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DifferentialGeometry[Tools]

  

DGinfo

  

obtain information about  DifferentialGeometry objects

 

Calling Sequence

Parameters

Description

Function, Vector, DifferentialForm, and Tensor Information

Transformation Information

Frame Information

Miscellaneous

Calling Sequence

DGinfo(X, keyword)

Parameters

X

-

a DifferentialGeometry object

keyword

-

a keyword string

Description

• 

The command DGinfo provides a user friendly interface to the internal representations of all the objects and frames created by the DifferentialGeometry software.

• 

The keyword strings accepted by the DGinfo command are:

AbstractForms

BiformDegree

CoefficientGrading

CoefficientList

CoefficientSet

CoframeLabels

CurrentFrame

DiffeqType

DiffeqVariables

DomainFrame

DomainOrder

ExteriorDerivativeFormStructureEquations

ExteriorDerivativeFunctionStructureEquations

FormDegree

FrameBaseDimension

FrameBaseForms

FrameBaseVectors

FrameDependentVariables

FrameFiberDimension

FrameFiberForms

FrameFiberVectors

FrameGlobals

FrameHorizontalBiforms

FrameIndependentVariables

FrameInformation

FrameJetDimension

FrameJetForms

FrameJetOrder

FrameJetVariables

FrameJetVectors

FrameLabels

FrameNames

FrameProtocol

FrameVerticalBiforms

FunctionOrder

Grading

HorizontalCoframeLabels

JacobianMatrix

JetIndets

JetNotation

JetSpaceDimension

Keywords

LieAlgebraDimension

LieBracketStructureEquations

LieBracketStructureFunction

LieBracketStructureFunctionList

LieDerivativeFunctionStructureEquations

NonJetIndets

ObjectAttributes

ObjectComponents

ObjectFrame

ObjectGenerators

ObjectOrder

ObjectType

RangeFrame

RangeOrder

RepresentationMatrices

TensorDensityType

TensorGenerators

TensorIndexPart1

TensorIndexPart2

TensorIndexType

TensorType

TransformationType

VectorType

VerticalCoframeLabels

Weight

 

 

• 

This command is part of the DifferentialGeometry:-Tools package and so can be used in the form DGinfo(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGinfo(...).

Function, Vector, DifferentialForm, and Tensor Information

"BiformDegree"

with(DifferentialGeometry): with(Tools):

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

 

Example 1.

alpha1 := evalDG(Dx &wedge Dy);

_DGbiform,E,2,0,1,2,1

(1)

DGinfo(alpha1, "BiformDegree");

2,0

(2)

Example 2.

alpha2 := evalDG(Dx &wedge Cu[]);

_DGbiform,E,1,1,1,3,1

(3)

DGinfo(alpha2, "BiformDegree");

1,1

(4)

Example 3.

alpha3 := evalDG(Cu[1] &wedge Cu[2]);

_DGbiform,E,0,2,4,5,1

(5)

DGinfo(alpha3, "BiformDegree");

0,2

(6)

"CoefficientList": list some or all of the coefficients of a vector, differential form, or tensor.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z, w], M):

alpha := evalDG(a*dx &w dy + b*dx &w dz + c*dy &w dz + d*dx &w dw + e*dz &w dw);

_DGform,M,2,1,2,a,1,3,b,1,4,d,2,3,c,3,4,e

(7)

DGinfo(alpha, "CoefficientList", "all");

a,b,d,c,e

(8)

DGinfo(alpha, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);

a,b,0

(9)

DGinfo(alpha, "CoefficientList", [[1,2], [1,3], [2,4]]);

a,b,0

(10)

"CoefficientSet": find the set of all the coefficients of a vector, differential form, or tensor.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z, w], M):

 

Example 1.

alpha := evalDG(a*dx &w dy + b*dx &w dz + b*dy &w dz + c*dx &w dw + a*dz &w dw);

_DGform,M,2,1,2,a,1,3,b,1,4,c,2,3,b,3,4,a

(11)

DGinfo(alpha, "CoefficientSet");

a,b,c

(12)

Example 2.

