 DGsetup - Maple Help

DifferentialGeometry

 DGsetup
 set up a coordinate system, a frame, a Lie algebra, define a set of abstract forms Calling Sequence DGsetup(varlist1, framename, options) DGsetup(varlist1, varlist2, framename, options) DGsetup(varlist1, varlist2, framename, jetorder, options) DGsetup(framedata, options) DGsetup(Liealgebradata, options) DGsetup(alg, rho, V) DGsetup(abstractforms, streqn, framename) Parameters

 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' Description

 • All computational sessions with the DifferentialGeometry package begin with a call to the DGsetup command.  This command fixes the coordinate names for the manifold being defined; specifies the vectors and 1-forms to be used as local frames and co-frames on the manifold; and stores all the computational rules needed to work with the given frame or co-frame.  It is also used by the JetCalculus package to initialize a jet space to any given order and by the LieAlgebras package to prepare for computations with Lie algebras.
 • The following table summarizes the different calling sequences for DGsetup.

 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): Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.  Use the first calling sequence to set up a coordinate system on a manifold.

In this first example, we create a two-dimensional manifold M with coordinates [x, y].

 > $\mathrm{DGsetup}\left(\left[x,y\right],M,\mathrm{verbose}\right)$
 ${\mathrm{The following coordinates have been protected:}}$
 $\left[{x}{,}{y}\right]$
 ${\mathrm{The following vector fields have been defined and protected:}}$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$
 ${\mathrm{The following differential 1-forms have been defined and protected:}}$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$