varlist1

-

a list of unassigned Maple names

varlist2

-

a list of unassigned Maple names

framename

-

an unassigned Maple name or a string

jetorder

-

a positive integer

framedata

-

the structure equations for an anholonomic frame, as calculated by the procedure FrameData

Liealgebradata

-

the structure equations for a Lie algebra, as calculated by the procedure LieAlgebraData

abstractforms

-

a list specifying a set of abstract forms, without reference to any underlying set of coordinates

streqn

-

a list of structure equations for the exterior derivatives and interior products of an abstract form

options

-

a list of frame labels, a list of co-frame labels, the keyword 'quiet' or 'verbose'

Ex 1.

Create a two-dimensional manifold $M$ with local coordinates $\left(x,y\right)$.

    > DGsetup([x, y], M)

Ex 2.

Create a fiber bundle $E$ with fiber coordinates $\left(u,v\right)$ over a three-dimensional base space with coordinates $\left(x,y,z\right)$.

    > DGsetup([x, y, z], [u,v], M);

Ex 3.

Create the 3rd order jet space $J$  for 2 independent variables $\left(x,y\right)$  and 1 dependent variable $\left(u\right)$.

    > DGsetup([x, y], [u], J, 3);

Ex 4a.

Perform calculations on a three-dimensional manifold $N$ in terms of an anholonomic frame $F$. The command FrameData is used to calculate the structure equations for the frame and this is passed to DGsetup.

    > DGsetup([x, y, z], M);

    > F := evalDG([D_x, x*D_x + D_y, y D_x + D_z]);

    > FD := FrameData(F, N);

    > DGsetup(FD);

Ex 4b.

Perform calculations on a three-dimensional manifold $N$ in terms of an anholonomic frame $F$. Label the frame vectors $\left[X,Y,Z\right]$ and the co-frame 1-forms $\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\sigma }\right]$.

    > DGsetup(FD, [X, Y, Z], [alpha, beta, sigma]);

Ex 5.

Perform calculations on a three-dimensional manifold $N$ in terms of an anholonomic co-frame $\mathrm{\Omega }$. The command FrameData is used to calculate the structure equations for the frame and this is passed to DGsetup.

    > DGsetup([x, y, z], M);

    > Omega := evalDG([dx, x*dx + dy, y*dx + dz]);

    > FD := FrameData(Omega, N);

    > DGsetup(FD);

 Ex 6a.

Initialize a Lie algebra alg1 defined by a set of 3 matrices $A=\left[M_1,M_2,M_3\right]$. Use LieAlgebraData to calculate the structure equations for the Lie algebra and pass this result to DGsetup.

    > A := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0],[0, 1]])];

    > LD := LieAlgebraData(A, alg1);

    > DGsetup(LD);

 Ex 6b.

Initialize a Lie algebra alg1 defined by a set of 3 matrices. Label the basis elements for the Lie algebra as $\left[f_1,f_2,f_3\right]$ and the dual 1-forms by $\left[{\mathrm{\xi }}_1,{\mathrm{\xi }}_2,{\mathrm{\xi }}_3\right]$

    > DGsetup(LD, [f], [xi]):

 Ex 7.

Initialize a Lie algebra alg1 defined by a set of 3 vector fields $\mathrm{\Gamma }=\left[X_1,X_2,X_3\right]$. Use LieAlgebraData to calculate the structure equations for the Lie algebra and passed this result to DGsetup.

    > DGsetup([x, y], M);

    > Gamma : = evalDG([D_x, D_y, y*D_x - x*D_y]);

    > LD := LieAlgebraData(A, alg1):

    > DGsetup(LD);

 Ex 8.

Initialize a Lie algebra alg1 from a set of structure equations. For other ways to initialize a Lie algebra, see LieAlgebraData.

    > B := [x, y, h];

    > S := [[h, x] = 2*x, [h, y] = -2*y, [x, y] = h];

    > LD := LieAlgebraData( S, B, alg);

    > DGsetup(LD);

 Ex 9.

Initialize a classical simple Lie algebra, say $\mathrm{sl}\left(3\right)$, the Lie algebra of trace-free matrices. See SimpleLieAlgebraData.

    > LD := SimpleLieAlgebraData(sl(3), alg);

    > DGsetup(LD);

Ex 10.

Initialize a Lie algebra with coefficients in a representation.

    > LD := LieAlgebraData( [h,x] = 2x, [h,y] = -2y, [x,y] =h], [x, y, h], alg);

    > DGsetup(LD);

    > DGsetup([x1, x2, x3], V);

    > A := Adjoint();

    > rho := Representation(alg, V, Adjoint());

    > DGsetup(alg, rho, R):

Ex 11.

Initialize a set of abstract differential forms with a given set of structure equations.

    > DGsetup([f = dgform(0), alpha = dgform(1), beta = dgfom(2)], [d(beta) = alpha &w beta], M)

Ex 12.

Initialize an abstract co-frame and set of abstract differential forms with a given set of structure equations.

    > DGsetup([[alpha, beta], f = dgform(0),  sigma = dgform(2)], [d(alpha) = f sigma, hook(D_alpha, sigma) = beta], M):