DGinfo - Maple Help

DifferentialGeometry[Tools]

 DGinfo

 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:

 • 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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (1)
 > DGinfo(alpha1, "BiformDegree");
 $\left[{2}{,}{0}\right]$ (2)

Example 2.

 > alpha2 := evalDG(Dx &wedge Cu[]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (3)
 > DGinfo(alpha2, "BiformDegree");
 $\left[{1}{,}{1}\right]$ (4)

Example 3.

 > alpha3 := evalDG(Cu[1] &wedge Cu[2]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{0}{,}{2}\right]\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)$ (5)
 > DGinfo(alpha3, "BiformDegree");
 $\left[{0}{,}{2}\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{a}\right]{,}\left[\left[{1}{,}{3}\right]{,}{b}\right]{,}\left[\left[{1}{,}{4}\right]{,}{d}\right]{,}\left[\left[{2}{,}{3}\right]{,}{c}\right]{,}\left[\left[{3}{,}{4}\right]{,}{e}\right]\right]\right]\right)$ (7)
 > DGinfo(alpha, "CoefficientList", "all");
 $\left[{a}{,}{b}{,}{d}{,}{c}{,}{e}\right]$ (8)
 > DGinfo(alpha, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);
 $\left[{a}{,}{b}{,}{0}\right]$ (9)
 > DGinfo(alpha, "CoefficientList", [[1,2], [1,3], [2,4]]);
 $\left[{a}{,}{b}{,}{0}\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{a}\right]{,}\left[\left[{1}{,}{3}\right]{,}{b}\right]{,}\left[\left[{1}{,}{4}\right]{,}{c}\right]{,}\left[\left[{2}{,}{3}\right]{,}{b}\right]{,}\left[\left[{3}{,}{4}\right]{,}{a}\right]\right]\right]\right)$ (11)
 > DGinfo(alpha, "CoefficientSet");
 $\left\{{a}{,}{b}{,}{c}\right\}$ (12)

Example 2.

 > X := DGzero("vector");
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)$ (13)
 > DGinfo(X, "CoefficientSet");
 $\left\{{0}\right\}$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{a}\right]{,}\left[\left[{1}{,}{3}\right]{,}{b}\right]{,}\left[\left[{1}{,}{4}\right]{,}{d}\right]{,}\left[\left[{2}{,}{3}\right]{,}{c}\right]{,}\left[\left[{3}{,}{4}\right]{,}{e}\right]\right]\right]\right)$ (15)
 > DGinfo(alpha, "CoefficientList", "all");
 $\left[{a}{,}{b}{,}{d}{,}{c}{,}{e}\right]$ (16)
 > DGinfo(alpha, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);
 $\left[{a}{,}{b}{,}{0}\right]$ (17)
 > DGinfo(alpha, "CoefficientList", [[1,2], [1,3], [2,4]]);
 $\left[{a}{,}{b}{,}{0}\right]$ (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]]);
 ${\mathrm{\Delta }}{≔}\left[\left\{{{u}}_{{1}}{,}{{v}}_{{1}}\right\}{,}\left[{{u}}_{{2}}{+}{{v}}_{{1}}{,}{{u}}_{{1}}{-}{{v}}_{{2}}\right]\right]$ (19)
 > DGinfo(Delta, "DiffeqType");
 $\left[{"evolutionary"}{,}{0}\right]$ (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]]);
 ${\mathrm{\Delta }}{≔}\left[\left\{{{u}}_{{1}}{,}{{v}}_{{1}}\right\}{,}\left[{{u}}_{{2}}{+}{{v}}_{{1}}{,}{{u}}_{{1}}{-}{{v}}_{{2}}\right]\right]$ (21)
 > DGinfo(Delta, "DiffeqVariables");
 $\left\{{{u}}_{{1}}{,}{{v}}_{{1}}\right\}$ (22)

"FormDegree": the degree of a differential form

 > with(DifferentialGeometry): with(Tools):
 > DGsetup([x, y, z], M):

Example 1.

 > alpha := evalDG(dx +dy);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)$ (23)
 > DGinfo(alpha, "FormDegree");
 ${1}$ (24)

Example 2.

 > beta := evalDG(3*dx &w dy + 4*dy &w dz - dx &w dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{3}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{4}\right]\right]\right]\right)$ (25)
 > DGinfo(beta, "FormDegree");
 ${2}$ (26)

Example 3.

 > nu := evalDG(dx &w dy &w dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (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;
 ${\mathrm{f1}}{≔}{u}{}{x}{+}{v}{}{y}$ (29)
 > DGinfo(f1, "FunctionOrder");
 ${0}$ (30)

Example 2.

