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Calling Sequence
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DGinfo(X, keyword)
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Parameters
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X
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-
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a DifferentialGeometry object
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keyword
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-
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a keyword string
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Description
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•
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The command DGinfo provides a user friendly interface to the internal representations of all the objects and frames created by the DifferentialGeometry software.
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•
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The keyword strings accepted by the DGinfo command are:
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•
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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(...).
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Function, Vector, DifferentialForm, and Tensor Information
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"BiformDegree"
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y], [u], E, 2):
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Example 1.
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alpha1 := evalDG(Dx &wedge Dy);
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>
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DGinfo(alpha1, "BiformDegree");
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Example 2.
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alpha2 := evalDG(Dx &wedge Cu[]);
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>
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DGinfo(alpha2, "BiformDegree");
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Example 3.
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alpha3 := evalDG(Cu[1] &wedge Cu[2]);
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>
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DGinfo(alpha3, "BiformDegree");
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"CoefficientList": list some or all of the coefficients of a vector, differential form, or tensor.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y, z, w], M):
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>
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alpha := evalDG(a*dx &w dy + b*dx &w dz + c*dy &w dz + d*dx &w dw + e*dz &w dw);
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| (7) |
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DGinfo(alpha, "CoefficientList", "all");
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| (8) |
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DGinfo(alpha, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);
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>
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DGinfo(alpha, "CoefficientList", [[1,2], [1,3], [2,4]]);
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"CoefficientSet": find the set of all the coefficients of a vector, differential form, or tensor.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y, z, w], M):
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Example 1.
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alpha := evalDG(a*dx &w dy + b*dx &w dz + b*dy &w dz + c*dx &w dw + a*dz &w dw);
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| (11) |
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DGinfo(alpha, "CoefficientSet");
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Example 2.
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DGinfo(X, "CoefficientSet");
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"CoefficientList": list some or all of the coefficients of a vector, differential form, or tensor.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y, z, w], M):
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>
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alpha := evalDG(a*dx &w dy + b*dx &w dz + c*dy &w dz + d*dx &w dw + e*dz &w dw);
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| (15) |
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DGinfo(alpha, "CoefficientList", "all");
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| (16) |
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DGinfo(alpha, "CoefficientList", [dx &w dy, dx &w dz, dy &w dw]);
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>
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DGinfo(alpha, "CoefficientList", [[1,2], [1,3], [2,4]]);
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"DiffeqType": find the type of the system of differential equations
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with(DifferentialGeometry): with(JetCalculus): with(Tools):
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>
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DGsetup([x, y], [u, v], E, 2):
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>
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Delta := DifferentialEquationData([u[2] + v[1], u[1] - v[2]], [u[1], v[1]]);
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| (19) |
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DGinfo(Delta, "DiffeqType");
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| (20) |
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"DiffeqVariables": list the jet variables in the differential equation to be solved for
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with(DifferentialGeometry): with(JetCalculus): with(Tools):
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>
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DGsetup([x, y], [u, v], E, 2):
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>
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Delta := DifferentialEquationData([u[2] + v[1], u[1] - v[2]], [u[1], v[1]]);
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| (21) |
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DGinfo(Delta, "DiffeqVariables");
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"FormDegree": the degree of a differential form
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with(DifferentialGeometry): with(Tools):
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Example 1.
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alpha := evalDG(dx +dy);
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DGinfo(alpha, "FormDegree");
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Example 2.
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beta := evalDG(3*dx &w dy + 4*dy &w dz - dx &w dz);
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| (25) |
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DGinfo(beta, "FormDegree");
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Example 3.
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nu := evalDG(dx &w dy &w dz);
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>
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DGinfo(nu, "FormDegree");
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"FunctionOrder": the order of the highest jet coordinate appearing in a Maple expression.
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>
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with(DifferentialGeometry): with(Tools):
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Example 1.
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DGsetup([x, y], [u, v], E):
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>
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DGinfo(f1, "FunctionOrder");
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Example 2.
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DGsetup([x, y], [u], J, 1):
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>
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DGinfo(f1, "FunctionOrder");
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Example 3.
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f1 := u[1, 1, 2]*x + u[2, 2, 2, 2];
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| (33) |
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DGinfo(f1, "FunctionOrder");
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"ObjectAttributes": list all the properties of a vector, differential form, tensor, or transformation.
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y, z], M): DGsetup([u, v], N):
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Example 1.
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X := evalDG(a*D_x + b*D_y + c*D_z);
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DGinfo(X, "ObjectAttributes");
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| (36) |
Example 2.