X := DGzero("vector");

_DGvector,M,,1,0

(13)

DGinfo(X, "CoefficientSet");

0

(14)

"CoefficientList": list some or all of the coefficients of a vector, differential form, or tensor.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z, w], M):

alpha := evalDG(a*dx &w dy + b*dx &w dz + c*dy &w dz + d*dx &w dw + e*dz &w dw);

_DGform,M,2,1,2,a,1,3,b,1,4,d,2,3,c,3,4,e

(15)

DGinfo(alpha, "CoefficientList", "all");

a,b,d,c,e

(16)

DGinfo(alpha, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);

a,b,0

(17)

DGinfo(alpha, "CoefficientList", [[1,2], [1,3], [2,4]]);

a,b,0

(18)

"DiffeqType": find the type of the system of differential equations

with(DifferentialGeometry): with(JetCalculus): with(Tools):

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

Delta := DifferentialEquationData([u[2] + v[1], u[1] - v[2]], [u[1], v[1]]);

Δu1,v1,u2+v1,u1v2

(19)

DGinfo(Delta, "DiffeqType");

evolutionary,0

(20)

"DiffeqVariables": list the jet variables in the differential equation to be solved for

with(DifferentialGeometry): with(JetCalculus): with(Tools):

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

Delta := DifferentialEquationData([u[2] + v[1], u[1] - v[2]], [u[1], v[1]]);

Δu1,v1,u2+v1,u1v2

(21)

DGinfo(Delta, "DiffeqVariables");

u1,v1

(22)

"FormDegree": the degree of a differential form

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M):

 

Example 1.

alpha := evalDG(dx +dy);

_DGform,M,1,1,1,2,1

(23)

DGinfo(alpha, "FormDegree");

1

(24)

Example 2.

beta := evalDG(3*dx &w dy + 4*dy &w dz - dx &w dz);

_DGform,M,2,1,2,3,1,3,1,2,3,4

(25)

DGinfo(beta, "FormDegree");

2

(26)

Example 3.

nu := evalDG(dx &w dy &w dz);

_DGform,M,3,1,2,3,1

(27)

DGinfo(nu, "FormDegree");

3

(28)

"FunctionOrder": the order of the highest jet coordinate appearing in a Maple expression.

with(DifferentialGeometry): with(Tools):

 

Example 1.

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

f1 := u*x + v*y;

f1ux+vy

(29)

DGinfo(f1, "FunctionOrder");

0

(30)

Example 2.

DGsetup([x, y], [u], J, 1):

f1 := u[1]*x + u[2];

f1u1x+u2

(31)

DGinfo(f1, "FunctionOrder");

1

(32)

Example 3.

f1 := u[1, 1, 2]*x + u[2, 2, 2, 2];

f1u1,1,2x+u2,2,2,2

(33)

DGinfo(f1, "FunctionOrder");

4

(34)

"ObjectAttributes": list all the properties of a vector, differential form, tensor, or transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M): DGsetup([u, v], N):

 

Example 1.

X := evalDG(a*D_x + b*D_y + c*D_z);

_DGvector,M,,1,a,2,b,3,c

(35)

DGinfo(X, "ObjectAttributes");

vector,M,

(36)

Example 2.

alpha := evalDG(d*dx &w dy + e*dx &w dz + f*dy &w dz);

_DGform,M,2,1,2,d,1,3,e,2,3,f

(37)

DGinfo(alpha, "ObjectAttributes");

form,M,2

(38)

Example 3.

T := evalDG(r*D_x &t dx + s*D_z&t dy + t*D_z &t dx);

_DGtensor,M,con_bas,cov_bas,,1,1,r,3,1,t,3,2,s

(39)

DGinfo(T, "ObjectAttributes");

tensor,M,con_bas,cov_bas,

(40)

Example 4.

Phi := Transformation(M, N, [u = x^2 + y^2 + z^2, v = x*y*z]);

_DGtransformation,M,0,N,0,,2x2y2zyzxzxy,x2+y2+z2,u,xyz,v

(41)

DGinfo(Phi, "ObjectAttributes");

transformation,M,0,N,0,,2x2y2zyzxzxy

(42)

Example 5.

DGsetup([[gamma1, gamma2, gamma3, gamma4], chi = dgform(3)], [], P):

ExteriorDerivative(chi);

_DGform,P,4,6,1

(43)

DGinfo(dchi, "ObjectAttributes");

form,P,4,d,5

(44)

Example 6.

Hook(D_gamma1, chi);

_DGform,P,2,7,1

(45)

DGinfo(i_1chi, "ObjectAttributes");

form,P,2,hook,1,5

(46)

Example 7.

Hook([D_gamma2, D_gamma3], chi);

_DGform,P,1,9,1

(47)

DGinfo(i_2_3chi, "ObjectAttributes");

form,P,1,hook,2,3,5

(48)

"ObjectComponents": list all the components of a vector, differential form, tensor, or transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z], M): DGsetup([u, v], N):

 

Example 1.

X := evalDG(a*D_x + b*D_y + c*D_z);

_DGvector,M,,1,a,2,b,3,c

(49)

DGinfo(X, "ObjectComponents");

1,a,2,b,3,c

(50)

Example 2.

alpha := evalDG(d*dx &w dy + e*dx &w dz + f*dy &w dz);

_DGform,M,2,1,2,d,1,3,e,2,3,f

(51)

DGinfo(alpha, "ObjectComponents");

1,2,d,1,3,e,2,3,f

(52)

Example 3.