 > DGsetup([x, y], [u], J, 1):
 > f1 := u[1]*x + u[2];
 ${\mathrm{f1}}{≔}{{u}}_{{1}}{}{x}{+}{{u}}_{{2}}$ (31)
 > DGinfo(f1, "FunctionOrder");
 ${1}$ (32)

Example 3.

 > f1 := u[1, 1, 2]*x + u[2, 2, 2, 2];
 ${\mathrm{f1}}{≔}{{u}}_{{1}{,}{1}{,}{2}}{}{x}{+}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]\right]\right]\right)$ (35)
 > DGinfo(X, "ObjectAttributes");
 $\left[{"vector"}{,}{M}{,}\left[\right]\right]$ (36)

Example 2.

 > alpha := evalDG(d*dx &w dy + e*dx &w dz + f*dy &w dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{d}\right]{,}\left[\left[{1}{,}{3}\right]{,}{e}\right]{,}\left[\left[{2}{,}{3}\right]{,}{f}\right]\right]\right]\right)$ (37)
 > DGinfo(alpha, "ObjectAttributes");
 $\left[{"form"}{,}{M}{,}{2}\right]$ (38)

Example 3.

 > T := evalDG(r*D_x &t dx + s*D_z&t dy + t*D_z &t dx);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{r}\right]{,}\left[\left[{3}{,}{1}\right]{,}{t}\right]{,}\left[\left[{3}{,}{2}\right]{,}{s}\right]\right]\right]\right)$ (39)
 > DGinfo(T, "ObjectAttributes");
 $\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]$ (40)

Example 4.

 > Phi := Transformation(M, N, [u = x^2 + y^2 + z^2, v = x*y*z]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}2{}x& 2{}y& 2{}z\\ y{}z& x{}z& x{}y\end{array}\right]\right]\right]{,}\left[\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{,}{u}\right]{,}\left[{x}{}{y}{}{z}{,}{v}\right]\right]\right]\right)$ (41)
 > DGinfo(Phi, "ObjectAttributes");
 $\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}2{}x& 2{}y& 2{}z\\ y{}z& x{}z& x{}y\end{array}\right]\right]\right]$ (42)

Example 5.

 > DGsetup([[gamma1, gamma2, gamma3, gamma4], chi = dgform(3)], [], P):
 > ExteriorDerivative(chi);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{4}\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right)$ (43)
 > DGinfo(dchi, "ObjectAttributes");
 $\left[{"form"}{,}{P}{,}{4}{,}{"d"}{,}{5}\right]$ (44)

Example 6.

 > Hook(D_gamma1, chi);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{7}\right]{,}{1}\right]\right]\right]\right)$ (45)
 > DGinfo(i_1chi, "ObjectAttributes");
 $\left[{"form"}{,}{P}{,}{2}{,}{"hook"}{,}\left[\left[{1}\right]{,}{5}\right]\right]$ (46)

Example 7.

 > Hook([D_gamma2, D_gamma3], chi);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{9}\right]{,}{1}\right]\right]\right]\right)$ (47)
 > DGinfo(i_2_3chi, "ObjectAttributes");
 $\left[{"form"}{,}{P}{,}{1}{,}{"hook"}{,}\left[\left[{2}{,}{3}\right]{,}{5}\right]\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]\right]\right]\right)$ (49)
 > DGinfo(X, "ObjectComponents");
 $\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]\right]$ (50)

Example 2.

 > alpha := evalDG(d*dx &w dy + e*dx &w dz + f*dy &w dz);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{d}\right]{,}\left[\left[{1}{,}{3}\right]{,}{e}\right]{,}\left[\left[{2}{,}{3}\right]{,}{f}\right]\right]\right]\right)$ (51)
 > DGinfo(alpha, "ObjectComponents");
 $\left[\left[\left[{1}{,}{2}\right]{,}{d}\right]{,}\left[\left[{1}{,}{3}\right]{,}{e}\right]{,}\left[\left[{2}{,}{3}\right]{,}{f}\right]\right]$ (52)

Example 3.

 > T := evalDG(r*D_x &t dx + s*D_z&t dy + t*D_z &t dx);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{r}\right]{,}\left[\left[{3}{,}{1}\right]{,}{t}\right]{,}\left[\left[{3}{,}{2}\right]{,}{s}\right]\right]\right]\right)$ (53)
 > DGinfo(T, "ObjectComponents");
 $\left[\left[\left[{1}{,}{1}\right]{,}{r}\right]{,}\left[\left[{3}{,}{1}\right]{,}{t}\right]{,}\left[\left[{3}{,}{2}\right]{,}{s}\right]\right]$ (54)

Example 4.