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alpha := evalDG(d*dx &w dy + e*dx &w dz + f*dy &w dz);
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| (37) |
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DGinfo(alpha, "ObjectAttributes");
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| (38) |
Example 3.
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T := evalDG(r*D_x &t dx + s*D_z&t dy + t*D_z &t dx);
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| (39) |
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DGinfo(T, "ObjectAttributes");
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| (40) |
Example 4.
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Phi := Transformation(M, N, [u = x^2 + y^2 + z^2, v = x*y*z]);
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| (41) |
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DGinfo(Phi, "ObjectAttributes");
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| (42) |
Example 5.
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DGsetup([[gamma1, gamma2, gamma3, gamma4], chi = dgform(3)], [], P):
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ExteriorDerivative(chi);
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DGinfo(dchi, "ObjectAttributes");
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| (44) |
Example 6.
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DGinfo(i_1chi, "ObjectAttributes");
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| (46) |
Example 7.
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Hook([D_gamma2, D_gamma3], chi);
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DGinfo(i_2_3chi, "ObjectAttributes");
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| (48) |
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"ObjectComponents": list all the components of a vector, differential form, tensor, or transformation.
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y, z], M): DGsetup([u, v], N):
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Example 1.
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X := evalDG(a*D_x + b*D_y + c*D_z);
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DGinfo(X, "ObjectComponents");
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| (50) |
Example 2.
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alpha := evalDG(d*dx &w dy + e*dx &w dz + f*dy &w dz);
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| (51) |
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DGinfo(alpha, "ObjectComponents");
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| (52) |
Example 3.
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T := evalDG(r*D_x &t dx + s*D_z&t dy + t*D_z &t dx);
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| (53) |
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DGinfo(T, "ObjectComponents");
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| (54) |
Example 4.
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Phi := Transformation(M, N, [u = x^2 + y^2 + z^2, v= x*y*z]);
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| (55) |
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DGinfo(Phi, "ObjectComponents");
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| (56) |
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"ObjectFrame": return the frame with respect to which the object is defined.
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with(DifferentialGeometry): with(Tools):
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DGsetup([x, y], [u], M, 1): DGsetup([r,s,t], N):
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Example 1.
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X := evalDG(D_x -3*D_y);
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DGinfo(X, "ObjectFrame");
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Example 2.
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T := evalDG(D_r &t D_s &t dt);
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DGinfo(T, "ObjectFrame");
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"ObjectGenerators": list the monomial vectors in a vector or the monomial 1-forms in a differential form.
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>
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with(DifferentialGeometry): with(Tools):
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DGsetup([x, y, z, w], M):
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Example 1.
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X := evalDG(y*D_x + z*D_z);
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DGinfo(X, "ObjectGenerators");
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This means that X has only the 1st and 3rd elements from the standard basis for the tangent bundle.
Example 2.
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alpha := evalDG(w*dx &w dy + x*dx &w dw + z^2*dy &w dw);
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| (63) |
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DGinfo(alpha, "ObjectGenerators");
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This means that alpha do not contain the 3rd element (dz) from the standard basis for the cotangent bundle.
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"ObjectOrder": the order of the jet space on which the object is defined.
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with(DifferentialGeometry): with(JetCalculus): with(Tools):
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Example 1.
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DGsetup([x, y], [u, v], E, 3):
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DGinfo(f, "ObjectOrder");
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Example 2.
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alpha := evalDG(du[1]*x + du[2,2]);
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DGinfo(alpha, "ObjectOrder");
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Example 3.
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beta := evalDG(du[1, 1, 2]*x + u[2, 2, 2, 2]*dy);
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| (69) |
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DGinfo(beta, "FunctionOrder");
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Example 4.
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X := evalDG(u[1]*D_u[]);
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| (72) |
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DGinfo(X3, "FunctionOrder");
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"ObjectType": the type of the DifferentialGeometry object.
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with(DifferentialGeometry): with(Tools): with(LieAlgebras):
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DGsetup([x, y], [u], M, 1):
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Example 1.
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DGinfo(f, "ObjectType");
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Example 2.
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X := evalDG(D_x + y*D_u[]);
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DGinfo(X, "ObjectType");
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Example 3.
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alpha := evalDG(dx &w dy + dx &w du[1]);
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DGinfo(alpha, "ObjectType");
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Example 4.
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theta := evalDG(Dx &w Cu[1]);
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DGinfo(theta, "ObjectType");
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Example 5.
