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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

 > with(DifferentialGeometry): with(LieAlgebras):

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].

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

The coordinates are now protected names which cannot be assigned values.

The standard coordinate vectors D_x, D_y and dual 1 forms dx, dy have been defined and protected.  We display the internal representation of these vectors just to show that they have been assigned values. (Knowledge of this internal representation of various objects used in the DifferentialGeometry package is not required of the user).

 > lprint(D_x):
 _DG([["vector", M, []], [[[1], 1]]])
 > lprint(dx);
 _DG([["form", M, 1], [[[1], 1]]])

Without the option 'verbose', the information concerning the definition and protection of variable is suppressed.  This behavior of the DGsetup command can be controlled globally with the Preferences command.

 > DGsetup([u, v], N);
 ${\mathrm{frame name: N}}$ (2)

Example 2.  Use the second calling sequence to set up coordinates on a vector or fiber bundle.

In many situations in differential geometry, one deals with vector bundles or fiber bundles pi: E -> M.  For these applications, use the second calling sequence to DGsetup.  The

first argument varlist1 should be coordinates for the basis manifold M and the second argument should be the list of coordinates for the fiber of the bundle E.  Here is the command

for creating a rank-2 bundle over a three-dimensional space.  The base coordinates are [x, y, z] and the fiber coordinates are [u, v].

 > DGsetup([x, y, z], [u, v], P, verbose);
 ${\mathrm{The following coordinates have been protected:}}$
 $\left[{x}{,}{y}{,}{z}{,}{u}{,}{v}\right]$
 ${\mathrm{The following vector fields have been defined and protected:}}$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{P}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{P}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{P}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{P}{,}\left[{}\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{P}{,}\left[{}\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]$
 ${\mathrm{The following differential 1-forms have been defined and protected:}}$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{P}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]$
 ${\mathrm{frame name: P}}$ (3)

This particular use of DGsetup could be used, for example, to study a three-dimensional manifold imbedded in a five-dimensional Riemannian manifold, where the vectors D_u and D_v define a local basis for the normal bundle.

Example 3.  Use the third calling sequence to set up coordinates on a jet bundle. The JetCalculus subpackage of the DifferentialGeometry package contains an extensive set of commands for symbolic computations on jet spaces.  These spaces provide the natural mathematical setting for the geometric approach to the calculus of variations and to differential equations.  The third calling sequence for the DGsetup command will create a jet space over a bundle pi: E -> M with prescribed lists of independent variables (coordinates on M) and dependent variables (fiber coordinates on E). In addition to the coordinate vectors and coordinate differential forms on jet space, DGsetup also defines and protects the contact forms on jet space.  See the JetCalculus overview page for additional information regarding these differential forms. The user may specify which of two notational conventions are to be used to denote the jet coordinates.  See the Preferences command for details.

First we initialize the 3rd order jet space for 1 independent and 2 dependent variables.

 > DGsetup([t], [x, y], J12, 3, verbose);
 ${\mathrm{The following coordinates have been protected:}}$
 $\left[{t}{,}{x}\left[\right]{,}{y}\left[\right]{,}{{x}}_{{1}}{,}{{y}}_{{1}}{,}{{x}}_{{1}{,}{1}}{,}{{y}}_{{1}{,}{1}}{,}{{x}}_{{1}{,}{1}{,}{1}}{,}{{y}}_{{1}{,}{1}{,}{1}}\right]$
 ${\mathrm{The following vector fields have been defined and protected:}}$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{7}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{8}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{J12}}{,}\left[{}\right]\right]{,}\left[\left[\left[{9}\right]{,}{1}\right]\right]\right]\right)\right]$
 ${\mathrm{The following differential 1-forms have been defined and protected:}}$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{J12}}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{J12}}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{J12}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{J12}}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{J12}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]$