 > Phi := Transformation(M, N, [u = x^2 + y^2 + z^2, v= x*y*z]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}2{}x& 2{}y& 2{}z\\ y{}z& x{}z& x{}y\end{array}\right]\right]\right]{,}\left[\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{,}{u}\right]{,}\left[{x}{}{y}{}{z}{,}{v}\right]\right]\right]\right)$ (55)
 > DGinfo(Phi, "ObjectComponents");
 $\left[\left[{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{,}{u}\right]{,}\left[{x}{}{y}{}{z}{,}{v}\right]\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}{-}{3}\right]\right]\right]\right)$ (57)
 > DGinfo(X, "ObjectFrame");
 ${M}$ (58)

Example 2.

 > T := evalDG(D_r &t D_s &t dt);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{N}{,}\left[\left[{"con_bas"}{,}{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{y}\right]{,}\left[\left[{3}\right]{,}{z}\right]\right]\right]\right)$ (61)
 > DGinfo(X, "ObjectGenerators");
 $\left\{{1}{,}{3}\right\}$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{w}\right]{,}\left[\left[{1}{,}{4}\right]{,}{x}\right]{,}\left[\left[{2}{,}{4}\right]{,}{{z}}^{{2}}\right]\right]\right]\right)$ (63)
 > DGinfo(alpha, "ObjectGenerators");
 $\left\{{1}{,}{2}{,}{4}\right\}$ (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;
 ${f}{≔}{{u}}_{{2}}{}{x}{+}{v}\left[\right]{}{y}$ (65)
 > DGinfo(f, "ObjectOrder");
 ${1}$ (66)

Example 2.

 > alpha := evalDG(du[1]*x + du[2,2]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{x}\right]{,}\left[\left[{11}\right]{,}{1}\right]\right]\right]\right)$ (67)
 > DGinfo(alpha, "ObjectOrder");
 ${2}$ (68)

Example 3.

 > beta := evalDG(du[1, 1, 2]*x + u[2, 2, 2, 2]*dy);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right]{,}\left[\left[{16}\right]{,}{x}\right]\right]\right]\right)$ (69)
 > DGinfo(beta, "FunctionOrder");
 ${4}$ (70)

Example 4.

 > X := evalDG(u[1]*D_u[]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{{u}}_{{1}}\right]\right]\right]\right)$ (71)
 > X3 := Prolong(X, 2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"evolutionary"}{,}{2}\right]\right]{,}\left[\left[\left[{3}\right]{,}{{u}}_{{1}}\right]{,}\left[\left[{5}\right]{,}{{u}}_{{1}{,}{1}}\right]{,}\left[\left[{6}\right]{,}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{9}\right]{,}{{u}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{10}\right]{,}{{u}}_{{1}{,}{1}{,}{2}}\right]{,}\left[\left[{11}\right]{,}{{u}}_{{1}{,}{2}{,}{2}}\right]\right]\right]\right)$ (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);
 ${f}{≔}{\mathrm{sin}}{}\left({x}\right){}{\mathrm{cos}}{}\left({y}\right)$ (74)
 > DGinfo(f, "ObjectType");
 ${"DGscalar"}$ (75)

Example 2.

 > X := evalDG(D_x + y*D_u[]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{y}\right]\right]\right]\right)$ (76)
 > DGinfo(X, "ObjectType");
 ${"vector"}$ (77)

Example 3.

 > alpha := evalDG(dx &w dy + dx &w du[1]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (78)
 > DGinfo(alpha, "ObjectType");
 ${"form"}$ (79)

Example 4.

 > theta := evalDG(Dx &w Cu[1]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{M}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (80)
 > DGinfo(theta, "ObjectType");
 ${"biform"}$ (81)

Example 5.

 > T := evalDG(dx &t D_y + dy &t D_y);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (82)
 > DGinfo(T, "ObjectType");
 ${"tensor"}$ (83)

Example 6.

 > A := Vector([1, 2]);
 $\left[\begin{array}{r}1\\ 2\end{array}\right]$ (84)
 > DGinfo(A, "ObjectType");
 ${"Vector"}$ (85)

Example 7.

 > B := Matrix([1, 2]);
 $\left[\begin{array}{rr}1& 2\end{array}\right]$ (86)
 > DGinfo(B, "ObjectType");
 ${"Matrix"}$ (87)

Example 8.

 > B:= Matrix([[1, 2], [3, 4]]);
 $\left[\begin{array}{rr}1& 2\\ 3& 4\end{array}\right]$ (88)
 > DGinfo(B, "ObjectType");
 ${"Matrix"}$ (89)

Example 9.

 > Phi := IdentityTransformation();
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rrr}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{x}\right]{,}\left[{y}{,}{y}\right]{,}\left[{{u}}_{{[}{]}}{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (90)
 > DGinfo(Phi, "ObjectType");
 ${"transformation"}$ (91)

Example 10.

