DifferentialGeometry - Maple Programming Help

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

DifferentialGeometry

 IntersectSubspaces
 find the intersection of a list of vector subspaces of vectors, forms or tensors

 Calling Sequence IntersectSubspaces(S)

Parameters

 S - a list [A1, A2, ...], where each Ai is a list of vectors, forms or tensors

Description

 • IntersectSubspaces(S) computes the intersection of the subspaces spanned by the elements of the list.
 • This command is part of the DifferentialGeometry package, and so can be used in the form IntersectSubspaces(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-IntersectSubspaces.

Examples

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

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

 > $\mathrm{DGsetup}\left(\left[x,y,z,w\right],M\right):$

Example 1.

Find the intersection of the three 3 dimensional subspaces spanned by A1, A2, A3.

 > $\mathrm{A1}≔\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${\mathrm{A1}}{≔}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (1)
 > $\mathrm{A2}≔\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_w}\right]$
 ${\mathrm{A2}}{≔}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_w}}\right]$ (2)
 > $\mathrm{A3}≔\mathrm{evalDG}\left(\left[\mathrm{D_y}+\mathrm{D_z},\mathrm{D_z}+\mathrm{D_w},\mathrm{D_w}\right]\right)$
 ${\mathrm{A3}}{≔}\left[{\mathrm{D_y}}{+}{\mathrm{D_z}}{,}{\mathrm{D_z}}{+}{\mathrm{D_w}}{,}{\mathrm{D_w}}\right]$ (3)
 > $\mathrm{IntersectSubspaces}\left(\left[\mathrm{A1},\mathrm{A2},\mathrm{A3}\right]\right)$
 $\left[{\mathrm{D_y}}\right]$ (4)

Example 2.

Find the intersection of the subspaces of 2-forms spanned by B1 and B2.  Check the result using the GetComponents command.

 > $\mathrm{B1}≔\mathrm{evalDG}\left(\left[\mathrm{dx}&w\mathrm{dy}+\mathrm{dy}&w\mathrm{dz},\mathrm{dx}&w\mathrm{dw}-\mathrm{dy}&w\mathrm{dz},\mathrm{dx}&w\mathrm{dw}+\mathrm{dy}&w\mathrm{dw},\mathrm{dx}&w\mathrm{dy}+\mathrm{dx}&w\mathrm{dz}-\mathrm{dz}&w\mathrm{dw}\right]\right)$
 ${\mathrm{B1}}{≔}\left[{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}{,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dw}}{-}\left({\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\right){,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dw}}{+}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dw}}{,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{-}\left({\mathrm{dz}}{}{\bigwedge }{}{\mathrm{dw}}\right)\right]$ (5)
 > $\mathrm{B2}≔\mathrm{evalDG}\left(\left[\mathrm{dx}&w\mathrm{dy}-\mathrm{dy}&w\mathrm{dz},\mathrm{dy}&w\mathrm{dz}+\mathrm{dz}&w\mathrm{dw},\mathrm{dx}&w\mathrm{dz}+\mathrm{dz}&w\mathrm{dw}\right]\right)$
 ${\mathrm{B2}}{≔}\left[{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{-}\left({\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\right){,}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\bigwedge }{}{\mathrm{dw}}{,}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\bigwedge }{}{\mathrm{dw}}\right]$ (6)
 > $C≔\mathrm{IntersectSubspaces}\left(\left[\mathrm{B1},\mathrm{B2}\right]\right)$
 ${C}{≔}\left[{-}\left(\frac{{\mathrm{dx}}}{{2}}{}{\bigwedge }{}{\mathrm{dy}}\right){+}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{-}\left(\frac{{3}{}{\mathrm{dy}}}{{2}}{}{\bigwedge }{}{\mathrm{dz}}\right){-}\left({\mathrm{dz}}{}{\bigwedge }{}{\mathrm{dw}}\right)\right]$ (7)

The command GetComponents returns the components of the 2-form in C with respect to the 2-forms in B1 and B2.  This proves that the 2-form in C does indeed belong to the intersection of the spans of B1 and B2.

 > $\mathrm{GetComponents}\left(C,\mathrm{B1}\right)$
 $\left[\left[{-}\frac{{3}}{{2}}{,}{0}{,}{0}{,}{1}\right]\right]$ (8)
 > $\mathrm{GetComponents}\left(C,\mathrm{B2}\right)$
 $\left[\left[{-}\frac{{1}}{{2}}{,}{-}{2}{,}{1}\right]\right]$ (9)
 M >