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T := evalDG(dx &t D_y + dy &t D_y);
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DGinfo(T, "ObjectType");
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Example 6.
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DGinfo(A, "ObjectType");
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Example 7.
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DGinfo(B, "ObjectType");
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Example 8.
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B:= Matrix([[1, 2], [3, 4]]);
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DGinfo(B, "ObjectType");
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Example 9.
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Phi := IdentityTransformation();
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| (90) |
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DGinfo(Phi, "ObjectType");
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Example 10.
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L := LieAlgebraData(evalDG([D_x, x*D_x, x^2*D_x]));
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| (92) |
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DGinfo(L, "ObjectType");
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Example 11.
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F := FrameData([y*dx, x*dy], N);
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| (95) |
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DGinfo(F, "ObjectType");
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Example 12.
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R := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [0, 1]])];
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| (98) |
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L := LieAlgebraData(R, Alg);
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| (99) |
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rho := Representation(Alg, V, R);
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| (101) |
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DGinfo(rho, "ObjectType");
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"TensorDensityType": the density weight of a tensor.
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with(DifferentialGeometry): with(Tools): with(Tensor):
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DGsetup([x, y],[u, v, w], E):
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g := evalDG(dx &t dx + dy &t dy);
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Example 1.
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rho := MetricDensity(g, 3);
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DGinfo(T1, "TensorDensityType");
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Example 2.
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T2 := PermutationSymbol("cov_vrt");
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| (107) |
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DGinfo(T2, "TensorDensityType");
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"TensorGenerators": a list of the monomial tensors which generate a given tensor.
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with(DifferentialGeometry): with(Tools): with(Tensor):
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DGsetup([x, y], [u, v, w], E):
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Example 1.
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T1 := evalDG(dx &t dx &t dx);
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DGinfo(T1, "TensorGenerators");
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Example 2.
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T2 := evalDG(dx &t D_y &t dy);
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DGinfo(T2, "TensorGenerators");
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| (112) |
Example 3.
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T3 := evalDG(du &t D_v &t dy);
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DGinfo(T3, "TensorGenerators");
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| (114) |
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"TensorIndexPart1": the tensor character of a tensor index type.
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with(DifferentialGeometry): with(Tools):
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DGinfo("con_bas", "TensorIndexPart1");
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DGinfo("cov_bas", "TensorIndexPart1");
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DGinfo("con_vrt", "TensorIndexPart1");
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DGinfo("cov_vrt", "TensorIndexPart1");
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"TensorIndexPart2": the spatial type of a tensor index type.
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with(DifferentialGeometry): with(Tools):
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DGinfo("con_bas", "TensorIndexPart2");
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DGinfo("cov_bas", "TensorIndexPart2");
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DGinfo("con_vrt", "TensorIndexPart2");
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DGinfo("cov_vrt", "TensorIndexPart2");
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"TensorIndexType": the full tensor index type of a tensor.
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with(DifferentialGeometry): with(Tools): with(Tensor):
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DGsetup([x, y], [u, v], E):
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Example 1.
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T1 := evalDG(D_x &t D_y);
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DGinfo(T1, "TensorIndexType");
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| (124) |
Example 2.
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T2 := evalDG(dx &t D_y);
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DGinfo(T2, "TensorIndexType");
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| (126) |
Example 3.
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T3 := evalDG(du &t D_v);
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DGinfo(T3, "TensorIndexType");
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| (128) |
Example 4.
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T4 := evalDG(D_x &t D_v);
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DGinfo(T4, "TensorIndexType");
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| (130) |
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"TensorType": the full tensor index type and weight of a tensor.
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with(DifferentialGeometry): with(Tools): with(Tensor):
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DGsetup([x, y], [u, v], E):
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Example 1.
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T1 := evalDG(D_x &t D_y);
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DGinfo(T1, "TensorType");
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| (132) |
Example 2.
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T2 := PermutationSymbol("con_bas");
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DGinfo(T2, "TensorType");
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| (134) |
Example 3.
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T3 := evalDG(du &t D_v);
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DGinfo(T3, "TensorType");
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| (136) |
Example 4.
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T4 := PermutationSymbol("cov_vrt");
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DGinfo(T4, "TensorType");
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| (138) |
Example 5.
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T5 := PermutationSymbol("cov_bas") &tensor PermutationSymbol("con_vrt");
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| (139) |
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DGinfo(T5, "TensorType");
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| (140) |
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"VectorType": the type (projectionable, point, contact, ...) of a vector field and its order of prolongation. See AssignVectorType for details.
