DifferentialGeometry - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : DifferentialGeometry/DGzip

DifferentialGeometry

 DGzip
 form a linear combination, wedge product or tensor product of a list of vectors, forms or tensors

 Calling Sequence DGzip(C, T, operand)

Parameters

 C - a list of Maple expressions T - a list of vectors, differential p-forms, or tensors (of the same type); the number of elements in C and T must be equal operand - a string, one of "plus", "wedge", or "tensor"

Description

 • DGzip(C, T, "plus") returns the additive linear combination of the objects in T with coefficients taken from the list C.
 • DGzip(T, "wedge") returns the wedge product of the forms in T.
 • DGzip(T, "tensor") returns the tensor product of the tensors in the list T.
 • This command is part of the DifferentialGeometry package, and so can be used in the form DGzip(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-DGzip.

Examples

 > with(DifferentialGeometry):

Initialize a 4-dimensional manifold M with coordinates [x, y, z, w].

 > DGsetup([x, y, z, w], M):

Example 1.

Form a vector on M with arbitrary coefficients.

 > C1 := [a, b, c, d];
 ${\mathrm{C1}}{≔}\left[{a}{,}{b}{,}{c}{,}{d}\right]$ (1)
 > T1 := [D_x, D_y, D_z, D_w];
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (2)
 > X := DGzip(C1, T1, "plus");
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{c}\right]{,}\left[\left[{4}\right]{,}{d}\right]\right]\right]\right)$ (3)

Example 2.

Generate a list of 6 coefficients using the seq command, generate a basis for the 2-forms on M using the GenerateForms command, and use DGzip to make a general 2-form on M.

 > C2 := [seq(a||i, i = 1 ..6)];
 ${\mathrm{C2}}{≔}\left[{\mathrm{a1}}{,}{\mathrm{a2}}{,}{\mathrm{a3}}{,}{\mathrm{a4}}{,}{\mathrm{a5}}{,}{\mathrm{a6}}\right]$ (4)
 > T2 := Tools:-GenerateForms([dx, dy, dz, dw], 2);
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{2}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (5)
 > omega := DGzip(C2, T2, "plus");
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{\mathrm{a1}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{\mathrm{a2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{\mathrm{a3}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{\mathrm{a4}}\right]{,}\left[\left[{2}{,}{4}\right]{,}{\mathrm{a5}}\right]{,}\left[\left[{3}{,}{4}\right]{,}{\mathrm{a6}}\right]\right]\right]\right)$ (6)

Example 3.

Define the standard volume form on M.

 > T3 := [dx, dy, dz, dw];
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (7)
 > nu := DGzip(T3, "wedge");
 ${\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{4}\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (8)

Example 4.

Create the tensor product of a list of tensors.

 > T4 := evalDG([dx, D_x, dy, D_y, dz, D_z, dw, D_w]);
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (9)
 > T := DGzip(T4, "tensor");
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"con_bas"}{,}{"cov_bas"}{,}{"con_bas"}\right]{,}\left[{}\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{2}{,}{2}{,}{3}{,}{3}{,}{4}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (10)
 M >