&MatrixMinus - Maple Help

DifferentialGeometry:-Tools[&MatrixMinus, &MatrixMult, &MatrixPlus, &MatrixWedge]

 Calling Sequence A &MatrixMinus B - subtract two Matrices/Vectors of vectors, differential forms or tensors A &MatrixMult C - multiply a Matrix/Vector A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars C &MatrixMult A - multiply a Matrix A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars A &MatrixPlus B - add two Matrices/Vectors of vectors, differential forms or tensors E &MatrixWedge F - calculate the Matrix wedge product of two Matrices/Vectors of differential forms.

Parameters

 A, B - two Matrices/Vectors of vectors, differential forms or tensors C - a scalar or a Matrix/Vector of scalars E, F - two Matrices/Vectors of differential forms

Description

 • These commands provide, within the DifferentialGeometry environment, the basic arithmetical operations for Matrices or Vectors of: vectors, differential forms, or tensors.  They are particularly useful for curvature calculations for connections on principle bundles of matrix groups.
 • These commands are part of the DifferentialGeometry:-Tools package, and so can be used in the form described above only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.

Examples

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

Define a 3-dimensional manifold M with coordinates [x, y, z].

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

Example 1

Define two column Vectors of 1 forms A, B; a 2x2 matrix C of scalars; a row Vector of 1 forms E and a 2x2 Matrix of 1 forms F.

 > $A≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}-\mathrm{dy},\mathrm{dy}+\mathrm{dx}\right]\right)\right)$
 ${A}{≔}\left[\begin{array}{c}{\mathrm{dx}}{-}{\mathrm{dy}}\\ {\mathrm{dx}}{+}{\mathrm{dy}}\end{array}\right]$ (1)
 > $B≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}+2\mathrm{dy},\mathrm{dx}+3\mathrm{dy}\right]\right)\right)$
 ${B}{≔}\left[\begin{array}{c}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}\\ {\mathrm{dx}}{+}{3}{}{\mathrm{dy}}\end{array}\right]$ (2)
 > $C≔\mathrm{Matrix}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)$
 ${C}{≔}\left[\begin{array}{cc}{1}& {2}\\ {3}& {4}\end{array}\right]$ (3)
 > $E≔\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)$
 ${E}{≔}\left[\begin{array}{cc}{\mathrm{dx}}{-}{\mathrm{dy}}& {\mathrm{dx}}{+}{\mathrm{dy}}\end{array}\right]$ (4)
 > $F≔\mathrm{Matrix}\left(\mathrm{evalDG}\left(\left[\left[\mathrm{dx}-\mathrm{dz},\mathrm{dy}\right],\left[\mathrm{dz},\mathrm{dx}+\mathrm{dy}+3\mathrm{dz}\right]\right]\right)\right)$
 ${F}{≔}\left[\begin{array}{cc}{\mathrm{dx}}{-}{\mathrm{dz}}& {\mathrm{dy}}\\ {\mathrm{dz}}& {\mathrm{dx}}{+}{\mathrm{dy}}{+}{3}{}{\mathrm{dz}}\end{array}\right]$ (5)

Perform various arithmetic operations with the quantities A, B, C, E, F.

 > $A\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixPlus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}B$
 $\left[\begin{array}{c}{2}{}{\mathrm{dx}}{+}{\mathrm{dy}}\\ {2}{}{\mathrm{dx}}{+}{4}{}{\mathrm{dy}}\end{array}\right]$ (6)
 > $A\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMinus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}B$
 $\left[\begin{array}{c}{-}{3}{}{\mathrm{dy}}\\ {-}{2}{}{\mathrm{dy}}\end{array}\right]$ (7)
 > $a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}A$
 $\left[\begin{array}{c}{a}{}{\mathrm{dx}}{-}{a}{}{\mathrm{dy}}\\ {a}{}{\mathrm{dx}}{+}{a}{}{\mathrm{dy}}\end{array}\right]$ (8)
 > $C\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}A$
 $\left[\begin{array}{c}{3}{}{\mathrm{dx}}{+}{\mathrm{dy}}\\ {7}{}{\mathrm{dx}}{+}{\mathrm{dy}}\end{array}\right]$ (9)
 > $E\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}C$
 $\left[\begin{array}{cc}{4}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}& {6}{}{\mathrm{dx}}{+}{2}{}{\mathrm{dy}}\end{array}\right]$ (10)
 > $E\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixWedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}B$
 $\left[\begin{array}{c}{5}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}\end{array}\right]$ (11)
 > $C\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMult\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$
 $\left[\begin{array}{cc}{\mathrm{dx}}{+}{\mathrm{dz}}& {2}{}{\mathrm{dx}}{+}{3}{}{\mathrm{dy}}{+}{6}{}{\mathrm{dz}}\\ {3}{}{\mathrm{dx}}{+}{\mathrm{dz}}& {4}{}{\mathrm{dx}}{+}{7}{}{\mathrm{dy}}{+}{12}{}{\mathrm{dz}}\end{array}\right]$ (12)
 > $F\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixWedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}A$
 $\left[\begin{array}{c}{-}{2}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dy}}{+}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\\ {-}{4}{}{\mathrm{dx}}{}{\bigwedge }{}{\mathrm{dz}}{-}{2}{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\end{array}\right]$ (13)
 > $F\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixWedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}F$
 $\left[\begin{array}{cc}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}& {4}{}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\\ {\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}& {-}{\mathrm{dy}}{}{\bigwedge }{}{\mathrm{dz}}\end{array}\right]$ (14)