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>
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with(DifferentialGeometry): with(Tools): with(JetCalculus):
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>
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DGsetup([x,y], [u], E, 1):
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Example 1.
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X1 := evalDG(D_x + u[] *D_u[]);
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>
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X1a := AssignVectorType(X1);
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DGinfo(X1a, "VectorType");
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| (143) |
Example 2.
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X2 := Prolong(u[]*D_x - x *D_u[], 1);
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| (144) |
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DGinfo(X2, "VectorType");
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Example 3.
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X3 := GeneratingFunctionToContactVector(u[]*u[1]*u[2]);
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| (146) |
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X3a := AssignVectorType(X3);
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| (147) |
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DGinfo(X3a, "VectorType");
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Weight": the weight of a monomial form in a graded Lie algebra with coefficients
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>
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with(DifferentialGeometry): with(Tools):
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>
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LD:=LieAlgebraData([[x3, x4] = e1, [x3, x5] = x2, [x4, x5] = x3], [x1,x2,x3,x4,x5], Alg, grading = [-3, -3, -2, -1, -1]);
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| (149) |
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DGsetup([w1, w2, w3, w4, w5], V, grading = [-3, -3, -2, -1, -1]):
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>
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rho:= Representation(Alg, V, Adjoint(Alg));
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| (150) |
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DGsetup(Alg, rho, AlgV);
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| (151) |
Example 1.
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alpha1 := evalDG(w3 *theta5);
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| (152) |
>
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DGinfo(alpha1, "Weight");
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Example 2.
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alpha2 := evalDG(w3 * theta1 &w theta5);
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| (154) |
>
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DGinfo(alpha2, "Weight");
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Example 3.
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alpha3 := evalDG(w1 * theta3 &w &w theta4 &w theta5);
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| (156) |
>
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DGinfo(alpha3, "Weight");
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Transformation Information
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"DomainFrame": the domain frame of a transformation.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y], M): DGsetup([u, v], N):
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>
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Phi := Transformation(M, N, [ u =x*y, v= 1/x + 1/y]);
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| (158) |
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DGinfo(Phi, "DomainFrame");
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"DomainOrder": the jet space order of the domain of a transformation.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y], M): DGsetup([u, v], N): DGsetup([t], [w], J, 1):
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Example 1.
>
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Phi1 := Transformation(M, N, [u =x*y, v= 1/x + 1/y]);
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| (160) |
>
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DGinfo(Phi1, "DomainOrder");
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Example 2.
>
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Phi2 := Transformation(J, M, [x = w[1], y = w[1, 1]]);
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| (162) |
>
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DGinfo(Phi2, "DomainOrder");
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"JacobianMatrix": the Jacobian Matrix of a transformation.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y], M): DGsetup([u, v, w], N):
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Example 1.
>
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Phi := Transformation(M, N, [u = x*y, v = 1/x + 1/y, w = x + y]);
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| (164) |
>
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DGinfo(Phi, "JacobianMatrix");
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"RangeFrame": the range frame of a transformation.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y], M): DGsetup([u, v], N):
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>
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Phi := Transformation(M, N, [u =x*y, v= 1/x + 1/y]);
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| (166) |
>
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DGinfo(Phi, "RangeFrame");
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|
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"RangeOrder": the jet space order of the range of a transformation.
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>
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with(DifferentialGeometry): with(Tools):
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>
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DGsetup([x, y], M): DGsetup([u, v], N): DGsetup([t], [w], J, 1):
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Example 1.
>
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Phi1 := Transformation(M, N, [u =x*y, v= 1/x + 1/y]);
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| (168) |
>
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DGinfo(Phi1, "RangeOrder");
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Example 2.
>
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Phi2 := Transformation(M, J, [t= x, w[]= y, w[1] = y^2]);
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| (170) |
>
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DGinfo(Phi2, "RangeOrder");
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|
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"RepresentationMatrices": the list of matrices defining a representation of a Lie Algebra
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>
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with(DifferentialGeometry): with(LieAlgebras): with(Tools):
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>
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R := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0], [0, 1]])];
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| (173) |
>
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L := LieAlgebraData(R, Alg);
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| (174) |
>
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rho := Representation(Alg, V, R);
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| (176) |
>
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DGinfo(rho, "RepresentationMatrices");
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| (177) |
|
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"TransformationType": the type (projectionable, point, contact, ...) of a transformation. See AssignTransformationType for details.