 > L := LieAlgebraData(evalDG([D_x, x*D_x, x^2*D_x]));
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}\right]$ (92)
 > DGinfo(L, "ObjectType");
 ${"LieAlgebra"}$ (93)

Example 11.

 > DGsetup([x, y], M);
 ${\mathrm{frame name: M}}$ (94)
 > F := FrameData([y*dx, x*dy], N);
 ${F}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{x}}{,}{d}{}{\mathrm{Θ2}}{=}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{x}}\right]$ (95)
 > DGinfo(F, "ObjectType");
 ${"moving_frame"}$ (96)

Example 12.

 > DGsetup([x, y], V);
 ${\mathrm{frame name: V}}$ (97)
 > R := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [0, 1]])];
 $\left[\left[\begin{array}{rr}1& 0\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 0\\ 0& 1\end{array}\right]\right]$ (98)
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (99)
 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg}}$ (100)
 > rho := Representation(Alg, V, R);
 ${\mathrm{_DG}}{}\left(\left[\left[{"Representation"}{,}\left[\left[{\mathrm{Alg}}{,}{0}\right]{,}\left[{V}{,}{0}\right]\right]{,}\left[{}\right]\right]{,}\left[\left[\begin{array}{rr}1& 0\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 0\\ 0& 1\end{array}\right]\right]\right]\right)$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (103)

Example 1.

 > rho := MetricDensity(g, 3);
 ${\mathrm{\rho }}{≔}{1}$ (104)
 > T1 := D_x &tensor rho;
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{3}\right]\right]\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right)$ (105)
 > DGinfo(T1, "TensorDensityType");
 $\left[\left[{"bas"}{,}{3}\right]\right]$ (106)

Example 2.

 > T2 := PermutationSymbol("cov_vrt");
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\left[{"vrt"}{,}{-}{1}\right]\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{1}\right]{,}\left[\left[{3}{,}{5}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{3}{,}{5}\right]{,}{-}{1}\right]{,}\left[\left[{4}{,}{5}{,}{3}\right]{,}{1}\right]{,}\left[\left[{5}{,}{3}{,}{4}\right]{,}{1}\right]{,}\left[\left[{5}{,}{4}{,}{3}\right]{,}{-}{1}\right]\right]\right]\right)$ (107)
 > DGinfo(T2, "TensorDensityType");
 $\left[\left[{"vrt"}{,}{-1}\right]\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}\right]{,}{1}\right]\right]\right]\right)$ (109)
 > DGinfo(T1, "TensorGenerators");
 $\left\{\left[{"cov_bas"}{,}{1}\right]\right\}$ (110)

Example 2.

 > T2 := evalDG(dx &t D_y &t dy);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (111)
 > DGinfo(T2, "TensorGenerators");
 $\left\{\left[{"con_bas"}{,}{2}\right]{,}\left[{"cov_bas"}{,}{1}\right]{,}\left[{"cov_bas"}{,}{2}\right]\right\}$ (112)

Example 3.

 > T3 := evalDG(du &t D_v &t dy);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"con_vrt"}{,}{"cov_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (113)
 > DGinfo(T3, "TensorGenerators");
 $\left\{\left[{"con_vrt"}{,}{4}\right]{,}\left[{"cov_bas"}{,}{2}\right]{,}\left[{"cov_vrt"}{,}{3}\right]\right\}$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (123)
 > DGinfo(T1, "TensorIndexType");
 $\left[{"con_bas"}{,}{"con_bas"}\right]$ (124)

Example 2.

 > T2 := evalDG(dx &t D_y);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (125)
 > DGinfo(T2, "TensorIndexType");
 $\left[{"cov_bas"}{,}{"con_bas"}\right]$ (126)

Example 3.

 > T3 := evalDG(du &t D_v);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"con_vrt"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (127)
 > DGinfo(T3, "TensorIndexType");
 $\left[{"cov_vrt"}{,}{"con_vrt"}\right]$ (128)

Example 4.

 > T4 := evalDG(D_x &t D_v);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}{,}{"con_vrt"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (129)
 > DGinfo(T4, "TensorIndexType");
 $\left[{"con_bas"}{,}{"con_vrt"}\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right)$ (131)
 > DGinfo(T1, "TensorType");
 $\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\right]\right]$ (132)

Example 2.

 > T2 := PermutationSymbol("con_bas");
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]{,}\left[\left[{2}{,}{1}\right]{,}{-}{1}\right]\right]\right]\right)$ (133)
 > DGinfo(T2, "TensorType");
 $\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\left[{"bas"}{,}{1}\right]\right]\right]$ (134)

Example 3.