T := evalDG(r*D_x &t dx + s*D_z&t dy + t*D_z &t dx);

_DGtensor,M,con_bas,cov_bas,,1,1,r,3,1,t,3,2,s

(53)

DGinfo(T, "ObjectComponents");

1,1,r,3,1,t,3,2,s

(54)

Example 4.

Phi := Transformation(M, N, [u = x^2 + y^2 + z^2, v= x*y*z]);

_DGtransformation,M,0,N,0,,2x2y2zyzxzxy,x2+y2+z2,u,xyz,v

(55)

DGinfo(Phi, "ObjectComponents");

x2+y2+z2,u,xyz,v

(56)

"ObjectFrame": return the frame with respect to which the object is defined.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], [u], M, 1): DGsetup([r,s,t], N):

Example 1.

X := evalDG(D_x -3*D_y);

_DGvector,M,,1,1,2,3

(57)

DGinfo(X, "ObjectFrame");

M

(58)

Example 2.

T := evalDG(D_r &t D_s &t dt);

_DGtensor,N,con_bas,con_bas,cov_bas,,1,2,3,1

(59)

DGinfo(T, "ObjectFrame");

N

(60)

"ObjectGenerators": list the monomial vectors in a vector or the monomial 1-forms in a differential form.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y, z, w], M):

Example 1.

X := evalDG(y*D_x + z*D_z);

_DGvector,M,,1,y,3,z

(61)

DGinfo(X, "ObjectGenerators");

1,3

(62)

This means that X has only the 1st and 3rd elements from the standard basis for the tangent bundle.

Example 2.

alpha := evalDG(w*dx &w dy + x*dx &w dw + z^2*dy &w dw);

_DGform,M,2,1,2,w,1,4,x,2,4,z2

(63)

DGinfo(alpha, "ObjectGenerators");

1,2,4

(64)

This means that alpha do not contain the 3rd element (dz) from the standard basis for the cotangent bundle.

"ObjectOrder": the order of the jet space on which the object is defined.

with(DifferentialGeometry): with(JetCalculus): with(Tools):

 

Example 1.

DGsetup([x, y], [u, v], E, 3):

f := u[2]*x + v[]*y;

fu2x+vy

(65)

DGinfo(f, "ObjectOrder");

1

(66)

Example 2.

alpha := evalDG(du[1]*x + du[2,2]);

_DGform,E,1,5,x,11,1

(67)

DGinfo(alpha, "ObjectOrder");

2

(68)

Example 3.

beta := evalDG(du[1, 1, 2]*x + u[2, 2, 2, 2]*dy);

_DGform,E,1,2,u2,2,2,2,16,x

(69)

DGinfo(beta, "FunctionOrder");

4

(70)

Example 4.

X := evalDG(u[1]*D_u[]);

_DGvector,E,,3,u1

(71)

X3 := Prolong(X, 2);

_DGvector,E,evolutionary,2,3,u1,5,u1,1,6,u1,2,9,u1,1,1,10,u1,1,2,11,u1,2,2

(72)

DGinfo(X3, "FunctionOrder");

3

(73)

"ObjectType": the type of the DifferentialGeometry object.

with(DifferentialGeometry): with(Tools): with(LieAlgebras):

DGsetup([x, y], [u], M, 1):

 

Example 1.

f := sin(x)*cos(y);

fsinxcosy

(74)

DGinfo(f, "ObjectType");

DGscalar

(75)

Example 2.

X := evalDG(D_x + y*D_u[]);

_DGvector,M,,1,1,3,y

(76)

DGinfo(X, "ObjectType");

vector

(77)

Example 3.

alpha := evalDG(dx &w dy + dx &w du[1]);

_DGform,M,2,1,2,1,1,4,1

(78)

DGinfo(alpha, "ObjectType");

form

(79)

Example 4.

theta := evalDG(Dx &w Cu[1]);

_DGbiform,M,1,1,1,4,1

(80)

DGinfo(theta, "ObjectType");

biform

(81)

Example 5.

T := evalDG(dx &t D_y + dy &t D_y);

_DGtensor,M,cov_bas,con_bas,,1,2,1,2,2,1

(82)

DGinfo(T, "ObjectType");

tensor

(83)

Example 6.

A := Vector([1, 2]);

12

(84)

DGinfo(A, "ObjectType");

Vector

(85)

Example 7.

B := Matrix([1, 2]);

12

(86)

DGinfo(B, "ObjectType");

Matrix

(87)

Example 8.

B:= Matrix([[1, 2], [3, 4]]);

1234

(88)

DGinfo(B, "ObjectType");

Matrix

(89)

Example 9.