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>
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with(DifferentialGeometry): with(Tools): with(JetCalculus):
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>
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DGsetup([x], [u], E, 1):
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Example 1.
>
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Phi1 := Transformation(E, E, [x = x^2, u[] = u[]*x]);
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| (178) |
>
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Phi1A := AssignTransformationType(Phi1);
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| (179) |
>
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DGinfo(Phi1A, "TransformationType");
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| (180) |
Example 2.
>
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Phi2 := Transformation(E, E, [x = u[], u[] = x]);
|
| (181) |
>
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Phi2A := AssignTransformationType(Phi2);
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| (182) |
>
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DGinfo(Phi2A, "TransformationType");
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Example 3.
>
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Phi3 := Transformation(E, E, [x = -2*u[1] + x, u[] = -u[1]^2 + u[], u[1] = u[1]]);
|
| (184) |
>
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Phi3A := AssignTransformationType(Phi3);
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| (185) |
>
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DGinfo(Phi3A, "TransformationType");
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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");
|
| (187) |
Example 2.
>
|
ExteriorDerivative(tau);
|
>
|
DGinfo(P, "AbstractForms");
|
| (189) |
Example 3.
>
|
Hook([D_sigma2, D_sigma3], xi);
|
>
|
DGinfo(P, "AbstractForms");
|
| (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]);
|
| (193) |
>
|
DGsetup([w1, w2, w3, w4, w5], V, grading = [-3, -3, -2, -1, -1]):
|
>
|
rho:= Representation(Alg, V, Adjoint(Alg));
|
| (194) |
>
|
DGsetup(Alg, rho, AlgV):
|
>
|
DGinfo(AlgV, "CoefficientGrading");
|
| (195) |
>
|
DGinfo(AlgV, "CoefficientGrading", output = "list");
|
| (196) |
>
|
DGinfo(output = "coefficients", "CoefficientGrading");
|
| (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.
>
|
DGinfo(M, "CoframeLabels");
|
| (198) |
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (199) |
>
|
DGsetup(F, [X], [omega]):
|
>
|
DGinfo(N, "CoframeLabels");
|
| (200) |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "CoframeLabels");
|
| (201) |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (202) |
>
|
DGinfo(Alg1, "CoframeLabels");
|
| (203) |
|
|
"CurrentFrame": the name of the currently active frame.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
>
|
DGsetup([x], M): DGsetup([y], N):
|
>
|
DGinfo("CurrentFrame");
|
>
|
LieBracket(D_x, x*D_x);
|
>
|
DGinfo("CurrentFrame");
|
|
|
"ExteriorDerivativeFormStructureEquations": list the formulas for the exterior derivatives of a frame with protocol "LieAlgebra" or "AnholonomicFrame".
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo("ExteriorDerivativeFormStructureEquations");
|
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (208) |
>
|
DGsetup(F, [X], [omega]):
|
>
|
DGinfo(N, "ExteriorDerivativeFormStructureEquations");
|
| (209) |
Example 3.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (210) |
>
|
DGinfo(Alg1, "ExteriorDerivativeFormStructureEquations");
|
| (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.
>
|
DGinfo("ExteriorDerivativeFunctionStructureEquations");
|
Example 2.
>
|
DGsetup([x, y, z], M):
F := FrameData([y*dx, z*dy, dz], N);
|
| (213) |
>
|
DGsetup(F, [X], [omega]):
|
>
|
DGinfo(N, "ExteriorDerivativeFunctionStructureEquations");
|
| (214) |
Example 3.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (215) |
>
|
DGinfo(Alg1, "ExteriorDerivativeFunctionStructureEquations");
|
|
|
"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.
>
|
DGinfo(M, "FrameBaseDimension");
|
Example 2.
>
|
DGsetup([x, y], [u, v, w], E):
|
>
|
DGinfo(E, "FrameBaseDimension");
|
Example 3.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (219) |
>
|
DGinfo(Alg1, "FrameBaseDimension");
|
|
|
"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.
>
|
DGinfo(M, "FrameBaseForms");
|
| (221) |
Example 2.
>
|
DGsetup([x, y, z], M):
F := FrameData([y*dx, z*dy, dz], N);
|
| (222) |
>
|
DGsetup(F, [X], [omega]):
|
>
|
DGinfo(N, "FrameBaseForms");
|
| (223) |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameBaseForms");
|
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (225) |
>
|
DGinfo(Alg1, "FrameBaseForms");
|
| (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.