 > T3 := evalDG(du &t D_v);
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"con_vrt"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (135)
 > DGinfo(T3, "TensorType");
 $\left[\left[{"cov_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]$ (136)

Example 4.

 > T4 := PermutationSymbol("cov_vrt");
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\left[{"vrt"}{,}{-}{1}\right]\right]\right]\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]{,}\left[\left[{4}{,}{3}\right]{,}{-}{1}\right]\right]\right]\right)$ (137)
 > DGinfo(T4, "TensorType");
 $\left[\left[{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\left[{"vrt"}{,}{-1}\right]\right]\right]$ (138)

Example 5.

 > T5 := PermutationSymbol("cov_bas") &tensor PermutationSymbol("con_vrt");
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{E}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\left[{"bas"}{,}{-}{1}\right]{,}\left[{"vrt"}{,}{1}\right]\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{1}\right]{,}\left[\left[{1}{,}{2}{,}{4}{,}{3}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}{,}{3}{,}{4}\right]{,}{-}{1}\right]{,}\left[\left[{2}{,}{1}{,}{4}{,}{3}\right]{,}{1}\right]\right]\right]\right)$ (139)
 > DGinfo(T5, "TensorType");
 $\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\left[{"bas"}{,}{-1}\right]{,}\left[{"vrt"}{,}{1}\right]\right]\right]$ (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[]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (141)
 > X1a := AssignVectorType(X1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"projectable"}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (142)
 > DGinfo(X1a, "VectorType");
 $\left[{"projectable"}{,}{0}\right]$ (143)

Example 2.

 > X2 := Prolong(u[]*D_x - x *D_u[], 1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"point"}{,}{1}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{[}{]}}\right]{,}\left[\left[{3}\right]{,}{-}{x}\right]{,}\left[\left[{4}\right]{,}{-}{{u}}_{{1}}^{{2}}{-}{1}\right]{,}\left[\left[{5}\right]{,}{-}{{u}}_{{2}}{}{{u}}_{{1}}\right]\right]\right]\right)$ (144)
 > DGinfo(X2, "VectorType");
 $\left[{"point"}{,}{1}\right]$ (145)

Example 3.

 > X3 := GeneratingFunctionToContactVector(u[]*u[1]*u[2]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{2}}\right]{,}\left[\left[{2}\right]{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{1}}\right]{,}\left[\left[{3}\right]{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{1}}{}{{u}}_{{2}}\right]{,}\left[\left[{4}\right]{,}{{u}}_{{1}}^{{2}}{}{{u}}_{{2}}\right]{,}\left[\left[{5}\right]{,}{{u}}_{{2}}^{{2}}{}{{u}}_{{1}}\right]\right]\right]\right)$ (146)
 > X3a := AssignVectorType(X3);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"contact"}{,}{1}\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{2}}\right]{,}\left[\left[{2}\right]{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{1}}\right]{,}\left[\left[{3}\right]{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{1}}{}{{u}}_{{2}}\right]{,}\left[\left[{4}\right]{,}{{u}}_{{1}}^{{2}}{}{{u}}_{{2}}\right]{,}\left[\left[{5}\right]{,}{{u}}_{{2}}^{{2}}{}{{u}}_{{1}}\right]\right]\right]\right)$ (147)
 > DGinfo(X3a, "VectorType");
 $\left[{"contact"}{,}{1}\right]$ (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]);
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (149)
 > DGsetup(LD):
 > DGsetup([w1, w2, w3, w4, w5], V, grading = [-3, -3, -2, -1, -1]):
 ${\mathrm{_DG}}{}\left(\left[\left[{"Representation"}{,}\left[\left[{\mathrm{Alg}}{,}{0}\right]{,}\left[{V}{,}{0}\right]\right]{,}\left[{}\right]\right]{,}\left[\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\right]\right]\right)$ (150)
 > DGsetup(Alg, rho, AlgV);
 ${\mathrm{Lie algebra with coefficients: AlgV}}$ (151)

Example 1.

 > alpha1 := evalDG(w3 *theta5);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{AlgV}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{\mathrm{w3}}\right]\right]\right]\right)$ (152)
 > DGinfo(alpha1, "Weight");
 ${-1}$ (153)

Example 2.

 > alpha2 := evalDG(w3 * theta1 &w theta5);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{AlgV}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{5}\right]{,}{\mathrm{w3}}\right]\right]\right]\right)$ (154)
 > DGinfo(alpha2, "Weight");
 ${2}$ (155)

Example 3.