Phi := IdentityTransformation();

_DGtransformation,M,0,M,0,,100010001,x,x,y,y,u[],u[]

(90)

DGinfo(Phi, "ObjectType");

transformation

(91)

Example 10.

L := LieAlgebraData(evalDG([D_x, x*D_x, x^2*D_x]));

Le1,e2=e1,e1,e3=2e2,e2,e3=e3

(92)

DGinfo(L, "ObjectType");

LieAlgebra

(93)

Example 11.

DGsetup([x, y], M);

frame name: M

(94)

F := FrameData([y*dx, x*dy], N);

FdΘ1=Θ1Θ2yx,dΘ2=Θ1Θ2yx

(95)

DGinfo(F, "ObjectType");

moving_frame

(96)

Example 12.

DGsetup([x, y], V);

frame name: V

(97)

R := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [0, 1]])];

1000,0100,0001

(98)

L := LieAlgebraData(R, Alg);

Le1,e2=e2,e2,e3=e2

(99)

DGsetup(L);

Lie algebra: Alg

(100)

rho := Representation(Alg, V, R);

_DGRepresentation,Alg,0,V,0,,1000,0100,0001

(101)

DGinfo(rho, "ObjectType");

Representation

(102)

"TensorDensityType": the density weight of a tensor.

with(DifferentialGeometry): with(Tools): with(Tensor):

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

g := evalDG(dx &t dx + dy &t dy);

_DGtensor,E,cov_bas,cov_bas,,1,1,1,2,2,1

(103)

Example 1.

rho := MetricDensity(g, 3);

ρ1

(104)

T1 := D_x &tensor rho;

_DGtensor,E,con_bas,bas,3,1,1

(105)

DGinfo(T1, "TensorDensityType");

bas,3

(106)

Example 2.

T2 := PermutationSymbol("cov_vrt");

_DGtensor,E,cov_vrt,cov_vrt,cov_vrt,vrt,1,3,4,5,1,3,5,4,1,4,3,5,1,4,5,3,1,5,3,4,1,5,4,3,1

(107)

DGinfo(T2, "TensorDensityType");

vrt,−1

(108)

"TensorGenerators": a list of the monomial tensors which generate a given tensor.

with(DifferentialGeometry): with(Tools): with(Tensor):

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

 

Example 1.

T1 := evalDG(dx &t dx &t dx);

_DGtensor,E,cov_bas,cov_bas,cov_bas,,1,1,1,1

(109)

DGinfo(T1, "TensorGenerators");

cov_bas,1

(110)

Example 2.

T2 := evalDG(dx &t D_y &t dy);

_DGtensor,E,cov_bas,con_bas,cov_bas,,1,2,2,1

(111)

DGinfo(T2, "TensorGenerators");

con_bas,2,cov_bas,1,cov_bas,2

(112)

Example 3.

T3 := evalDG(du &t D_v &t dy);

_DGtensor,E,cov_vrt,con_vrt,cov_bas,,3,4,2,1

(113)

DGinfo(T3, "TensorGenerators");

con_vrt,4,cov_bas,2,cov_vrt,3

(114)

"TensorIndexPart1": the tensor character of a tensor index type.

with(DifferentialGeometry): with(Tools):

DGinfo("con_bas", "TensorIndexPart1");

con

(115)

DGinfo("cov_bas", "TensorIndexPart1");

cov

(116)

DGinfo("con_vrt", "TensorIndexPart1");

con

(117)

DGinfo("cov_vrt", "TensorIndexPart1");

cov

(118)

"TensorIndexPart2": the spatial type of a tensor index type.

with(DifferentialGeometry): with(Tools):

DGinfo("con_bas", "TensorIndexPart2");

bas

(119)

DGinfo("cov_bas", "TensorIndexPart2");

bas

(120)

DGinfo("con_vrt", "TensorIndexPart2");

vrt

(121)

DGinfo("cov_vrt", "TensorIndexPart2");

vrt

(122)

"TensorIndexType": the full tensor index type of a tensor.

with(DifferentialGeometry): with(Tools): with(Tensor):

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

Example 1.

T1 := evalDG(D_x &t D_y);

_DGtensor,E,con_bas,con_bas,,1,2,1

(123)

DGinfo(T1, "TensorIndexType");

con_bas,con_bas

(124)

Example 2.

T2 := evalDG(dx &t D_y);

_DGtensor,E,cov_bas,con_bas,,1,2,1

(125)

DGinfo(T2, "TensorIndexType");

cov_bas,con_bas

(126)

Example 3.

T3 := evalDG(du &t D_v);

_DGtensor,E,cov_vrt,con_vrt,,3,4,1

(127)

DGinfo(T3, "TensorIndexType");

cov_vrt,con_vrt

(128)

Example 4.