>
|
DGinfo(M, "FrameBaseVectors");
|
| (227) |
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (228) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
DGinfo(N, "FrameBaseVectors");
|
| (229) |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameBaseVectors");
|
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (231) |
>
|
DGinfo(Alg1, "FrameBaseVectors");
|
| (232) |
|
|
"FrameDependentVariables": the dependent or fiber variables for a frame defining a bundle E -> M.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameDependentVariables");
|
Example 2.
>
|
DGsetup([x, y, z], [u, v], E):
|
>
|
DGinfo(M, "FrameDependentVariables");
|
Example 3.
>
|
DGsetup([x, y, z], [u, v], J, 1):
|
>
|
DGinfo(M, "FrameDependentVariables");
|
|
|
"FrameFiberDimension": the dimension of the fiber for a frame defining a bundle E -> M.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameFiberDimension");
|
Example 2.
>
|
DGsetup([x, y, z], [u, v], E):
|
>
|
DGinfo(E, "FrameFiberDimension");
|
Example 3.
>
|
DGsetup([x, y, z], [u, v], J, 1):
|
>
|
DGinfo(J, "FrameFiberDimension");
|
|
|
"FrameFiberForms": the coordinate basis of vertical 1-forms for a fiber bundle E -> M.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameFiberForms");
|
Example 2
>
|
DGsetup([x, y, z], [u, v], E):
|
>
|
DGinfo(E, "FrameFiberForms");
|
Example 3
>
|
DGsetup([x, y, z], [u, v], J, 1):
|
>
|
DGinfo(J, "FrameFiberForms");
|
|
|
"FrameFiberVectors": the coordinate basis of vertical vectors for a fiber bundle E -> M.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameFiberVectors");
|
Example 2.
>
|
DGsetup([x, y, z], [u, v], E):
|
>
|
DGinfo(E, "FrameFiberVectors");
|
Example 3.
>
|
DGsetup([x, y, z], [u, v], J, 1):
|
>
|
DGinfo(J, "FrameFiberVectors");
|
|
|
"FrameGlobals": the list of Maple names that are assigned or protected when a frame is defined using the DGsetup command.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameGlobals");
|
| (245) |
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz],N);
|
| (246) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
DGinfo(N, "FrameGlobals");
|
| (247) |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameGlobals");
|
| (248) |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (249) |
>
|
DGinfo(Alg1, "FrameGlobals");
|
| (250) |
|
|
"FrameHorizontalBiforms": the list of horizontal biforms on a jet space
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameHorizontalBiforms");
|
Example 2.
>
|
DGsetup([x, y, z], [u, v], J, 1):
|
>
|
DGinfo(J, "FrameHorizontalBiforms");
|
| (252) |
|
|
"FrameIndependentVariables": the coordinate variables of the base manifold M for a frame defining a bundle E -> M; the coordinate variables of M for a frame defining a manifold M; the internally defining coordinate variables of the Lie algebra for a frame defining a Lie algebra.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameIndependentVariables");
|
Example 2.
>
|
Omega := evalDG([y*dx, z*dy, dz]);
|
| (254) |
>
|
F := FrameData(Omega, N);
|
| (255) |
>
|
DGsetup(F, [X], [omega]):
|
>
|
DGinfo(N, "FrameIndependentVariables");
|
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameIndependentVariables");
|
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (258) |
>
|
DGinfo(Alg1, "FrameIndependentVariables");
|
| (259) |
|
|
"FrameInformation": a complete display of all information pertaining to a given frame.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameInformation");
|
| |
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (261) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
DGinfo(N, "FrameInformation");
|
| |
| |
| |
| |
| |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameInformation");
|
| |
| |
| |
| |
| |
| |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (264) |
>
|
DGinfo(Alg1, "FrameInformation");
|
| |
| |
| |
| |
| |
| |
| |
| |
|
|
"FrameJetDimension": the total dimension of the frame.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameJetDimension");
|
Example 2.
>
|
DGsetup([x, y, z], [u, v], N):
|
>
|
DGinfo(N, "FrameJetDimension");
|
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(J, "FrameJetDimension");
|
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (269) |
>
|
DGinfo(Alg1, "FrameJetDimension");
|
|
|
"FrameJetForms": the basis of all 1-forms for the frame.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameJetForms");
|
| (271) |
Example 2.
>
|
DGsetup([x, y, z], [u, v], N):
|
>
|
DGinfo(N, "FrameJetForms");
|
| (272) |
Example 3.
