DifferentialGeometry:-Tools[DGbiform, DGform, DGtensor, DGvector] - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : Tools : DifferentialGeometry/Tools/DGform

DifferentialGeometry:-Tools[DGbiform, DGform, DGtensor, DGvector]

 Calling Sequence DGbiform(x, M) DGform(x, M) DGtensor(x, indexType, M) DGvector(y, M)

Parameters

 x - a positive integer, a list of positive integers, a coordinate variable, or a list of coordinate variables M - (optional) the name of defined frame indexType - specifying the index type of the tensor y - a positive integer or a coordinate variable

Description

 • The command DGform will create a single term differential form.  Let Theta = [theta_1, theta_2, theta_3, ...] denote the coframe for the current frame or, if the optional argument M is given, the frame M.  The list Theta can be obtained from the command DGinfo with the keyword "frameBaseForms" or "frameJetForms".  Let V = [x_1, x_2, x_3, ...] denote the local coordinates for the current frame or, if the optional argument M is given, the frame M.  The list V can be obtained from the command DGinfo with the keyword "frameIndependentVariables" or "frameJetVariables". If the integer i or coordinate x_i is given, the command returns the corresponding 1-form theta_i.  If a list of p integers [i, j, k, ...] or coordinates [x_i, x_j, x_k, ...] is given, the command returns the p-form  theta_i &w theta_j &w theta_k...
 • The commands DGbiform, DGtensor, and DGvector work in a similar fashion.
 • The command DGform is part of the DifferentialGeometry:-Tools package and so can be used in the form DGform(...) only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.  It can always be used in the long form DifferentialGeometry:-Tools:-DGform.  DGbiform, DGtensor, and DGvector work in the same way.

Examples

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

Example 1.

Define a manifold M with coordinates [x, y, z, w].

 > $\mathrm{DGsetup}\left(\left[x,y,z,w\right],M\right):$
 > $\mathrm{DGvector}\left(x\right)$
 ${\mathrm{D_x}}$ (1)
 > $\mathrm{DGvector}\left(3\right)$
 ${\mathrm{D_z}}$ (2)
 > $\mathrm{DGform}\left(y\right)$
 ${\mathrm{dy}}$ (3)
 > $\mathrm{DGform}\left(4\right)$
 ${\mathrm{dw}}$ (4)
 > $\mathrm{DGform}\left(\left[x,y\right]\right)$
 ${\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}$ (5)
 > $\mathrm{DGform}\left(\left[1,2,3,4\right]\right)$
 $\left(\left({\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}\right){}{\bigwedge }{}{\mathrm{dz}}\right){}{\bigwedge }{}{\mathrm{dw}}$ (6)
 > $\mathrm{DGtensor}\left(x,"con_bas"\right)$
 ${\mathrm{D_x}}$ (7)
 > $\mathrm{DGtensor}\left(2,"cov_bas"\right)$
 ${\mathrm{dy}}$ (8)
 > $\mathrm{DGtensor}\left(\left[1,1,1\right],\left[\left["cov_bas","con_bas","cov_bas"\right],\left[\right]\right]\right)$
 $\left({\mathrm{dx}}{}{\mathrm{D_x}}\right){}{\mathrm{dx}}$ (9)

Example 2.

Define a rank 3 vector bundle E with coordinates [x, y, u, v, w] over a two dimensional base with coordinates [x, y].

 > $\mathrm{DGsetup}\left(\left[x,y,u,v,w\right],E\right)$
 ${\mathrm{frame name: E}}$ (10)
 > $\mathrm{DGtensor}\left(\left[x,v,v\right],\left[\left["con_bas","cov_bas","con_bas"\right],\left[\right]\right]\right)$
 $\left({\mathrm{D_x}}{}{\mathrm{dv}}\right){}{\mathrm{D_v}}$ (11)

Example 2.

Define the jet space J^2(R^2, R^2) for two functions u and v of 2 independent variables x and y.

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],J,2\right):$
 > $\mathrm{ω}≔\mathrm{DGbiform}\left({u}_{1,1}\right)$
 ${\mathrm{ω}}{≔}{{\mathrm{Cu}}}_{{1}{,}{1}}$ (12)
 > $\mathrm{convert}\left(\mathrm{ω},\mathrm{DGform}\right)$
 ${-}\left({{u}}_{{1}{,}{1}{,}{1}}{}{\mathrm{dx}}\right){-}\left({{u}}_{{1}{,}{1}{,}{2}}{}{\mathrm{dy}}\right){+}{{\mathrm{du}}}_{{1}{,}{1}}$ (13)
 J >