T4 := evalDG(D_x &t D_v);

_DGtensor,E,con_bas,con_vrt,,1,4,1

(129)

DGinfo(T4, "TensorIndexType");

con_bas,con_vrt

(130)

"TensorType": the full tensor index type and weight of a tensor.

with(DifferentialGeometry): with(Tools): with(Tensor):

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

 

Example 1.

T1 := evalDG(D_x &t D_y);

_DGtensor,E,con_bas,con_bas,,1,2,1

(131)

DGinfo(T1, "TensorType");

con_bas,con_bas,

(132)

Example 2.

T2 := PermutationSymbol("con_bas");

_DGtensor,E,con_bas,con_bas,bas,1,1,2,1,2,1,1

(133)

DGinfo(T2, "TensorType");

con_bas,con_bas,bas,1

(134)

Example 3.

T3 := evalDG(du &t D_v);

_DGtensor,E,cov_vrt,con_vrt,,3,4,1

(135)

DGinfo(T3, "TensorType");

cov_vrt,con_vrt,

(136)

Example 4.

T4 := PermutationSymbol("cov_vrt");

_DGtensor,E,cov_vrt,cov_vrt,vrt,1,3,4,1,4,3,1

(137)

DGinfo(T4, "TensorType");

cov_vrt,cov_vrt,vrt,−1

(138)

Example 5.

T5 := PermutationSymbol("cov_bas") &tensor PermutationSymbol("con_vrt");

_DGtensor,E,cov_bas,cov_bas,con_vrt,con_vrt,bas,1,vrt,1,1,2,3,4,1,1,2,4,3,1,2,1,3,4,1,2,1,4,3,1

(139)

DGinfo(T5, "TensorType");

cov_bas,cov_bas,con_vrt,con_vrt,bas,−1,vrt,1

(140)

"VectorType": the type (projectionable, point, contact, ...) of a vector field and its order of prolongation.  See AssignVectorType for details.

with(DifferentialGeometry): with(Tools): with(JetCalculus):

DGsetup([x,y], [u], E, 1):

Example 1.

X1 := evalDG(D_x + u[] *D_u[]);

_DGvector,E,,1,1,3,u[]

(141)

X1a := AssignVectorType(X1);

_DGvector,E,projectable,0,1,1,3,u[]

(142)

DGinfo(X1a, "VectorType");

projectable,0

(143)

Example 2.

X2 := Prolong(u[]*D_x - x *D_u[], 1);

_DGvector,E,point,1,1,u[],3,x,4,u121,5,u2u1

(144)

DGinfo(X2, "VectorType");

point,1

(145)

Example 3.

X3 := GeneratingFunctionToContactVector(u[]*u[1]*u[2]);

_DGvector,E,,1,u[]u2,2,u[]u1,3,u[]u1u2,4,u12u2,5,u22u1

(146)

X3a := AssignVectorType(X3);

_DGvector,E,contact,1,1,u[]u2,2,u[]u1,3,u[]u1u2,4,u12u2,5,u22u1

(147)

DGinfo(X3a, "VectorType");

contact,1

(148)

Weight": the weight of a monomial form in a graded Lie algebra with coefficients

with(DifferentialGeometry): with(Tools):

 

LD:=LieAlgebraData([[x3, x4] = e1, [x3, x5] = x2, [x4, x5] = x3], [x1,x2,x3,x4,x5], Alg, grading = [-3, -3, -2, -1, -1]);

LDe3,e5=e2,e4,e5=e3

(149)

DGsetup(LD):

DGsetup([w1, w2, w3, w4, w5], V, grading = [-3, -3, -2, -1, -1]):

rho:= Representation(Alg, V, Adjoint(Alg));

_DGRepresentation,Alg,0,V,0,,0000000000000000000000000,0000000000000000000000000,0000000001000000000000000,0000000000000010000000000,0000000100000100000000000

(150)

DGsetup(Alg, rho, AlgV);

Lie algebra with coefficients: AlgV

(151)

Example 1.

alpha1 := evalDG(w3 *theta5);

_DGform,AlgV,1,5,w3

(152)

DGinfo(alpha1, "Weight");

−1

(153)

Example 2.

alpha2 := evalDG(w3 * theta1 &w theta5);

_DGform,AlgV,2,1,5,w3

(154)

DGinfo(alpha2, "Weight");

2

(155)

Example 3.

alpha3 := evalDG(w1 * theta3 &w  &w theta4 &w theta5);

_DGform,AlgV,3,3,4,5,w1

(156)

DGinfo(alpha3, "Weight");

1

(157)

Transformation Information

"DomainFrame": the domain frame of a transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M): DGsetup([u, v], N):

Phi := Transformation(M, N, [ u =x*y, v= 1/x + 1/y]);

_DGtransformation,M,0,N,0,,yx1x21y2,xy,u,1x+1y,v

(158)

DGinfo(Phi, "DomainFrame");

M

(159)

"DomainOrder": the jet space order of the domain of a transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M): DGsetup([u, v], N): DGsetup([t], [w], J, 1):

 

Example 1.