>
|
DGsetup([x, y], [u], J, 2):
|
>
|
DGinfo(J, "FrameJetForms");
|
| (273) |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (274) |
>
|
DGinfo(Alg1, "FrameJetForms");
|
| (275) |
|
|
"FrameJetOrder": the jet space order of the frame.
|
|
>
|
with(DifferentialGeometry): with(Tools): with(JetCalculus):
|
Example 1.
>
|
DGsetup([x, y, z], [u, v], N):
|
>
|
DGinfo(N, "FrameJetOrder");
|
Example 2.
>
|
DGsetup([x, y], [u], J, 2):
|
>
|
DGinfo(J, "FrameJetOrder");
|
Example 3.
>
|
DGsetup([x, y], [u], J, 2):
|
>
|
DGinfo(J, "FrameJetOrder");
|
|
|
"FrameJetVariables": the list of all variables for the frame.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameJetVariables");
|
Example 2.
>
|
DGsetup([x, y, z], [u, v], N):
|
>
|
DGinfo(N, "FrameJetVariables");
|
| (281) |
Example 3.
>
|
DGsetup([x, y], [u], J, 2):
|
>
|
DGinfo(J, "FrameJetVariables");
|
| (282) |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (283) |
>
|
DGinfo(Alg1, "FrameJetVariables");
|
| (284) |
|
|
"FrameJetVectors": the list of all basis vectors for the frame.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameJetVectors");
|
| (285) |
Example 2.
>
|
DGsetup([x, y, z], [u, v], N):
|
>
|
DGinfo(N, "FrameJetVectors");
|
| (286) |
Example 3.
>
|
DGsetup([x, y], [u], J, 2):
|
>
|
DGinfo(J, "FrameJetVectors");
|
| (287) |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (288) |
>
|
DGinfo(Alg1, "FrameJetVectors");
|
| (289) |
|
|
"FrameLabels": list the labels used to input and display the basis of vectors for the tangent space.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameLabels");
|
| (290) |
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (291) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
DGinfo(N, "FrameLabels");
|
| (292) |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameLabels");
|
| (293) |
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (294) |
>
|
DGinfo(Alg1, "FrameLabels");
|
| (295) |
|
|
"FrameNames": the list of initialized frames
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
| (296) |
|
|
"FrameProtocol": the name of the frame protocol; this protocol controls the computation rules for many operations in DifferentialGeometry.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameProtocol");
|
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (298) |
>
|
DGsetup(F, [X], [omega]):
|
>
|
DGinfo(N, "FrameProtocol");
|
| (299) |
Example 3.
>
|
DGsetup([x, y], [u], E, 2):
|
>
|
DGinfo(E, "FrameProtocol");
|
Example 4.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (301) |
>
|
DGinfo(Alg1, "FrameProtocol");
|
|
|
"FrameVerticalBiforms": the list of the contact 1-forms for a frame defining a jet space.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "FrameVerticalBiforms");
|
>
|
DGsetup([x, y], [u, v], J, 2):
|
>
|
DGinfo(J, "FrameVerticalBiforms");
|
| (304) |
|
|
"Grading": the grading of a Lie algebra
|
|
>
|
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]);
|
| (305) |
>
|
DGinfo(Alg, "Grading");
|
| (307) |
|
|
"HorizontalCoframeLabels": the list of labels, used for display purposes, for the horizontal biforms defined on a jet space.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "HorizontalCoframeLabels");
|
>
|
DGsetup([x, y], [u, v], J, 2):
|
>
|
DGinfo(J, "HorizontalCoframeLabels");
|
| (309) |
|
|
"LieBracketStructureEquations": a matrix giving the Lie brackets of the frame basis vectors for a frame with protocol "LieAlgebra" or "AnholonomicFrame".
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "LieBracketStructureEquations");
|
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (311) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
DGinfo(N, "LieBracketStructureEquations");
|
| (312) |
Example 3.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (313) |
>
|
DGinfo(Alg1, "LieBracketStructureEquations");
|
| (314) |
|
|
"LieBracketStructureFunction": a specific structure function for a frame with "LieAlgebra" or "AnholonomicFrame" protocol.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGsetup([x, y, z], M):
F := FrameData([y*dx, z*dy, dz], N);
|
| (315) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
DGinfo(N, "LieBracketStructureEquations");
|
| (316) |
>
|
DGinfo([1,2,1], "LieBracketStructureFunction");
|
>
|
DGinfo([2,3,2], "LieBracketStructureFunction");
|
>
|
DGinfo([2,3,3], "LieBracketStructureFunction");
|
Example 2.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (320) |
>
|
DGinfo(Alg1, "LieBracketStructureEquations");
|
| (321) |
|
|
"LieBracketStructureFunctionList ": a list of structure function for a frame with "LieAlgebra" or "AnholonomicFrame" protocol.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGinfo(M, "LieBracketStructureFunctionList");
|
| (322) |
Example 2.