 > alpha3 := evalDG(w1 * theta3 &w  &w theta4 &w theta5);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{AlgV}}{,}{3}\right]{,}\left[\left[\left[{3}{,}{4}{,}{5}\right]{,}{\mathrm{w1}}\right]\right]\right]\right)$ (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]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}y& x\\ -\frac{1}{{x}^{2}}& -\frac{1}{{y}^{2}}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{u}\right]{,}\left[\frac{{1}}{{x}}{+}\frac{{1}}{{y}}{,}{v}\right]\right]\right]\right)$ (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]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}y& x\\ -\frac{1}{{x}^{2}}& -\frac{1}{{y}^{2}}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{u}\right]{,}\left[\frac{{1}}{{x}}{+}\frac{{1}}{{y}}{,}{v}\right]\right]\right]\right)$ (160)
 > DGinfo(Phi1, "DomainOrder");
 ${0}$ (161)

Example 2.

 > Phi2 := Transformation(J, M, [x = w[1], y = w[1, 1]]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{J}{,}{2}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rrrr}0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\right]\right]{,}\left[\left[{{w}}_{{1}}{,}{x}\right]{,}\left[{{w}}_{{1}{,}{1}}{,}{y}\right]\right]\right]\right)$ (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]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}y& x\\ -\frac{1}{{x}^{2}}& -\frac{1}{{y}^{2}}\\ 1& 1\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{u}\right]{,}\left[\frac{{1}}{{x}}{+}\frac{{1}}{{y}}{,}{v}\right]{,}\left[{x}{+}{y}{,}{w}\right]\right]\right]\right)$ (164)
 > DGinfo(Phi, "JacobianMatrix");
 $\left[\begin{array}{cc}y& x\\ -\frac{1}{{x}^{2}}& -\frac{1}{{y}^{2}}\\ 1& 1\end{array}\right]$ (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]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}y& x\\ -\frac{1}{{x}^{2}}& -\frac{1}{{y}^{2}}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{u}\right]{,}\left[\frac{{1}}{{x}}{+}\frac{{1}}{{y}}{,}{v}\right]\right]\right]\right)$ (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]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}y& x\\ -\frac{1}{{x}^{2}}& -\frac{1}{{y}^{2}}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{u}\right]{,}\left[\frac{{1}}{{x}}{+}\frac{{1}}{{y}}{,}{v}\right]\right]\right]\right)$ (168)
 > DGinfo(Phi1, "RangeOrder");
 ${0}$ (169)

Example 2.

 > Phi2 := Transformation(M, J, [t= x, w[]= y, w[1] = y^2]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{J}{,}{1}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 2{}y\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{t}\right]{,}\left[{y}{,}{{w}}_{{[}{]}}\right]{,}\left[{{y}}^{{2}}{,}{{w}}_{{1}}\right]\right]\right]\right)$ (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);
 ${\mathrm{frame name: V}}$ (172)
 > R := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [0, 1]])];
 $\left[\left[\begin{array}{rr}1& 0\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 0\\ 0& 1\end{array}\right]\right]$ (173)
 ${L}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}\right]$ (174)
 > DGsetup(L);
 ${\mathrm{Lie algebra: Alg}}$ (175)
 > rho := Representation(Alg, V, R);
 ${\mathrm{_DG}}{}\left(\left[\left[{"Representation"}{,}\left[\left[{\mathrm{Alg}}{,}{0}\right]{,}\left[{V}{,}{0}\right]\right]{,}\left[{}\right]\right]{,}\left[\left[\begin{array}{rr}1& 0\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 0\\ 0& 1\end{array}\right]\right]\right]\right)$ (176)
 > DGinfo(rho, "RepresentationMatrices");
 $\left[\left[\begin{array}{rr}1& 0\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 1\\ 0& 0\end{array}\right]{,}\left[\begin{array}{rr}0& 0\\ 0& 1\end{array}\right]\right]$ (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]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{E}{,}{0}\right]{,}\left[{E}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}2{}x& 0\\ {u}_{[]}& x\end{array}\right]\right]\right]{,}\left[\left[{{x}}^{{2}}{,}{x}\right]{,}\left[{{u}}_{{[}{]}}{}{x}{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (178)
 > Phi1A := AssignTransformationType(Phi1);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{E}{,}{0}\right]{,}\left[{E}{,}{0}\right]\right]{,}\left[{"projectable"}{,}{0}\right]{,}\left[\left[\begin{array}{cc}2{}x& 0\\ {u}_{[]}& x\end{array}\right]\right]\right]{,}\left[\left[{{x}}^{{2}}{,}{x}\right]{,}\left[{{u}}_{{[}{]}}{}{x}{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (179)
 > DGinfo(Phi1A, "TransformationType");
 $\left[{"projectable"}{,}{0}\right]$ (180)

Example 2.