Phi1 := Transformation(M, N, [u =x*y, v= 1/x + 1/y]);

_DGtransformation,M,0,N,0,,yx1x21y2,xy,u,1x+1y,v

(160)

DGinfo(Phi1, "DomainOrder");

0

(161)

Example 2.

Phi2 := Transformation(J, M, [x = w[1], y = w[1, 1]]);

_DGtransformation,J,2,M,0,,00100001,w1,x,w1,1,y

(162)

DGinfo(Phi2, "DomainOrder");

2

(163)

"JacobianMatrix": the Jacobian Matrix of a transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M): DGsetup([u, v, w], N):

 

Example 1.

Phi := Transformation(M, N, [u = x*y, v = 1/x + 1/y, w = x + y]);

_DGtransformation,M,0,N,0,,yx1x21y211,xy,u,1x+1y,v,x+y,w

(164)

DGinfo(Phi, "JacobianMatrix");

yx1x21y211

(165)

"RangeFrame": the range frame of a transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M): DGsetup([u, v], N):

Phi := Transformation(M, N, [u =x*y, v= 1/x + 1/y]);

_DGtransformation,M,0,N,0,,yx1x21y2,xy,u,1x+1y,v

(166)

DGinfo(Phi, "RangeFrame");

N

(167)

"RangeOrder": the jet space order of the range of a transformation.

with(DifferentialGeometry): with(Tools):

DGsetup([x, y], M): DGsetup([u, v], N): DGsetup([t], [w], J, 1):

 

Example 1.

Phi1 := Transformation(M, N, [u =x*y, v= 1/x + 1/y]);

_DGtransformation,M,0,N,0,,yx1x21y2,xy,u,1x+1y,v

(168)

DGinfo(Phi1, "RangeOrder");

0

(169)

Example 2.

Phi2 := Transformation(M, J, [t= x, w[]= y, w[1] = y^2]);

_DGtransformation,M,0,J,1,,100102y,x,t,y,w[],y2,w1

(170)

DGinfo(Phi2, "RangeOrder");

1

(171)

"RepresentationMatrices": the list of matrices defining a representation of a Lie Algebra

with(DifferentialGeometry): with(LieAlgebras): with(Tools):

 

DGsetup([x, y], V);

frame name: V

(172)

R := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [0, 1]])];

1000,0100,0001

(173)

L := LieAlgebraData(R, Alg);

Le1,e2=e2,e2,e3=e2

(174)

DGsetup(L);

Lie algebra: Alg

(175)

rho := Representation(Alg, V, R);

_DGRepresentation,Alg,0,V,0,,1000,0100,0001

(176)

DGinfo(rho, "RepresentationMatrices");

1000,0100,0001

(177)

"TransformationType": the type (projectionable, point, contact, ...) of a transformation.  See AssignTransformationType for details.

with(DifferentialGeometry): with(Tools): with(JetCalculus):

DGsetup([x], [u], E, 1):

 

Example 1.

Phi1 := Transformation(E, E, [x = x^2, u[] = u[]*x]);

_DGtransformation,E,0,E,0,,2x0u[]x,x2,x,u[]x,u[]

(178)

Phi1A := AssignTransformationType(Phi1);

_DGtransformation,E,0,E,0,projectable,0,2x0u[]x,x2,x,u[]x,u[]

(179)

DGinfo(Phi1A, "TransformationType");

projectable,0

(180)

Example 2.

Phi2 := Transformation(E, E, [x = u[], u[] = x]);

_DGtransformation,E,0,E,0,,0110,u[],x,x,u[]

(181)

Phi2A := AssignTransformationType(Phi2);

_DGtransformation,E,0,E,0,point,0,0110,u[],x,x,u[]

(182)

DGinfo(Phi2A, "TransformationType");

point,0

(183)

Example 3.

Phi3 := Transformation(E, E, [x = -2*u[1] + x, u[] = -u[1]^2 + u[], u[1] = u[1]]);

_DGtransformation,E,1,E,1,,102012u1001,2u1+x,x,u12+u[],u[],u1,u1

(184)

Phi3A := AssignTransformationType(Phi3);

_DGtransformation,E,1,E,1,contact,1,102012u1001,2u1+x,x,u12+u[],u[],u1,u1

(185)

DGinfo(Phi3A, "TransformationType");

contact,1

(186)

Frame Information

"AbstractForms": the list of forms defined in an abstract frame.

with(DifferentialGeometry): with(Tools):

 

DGsetup([[sigma1, sigma2, sigma3], tau = dgform(2), xi = dgform(3)], [], P):

 

Example 1.

