorX,3

totXaD_x+bD_y+u1a+u2bD_u+au1,1+bu1,2D_u1+au1,2+bu2,2D_u2+au1,1,1+bu1,1,2D_u1,1+au1,1,2+bu1,2,2D_u1,2+au1,2,2+bu2,2,2D_u2,2+au1,1,1,1+bu1,1,1,2D_u1,1,1+au1,1,1,2+bu1,1,2,2D_u1,1,2+au1,1,2,2+bu1,2,2,2D_u1,2,2+au1,2,2,2+bu2,2,2,2D_u2,2,2

(2.4)

 

Push forward totX to a vector field on J4E2 

E1 > 

PushforwardTotalVectorφ,totX

u0,1,2a+u0,2,2bD_t+v1u0,1,2a+u0,2,2bD_v+u0,1,2a+u0,2,2bv1,1D_v1+u0,1,2a+u0,2,2bv1,1,1D_v1,1+u0,1,2a+u0,2,2bv1,1,1,1D_v1,1,1

(2.5)

See Also

DifferentialGeometry

JetCalculus

Prolong

TotalJacobian

Transformation

 


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