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JetCalculus[TotalJacobian] - find the Jacobian of a transformation using total derivatives

Calling Sequences

     TotalJacobian(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.

• 

TotalJacobian returns the m ×n  matrix Diφa.

• 

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

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

First initialize several different jet spaces over bundles E1M1, E2M2, E3M3. The dimension of the base spaces are dimM1 =2, dimM2 =1, dimM3 =3.

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

 

Define a transformation φ1:J2E1  E2 and compute its total Jacobian (a 1 ×2 matrix).

E3 > 

φ1TransformationE1,E2,t=u1,1,v[]=xy

φ1t=u1,1,v=xy

(2.1)
E1 > 

J1TotalJacobianφ1

J1u1,1,1u1,1,2

(2.2)

 

Define a transformation φ2:J2E1  E3 and compute its total Jacobian (a 3×2 matrix).

E1 > 

φ2TransformationE1,E3,p=xu1,q=yu[],r=u2,2,w[]=u1

φ2p=xu1,q=yu,r=u2,2,w=u1

(2.3)
E1 > 

J2TotalJacobianφ2

J2xu1,1+u1xu1,2yu1yu2+uu1,2,2u2,2,2

(2.4)

 

Define a transformation φ3:J1E1  E1 and compute its total Jacobian (a 2×2 matrix).

E1 > 

φ3TransformationE1,E1,x=xy,y=u[]u2,u[]=y

φ3x=xy,y=uu2,u=y

(2.5)
E1 > 

J3TotalJacobianφ3

J3yxuu1,2+u2u1uu2,2+u22

(2.6)

See Also

DifferentialGeometry

JetCalculus

PushforwardTotalVector

TotalDiff

Transformation

 


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