DGinfo(P, "AbstractForms");

_DGform,P,1,1,1,_DGform,P,1,2,1,_DGform,P,1,3,1,_DGform,P,2,4,1,_DGform,P,3,5,1

(187)

Example 2.

ExteriorDerivative(tau);

_DGform,P,3,6,1

(188)

DGinfo(P, "AbstractForms");

_DGform,P,1,1,1,_DGform,P,1,2,1,_DGform,P,1,3,1,_DGform,P,2,4,1,_DGform,P,3,5,1,_DGform,P,3,d,4,6,1

(189)

Example 3.

Hook(D_sigma1, xi);

_DGform,P,2,7,1

(190)

Hook([D_sigma2, D_sigma3], xi);

_DGform,P,1,9,1

(191)

DGinfo(P, "AbstractForms");

_DGform,P,1,1,1,_DGform,P,1,2,1,_DGform,P,1,3,1,_DGform,P,2,4,1,_DGform,P,3,5,1,_DGform,P,3,d,4,6,1,_DGform,P,2,hook,1,5,7,1,_DGform,P,2,hook,2,5,8,1,_DGform,P,1,hook,2,3,5,9,1

(192)

"CoefficientGrading": the grading of the coefficients in a Lie algebra with coefficients

with(DifferentialGeometry): with(LieAlgebras): with(Tools):

 

Example 1.

LD:=LieAlgebraData([[x3, x4] = e1, [x3, x5] = x2, [x4, x5] = x3], [x1,x2,x3,x4,x5], Alg, grading = [-3, -3, -2, -1, -1]);

LDe3,e5=e2,e4,e5=e3

(193)

DGsetup(LD):

DGsetup([w1, w2, w3, w4, w5], V, grading = [-3, -3, -2, -1, -1]):

rho:= Representation(Alg, V, Adjoint(Alg));

_DGRepresentation,Alg,0,V,0,,0000000000000000000000000,0000000000000000000000000,0000000001000000000000000,0000000000000010000000000,0000000100000100000000000

(194)

DGsetup(Alg, rho, AlgV):

DGinfo(AlgV, "CoefficientGrading");

tablew1=−3,w3=−2,w2=−3,w4=−1,w5=−1

(195)

DGinfo(AlgV, "CoefficientGrading", output = "list");

−3,−3,−2,−1,−1

(196)

DGinfo(output = "coefficients", "CoefficientGrading");

w1−3,w2−3,w3−2,w4−1,w5−1

(197)

"CoframeLabels": list the labels used to input and display the basis of 1-forms for the cotangent space.

with(DifferentialGeometry): with(Tools):

 

Example 1.

DGsetup([x, y, z], M):

DGinfo(M, "CoframeLabels");

dx,,dy,,dz,

(198)

Example 2.

DGsetup([x, y, z], M):

F := FrameData([y*dx, z*dy, dz], N);

FdΘ1=Θ1Θ2yz,dΘ2=Θ2Θ3z,dΘ3=0

(199)

DGsetup(F, [X], [omega]):

DGinfo(N, "CoframeLabels");

omega1,,omega2,,omega3,

(200)

Example 3.

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

DGinfo(E, "CoframeLabels");

dx,,dy,,du,,du,1,du,2,du,1,1,du,1,2,du,2,2

(201)

Example 4.

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

L1e2,e4=e1,e3,e4=e3

(202)

DGsetup(L1):

DGinfo(Alg1, "CoframeLabels");

theta1,,theta2,,theta3,,theta4,

(203)

"CurrentFrame": the name of the currently active frame.

with(DifferentialGeometry): with(Tools):

DGsetup([x], M): DGsetup([y], N):

DGinfo("CurrentFrame");

N

(204)

LieBracket(D_x, x*D_x);

_DGvector,M,,1,1

(205)

DGinfo("CurrentFrame");

M

(206)

"ExteriorDerivativeFormStructureEquations": list the formulas for the exterior derivatives of a frame with protocol "LieAlgebra" or "AnholonomicFrame".

with(DifferentialGeometry): with(Tools):

 

Example 1.

DGsetup([x, y, z], M):

DGinfo("ExteriorDerivativeFormStructureEquations");

(207)

Example 2.