 > Phi2 := Transformation(E, E, [x = u[], u[] = x]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{E}{,}{0}\right]{,}\left[{E}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{[}{]}}{,}{x}\right]{,}\left[{x}{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (181)
 > Phi2A := AssignTransformationType(Phi2);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{E}{,}{0}\right]{,}\left[{E}{,}{0}\right]\right]{,}\left[{"point"}{,}{0}\right]{,}\left[\left[\begin{array}{rr}0& 1\\ 1& 0\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{[}{]}}{,}{x}\right]{,}\left[{x}{,}{{u}}_{{[}{]}}\right]\right]\right]\right)$ (182)
 > DGinfo(Phi2A, "TransformationType");
 $\left[{"point"}{,}{0}\right]$ (183)

Example 3.

 > Phi3 := Transformation(E, E, [x = -2*u[1] + x, u[] = -u[1]^2 + u[], u[1] = u[1]]);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{E}{,}{1}\right]{,}\left[{E}{,}{1}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& -2{}{u}_{1}\\ 0& 0& 1\end{array}\right]\right]\right]{,}\left[\left[{-}{2}{}{{u}}_{{1}}{+}{x}{,}{x}\right]{,}\left[{-}{{u}}_{{1}}^{{2}}{+}{{u}}_{{[}{]}}{,}{{u}}_{{[}{]}}\right]{,}\left[{{u}}_{{1}}{,}{{u}}_{{1}}\right]\right]\right]\right)$ (184)
 > Phi3A := AssignTransformationType(Phi3);
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{E}{,}{1}\right]{,}\left[{E}{,}{1}\right]\right]{,}\left[{"contact"}{,}{1}\right]{,}\left[\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& -2{}{u}_{1}\\ 0& 0& 1\end{array}\right]\right]\right]{,}\left[\left[{-}{2}{}{{u}}_{{1}}{+}{x}{,}{x}\right]{,}\left[{-}{{u}}_{{1}}^{{2}}{+}{{u}}_{{[}{]}}{,}{{u}}_{{[}{]}}\right]{,}\left[{{u}}_{{1}}{,}{{u}}_{{1}}\right]\right]\right]\right)$ (185)
 > DGinfo(Phi3A, "TransformationType");
 $\left[{"contact"}{,}{1}\right]$ (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");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{3}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]$ (187)

Example 2.

 > ExteriorDerivative(tau);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{3}\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right)$ (188)
 > DGinfo(P, "AbstractForms");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{3}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{3}{,}{"d"}{,}{4}\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right)\right]$ (189)

Example 3.

 > Hook(D_sigma1, xi);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{7}\right]{,}{1}\right]\right]\right]\right)$ (190)
 > Hook([D_sigma2, D_sigma3], xi);
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{9}\right]{,}{1}\right]\right]\right]\right)$ (191)
 > DGinfo(P, "AbstractForms");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{3}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{3}{,}{"d"}{,}{4}\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}{,}{"hook"}{,}\left[\left[{1}\right]{,}{5}\right]\right]{,}\left[\left[\left[{7}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{2}{,}{"hook"}{,}\left[\left[{2}\right]{,}{5}\right]\right]{,}\left[\left[\left[{8}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}{,}{"hook"}{,}\left[\left[{2}{,}{3}\right]{,}{5}\right]\right]{,}\left[\left[\left[{9}\right]{,}{1}\right]\right]\right]\right)\right]$ (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]);
 ${\mathrm{LD}}{≔}\left[\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (193)
 > DGsetup(LD):
 > DGsetup([w1, w2, w3, w4, w5], V, grading = [-3, -3, -2, -1, -1]):
 ${\mathrm{_DG}}{}\left(\left[\left[{"Representation"}{,}\left[\left[{\mathrm{Alg}}{,}{0}\right]{,}\left[{V}{,}{0}\right]\right]{,}\left[{}\right]\right]{,}\left[\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]{,}\left[\begin{array}{rrrrr}0& 0& 0& 0& 0\\ 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]\right]\right]\right)$ (194)
 > DGsetup(Alg, rho, AlgV):
 ${table}{}\left(\left[{\mathrm{w1}}{=}{-3}{,}{\mathrm{w3}}{=}{-2}{,}{\mathrm{w2}}{=}{-3}{,}{\mathrm{w4}}{=}{-1}{,}{\mathrm{w5}}{=}{-1}\right]\right)$ (195)
 > DGinfo(AlgV, "CoefficientGrading", output = "list");
 $\left[{-3}{,}{-3}{,}{-2}{,}{-1}{,}{-1}\right]$ (196)
 $\left[{{\mathrm{w1}}}_{{-3}}{,}{{\mathrm{w2}}}_{{-3}}{,}{{\mathrm{w3}}}_{{-2}}{,}{{\mathrm{w4}}}_{{-1}}{,}{{\mathrm{w5}}}_{{-1}}\right]$ (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");
 $\left[\left[{"dx"}{,}\left[\right]\right]{,}\left[{"dy"}{,}\left[\right]\right]{,}\left[{"dz"}{,}\left[\right]\right]\right]$ (198)

Example 2.

