GenerateForms - Maple Help

DifferentialGeometry[Tools]

 GenerateForms

 Calling Sequence GenerateForms(Omega, deg)

Parameters

 Omega - a list of lists of differential 1-forms deg - a list of positive integers

Description

 • Let Omega = [Omega_1, Omega_2, Omega_3, ...] and let deg = [p_1, p_2, p_3, ...].  Then GenerateForm(Omega, deg) returns a list of differential forms of degree p = p_1 + p_2 + p_3 + ..., where each form omega in the list is of the form omega = omega_1 &w omega_2 &w omega_3 ....  and where omega_i  is a p_i-fold wedge product of forms in Omega_i.
 • The command GenerateForms is part of the DifferentialGeometry:-Tools package, and so can be used in the form GenerateForms(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-GenerateForms.

Examples

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

Define a 6-dimensional manifold M with coordinates [x1, x2, y1, y2, y3, z1].  (This choice of coordinate names is simply to help understand the output of the commands that follow).

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{y1},\mathrm{y2},\mathrm{y3},\mathrm{z1}\right],M\right)$
 ${\mathrm{frame name: M}}$ (1)

Example 1.

Find all 2 -forms generated from [dy1, dy2, dy3].

 > $\mathrm{GenerateForms}\left(\left[\left[\mathrm{dy1},\mathrm{dy2},\mathrm{dy3}\right]\right],\left[2\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{3}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]$ (2)

Example 2.

Find all 2-forms obtained by choosing 1 from [dx1, dx2] and 1 from [dy1, dy2, dy3].

 > $\mathrm{GenerateForms}\left(\left[\left[\mathrm{dx1},\mathrm{dx2}\right],\left[\mathrm{dy1},\mathrm{dy2},\mathrm{dy3}\right]\right],\left[1,1\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]$ (3)

Example 3.

Find all 5-forms obtained by choosing 2 from [dx1, dx2] and 3 from [dy1, dy2, dy3].

 > $\mathrm{GenerateForms}\left(\left[\left[\mathrm{dx1},\mathrm{dx2}\right],\left[\mathrm{dy1},\mathrm{dy2},\mathrm{dy3}\right]\right],\left[2,3\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{5}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}{,}{5}\right]{,}{1}\right]\right]\right]\right)\right]$ (4)

Example 4.

Find all 3-forms obtained by choosing 1 from [dx1, dx2], 1 from [dy1, dy2], and 1 from [dz1].

 > $\mathrm{GenerateForms}\left(\left[\left[\mathrm{dx1},\mathrm{dx2}\right],\left[\mathrm{dy1},\mathrm{dy2}\right],\left[\mathrm{dz1}\right]\right],\left[1,1,1\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{3}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{1}{,}{4}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{2}{,}{3}{,}{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{3}\right]{,}\left[\left[\left[{2}{,}{4}{,}{6}\right]{,}{1}\right]\right]\right]\right)\right]$ (5)