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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

with(DifferentialGeometry): with(JetCalculus):

 

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.

DGsetup([x, y], [u], E1, 2): DGsetup([t], [v], E2, 2): DGsetup([p, q, r], [w], E3, 2):

 

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

E3 > 

phi1 := Transformation(E1, E2, [t = u[1, 1], v[] = x*y]);

φ1_DGtransformation,E1,2,E2,0,,00000100yx000000,u1,1,t,xy,v

(2.1)
E1 > 

J1 := TotalJacobian(phi1);

J1u1,1,1u1,1,2

(2.2)

 

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

E1 > 

phi2 := Transformation(E1, E3, [p = x*u[1], q = y*u[], r = u[2, 2], w[] = u[1]]);

φ2_DGtransformation,E1,2,E3,0,,u100x00000uy000000000000100010000,xu1,p,yu,q,u2,2,r,u1,w

(2.3)
E1 > 

J2 := TotalJacobian(phi2);

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 > 

phi3 := Transformation(E1, E1, [x = x*y, y = u[]*u[2], u[] = y]);

φ3_DGtransformation,E1,1,E1,0,,yx00000u20u01000,xy,x,uu2,y,y,u

(2.5)
E1 > 

J3 := TotalJacobian(phi3);

J3yxuu1,2+u2u1uu2,2+u22

(2.6)

See Also

DifferentialGeometry

JetCalculus

PushforwardTotalVector

TotalDiff

Transformation