 > DGsetup([x, y, z], M):
 > F := FrameData([y*dx, z*dy, dz], N);
 ${F}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{z}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{z}}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (199)
 > DGsetup(F, [X], [omega]):
 > DGinfo(N, "CoframeLabels");
 $\left[\left[{"omega1"}{,}\left[\right]\right]{,}\left[{"omega2"}{,}\left[\right]\right]{,}\left[{"omega3"}{,}\left[\right]\right]\right]$ (200)

Example 3.

 > DGsetup([x, y], [u], E, 2):
 > DGinfo(E, "CoframeLabels");
 $\left[\left[{"dx"}{,}\left[\right]\right]{,}\left[{"dy"}{,}\left[\right]\right]{,}\left[{"du"}{,}\left[\left[\right]\right]\right]{,}\left[{"du"}{,}\left[\left[{1}\right]\right]\right]{,}\left[{"du"}{,}\left[\left[{2}\right]\right]\right]{,}\left[{"du"}{,}\left[\left[{1}{,}{1}\right]\right]\right]{,}\left[{"du"}{,}\left[\left[{1}{,}{2}\right]\right]\right]{,}\left[{"du"}{,}\left[\left[{2}{,}{2}\right]\right]\right]\right]$ (201)

Example 4.

 > L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (202)
 > DGsetup(L1):
 > DGinfo(Alg1, "CoframeLabels");
 $\left[\left[{"theta1"}{,}\left[\right]\right]{,}\left[{"theta2"}{,}\left[\right]\right]{,}\left[{"theta3"}{,}\left[\right]\right]{,}\left[{"theta4"}{,}\left[\right]\right]\right]$ (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);
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right)$ (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");
 $\left[\right]$ (207)

Example 2.

 > DGsetup([x, y, z], M):
 > F := FrameData([y*dx, z*dy, dz], N);
 ${F}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{z}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{z}}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (208)
 > DGsetup(F, [X], [omega]):
 > DGinfo(N, "ExteriorDerivativeFormStructureEquations");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}\frac{{1}}{{y}{}{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{-}\frac{{1}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{0}\right]\right]\right]\right)\right]$ (209)

Example 3.

 > L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (210)
 > DGsetup(L1):
 > DGinfo(Alg1, "ExteriorDerivativeFormStructureEquations");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{-}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{0}\right]\right]\right]\right)\right]$ (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");
 $\left[\right]$ (212)

Example 2.

 > DGsetup([x, y, z], M): F := FrameData([y*dx, z*dy, dz], N);
 ${F}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{z}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{z}}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (213)
 > DGsetup(F, [X], [omega]):
 > DGinfo(N, "ExteriorDerivativeFunctionStructureEquations");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{{y}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}\frac{{1}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (214)

Example 3.

 > L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (215)
 > DGsetup(L1):
 > DGinfo(Alg1, "ExteriorDerivativeFunctionStructureEquations");
 $\left[\right]$ (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]]]);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (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");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (221)

Example 2.

 > DGsetup([x, y, z], M): F := FrameData([y*dx, z*dy, dz], N);
 ${F}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{z}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{z}}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (222)
 > DGsetup(F, [X], [omega]):
 > DGinfo(N, "FrameBaseForms");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{N}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (223)

Example 3.

 > DGsetup([x, y], [u], E, 2):
 > DGinfo(E, "FrameBaseForms");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{E}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (224)

Example 4.

 > L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (225)
 > DGsetup(L1):
 > DGinfo(Alg1, "FrameBaseForms");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{Alg1}}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (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");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (227)

Example 2.

 > DGsetup([x, y, z], M):
 > F := FrameData([y*dx, z*dy, dz], N);
 ${F}{≔}\left[{d}{}{\mathrm{Θ1}}{=}{-}\frac{{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}{}{z}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{\mathrm{Θ2}}{}{\bigwedge }{}{\mathrm{Θ3}}}{{z}}{,}{d}{}{\mathrm{Θ3}}{=}{0}\right]$ (228)
 > DGsetup(F, [W], [omega]):
 > DGinfo(N, "FrameBaseVectors");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{N}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (229)

Example 3.

 > DGsetup([x, y], [u], E, 2):
 > DGinfo(E, "FrameBaseVectors");
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (230)

Example 4.

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