>
|
F := FrameData([y*dx, z*dy, dz], N);
|
| (323) |
>
|
DGsetup(F, [W], [omega]):
|
>
|
[LieBracket(W1, W2), LieBracket(W1, W3), LieBracket(W2, W3)];
|
| (324) |
>
|
DGinfo(N, "LieBracketStructureFunctionList");
|
| (325) |
Example 3.
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (326) |
>
|
DGinfo(Alg1, "LieBracketStructureFunctionList");
|
| (327) |
|
|
"LieDerivativeFunctionStructureEquations": a matrix specifying the Lie derivatives of the coordinate variables with respect to the frame basis vectors for a frame with "AnholonomicFrame" protocol.
|
|
>
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with(DifferentialGeometry): with(Tools):
|
Example 1.
>
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F := FrameData([y*dx, z*dy, dz], N);
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| (328) |
>
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DGsetup(F, [X], [omega]):
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>
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DGinfo(N, "LieDerivativeFunctionStructureEquations");
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|
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"VerticalCoframeLabels": the list of labels, used for display proposes, for the vertical biforms defined on jet space.
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>
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with(DifferentialGeometry): with(Tools):
|
Example 1.
>
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DGinfo(M, "VerticalCoframeLabels");
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>
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DGsetup([x, y], [u, v], J, 1):
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>
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DGinfo(J, "VerticalCoframeLabels");
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| (331) |
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|
|
Miscellaneous
|
|
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"JetNotation": the choice of jet notation for a frame defining the jet space or fiber bundle.
|
|
>
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with(DifferentialGeometry): with(Tools):
|
>
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DGsetup([x, y], [u], J1, 2):
|
>
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DGinfo(J1, "FrameJetVariables");
|
| (332) |
>
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DGinfo(J1, "JetNotation");
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>
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Preferences("JetNotation", "JetNotation2");
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>
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DGsetup([x, y], [u], J2, 2):
|
>
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DGinfo(J2, "FrameJetVariables");
|
| (335) |
>
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DGinfo(J2, "JetNotation");
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|
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"JetIndets": the list of frame variables appearing in a Maple expression. See the Maple indets command.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
| (338) |
>
|
DGinfo(f1, "JetIndets");
|
Example 2.
>
|
DGsetup([x, y], [u,v], E, 2):
|
>
|
f2 := u[2,2]*c + exp(v[])*d;
|
| (340) |
>
|
DGinfo(f2, "JetIndets");
|
|
|
"JetSpaceDimension": the dimension of a jet space of a given order for a given fiber bundle E.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
>
|
DGsetup([x], [u], J1, 1):
|
>
|
DGinfo(3, "JetSpaceDimension", J1);
|
>
|
DGinfo(4, "JetSpaceDimension", J1);
|
Example 2.
>
|
DGsetup([x, y], [u], J2, 1):
|
>
|
DGinfo(2, "JetSpaceDimension", J2);
|
>
|
DGinfo(3, "JetSpaceDimension", J2);
|
|
|
"Keywords": the keyword strings accepted by the DGinfo command.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
| (346) |
|
|
"LieAlgebraDimension": the dimension of a Lie algebra, defined by a LieAlgebraData structure.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
>
|
L1 := _DG([["LieAlgebra", Alg1, [4]], [[[2, 4, 1], 1], [[3, 4, 3], 1]]]);
|
| (347) |
>
|
DGinfo(L1, "LieAlgebraDimension");
|
|
|
"NonJetIndets": the list of Maple names other than frame variables appearing in a Maple expression.
|
|
>
|
with(DifferentialGeometry): with(Tools):
|
Example 1.
| (350) |
>
|
DGinfo(f1, "NonJetIndets");
|
Example 2.
>
|
DGsetup([x, y], [u,v], E, 2):
|
>
|
f2 := u[2,2]*c + exp(v[])*d;
|
| (352) |
>
|
DGinfo(f2, "NonJetIndets");
|
|
|
|
|