DGsetup([x, y, z], M):

F := FrameData([y*dx, z*dy, dz], N);

FdΘ1=Θ1Θ2yz,dΘ2=Θ2Θ3z,dΘ3=0

(208)

DGsetup(F, [X], [omega]):

DGinfo(N, "ExteriorDerivativeFormStructureEquations");

_DGform,N,2,1,2,1yz,_DGform,N,2,2,3,1z,_DGform,N,2,1,2,0

(209)

Example 3.

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

L1e2,e4=e1,e3,e4=e3

(210)

DGsetup(L1):

DGinfo(Alg1, "ExteriorDerivativeFormStructureEquations");

_DGform,Alg1,2,2,4,1,_DGform,Alg1,2,1,2,0,_DGform,Alg1,2,3,4,1,_DGform,Alg1,2,1,2,0

(211)

"ExteriorDerivativeFunctionStructureEquations": list the formulas for the exterior derivatives of the coordinate functions for a frame with protocol or "moving_frame".

with(DifferentialGeometry): with(Tools):

 

Example 1.

DGsetup([x, y, z], M):

DGinfo("ExteriorDerivativeFunctionStructureEquations");

(212)

Example 2.

DGsetup([x, y, z], M):
F := FrameData([y*dx, z*dy, dz], N);

FdΘ1=Θ1Θ2yz,dΘ2=Θ2Θ3z,dΘ3=0

(213)

DGsetup(F, [X], [omega]):

DGinfo(N, "ExteriorDerivativeFunctionStructureEquations");

_DGform,N,1,1,1y,_DGform,N,1,2,1z,_DGform,N,1,3,1

(214)

Example 3.

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

L1e2,e4=e1,e3,e4=e3

(215)

DGsetup(L1):

DGinfo(Alg1, "ExteriorDerivativeFunctionStructureEquations");

(216)

"FrameBaseDimension": the dimension of the base manifold M for a frame defining a bundle E -> M; the dimension of M for a frame defining a manifold M; the dimension of the Lie algebra for a frame defining a Lie algebra.

with(DifferentialGeometry): with(Tools):

 

Example 1.

DGsetup([x, y, z], M):

DGinfo(M, "FrameBaseDimension");

3

(217)

 

Example 2.

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

DGinfo(E, "FrameBaseDimension");

2

(218)

Example 3.

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

L1e2,e4=e1,e3,e4=e3

(219)

DGsetup(L1):

DGinfo(Alg1, "FrameBaseDimension");

4

(220)

"FrameBaseForms": the basis 1-forms for the cotangent space of the base manifold M for a frame defining a bundle E -> M; the basis 1-forms for the cotangent space M for a frame defining a manifold M; the dual 1-forms of a Lie algebra for a frame defining a Lie algebra.

with(DifferentialGeometry): with(Tools):

 

Example 1.

DGsetup([x, y, z], M):

DGinfo(M, "FrameBaseForms");

_DGform,M,1,1,1,_DGform,M,1,2,1,_DGform,M,1,3,1

(221)

 

Example 2.

DGsetup([x, y, z], M):
F := FrameData([y*dx, z*dy, dz], N);

FdΘ1=Θ1Θ2yz,dΘ2=Θ2Θ3z,dΘ3=0

(222)

DGsetup(F, [X], [omega]):

DGinfo(N, "FrameBaseForms");

_DGform,N,1,1,1,_DGform,N,1,2,1,_DGform,N,1,3,1

(223)

 

Example 3.

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

DGinfo(E, "FrameBaseForms");

_DGform,E,1,1,1,_DGform,E,1,2,1

(224)

 

Example 4.

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

L1e2,e4=e1,e3,e4=e3

(225)

DGsetup(L1):

DGinfo(Alg1, "FrameBaseForms");

_DGform,Alg1,1,1,1,_DGform,Alg1,1,2,1,_DGform,Alg1,1,3,1,_DGform,Alg1,1,4,1

(226)

"FrameBaseVectors": the basis vectors for the tangent space of the base manifold M for a frame defining a bundle E -> M; the basis vectors for the tangent space M for a frame defining a manifold M; the basis vectors of a Lie algebra for a frame defining a Lie algebra.

with(DifferentialGeometry): with(Tools):

 

Example 1.

DGsetup([x, y, z], M):

DGinfo(M, "FrameBaseVectors");

_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,3,1

(227)

Example 2.

DGsetup([x, y, z], M):

F := FrameData([y*dx, z*dy, dz], N);

FdΘ1=Θ1Θ2yz,dΘ2=Θ2Θ3z,dΘ3=0

(228)

DGsetup(F, [W], [omega]):

DGinfo(N, "FrameBaseVectors");

_DGvector,N,,1,1,_DGvector,N,,2,1,_DGvector,N,,3,1

(229)

Example 3.

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

DGinfo(E, "FrameBaseVectors");

_DGvector,E,,1,1,_DGvector,E,,2,1

(230)

Example 4.

L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);

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