 MultiVector - Maple Help

Tensor[MultiVector] - compute the alternating sum of the tensor product of a list of vector fields

Calling Sequences

MultiVector(V)

Parameters

V     - a list of vector fields Description

 • The bi-vector defined by vector fields $X$ and $Y$ is the rank 2, skew-symmetric, contravariant tensor field $T=X\otimes Y-Y\otimes X$. More generally, the multi-vector defined by vector fields ${X}_{1},{X}_{2},...,{X}_{r}$ is the rank $r$, skew-symmetric contravariant tensor field defined as the alternating sum of the tensor products of ${X}_{1},{X}_{2},...,{X}_{r}$.
 • The vector fields ${X}_{1},{X}_{2},...,{X}_{r}$ are linearly dependent if and only if the associated multi-vector vanishes.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form MultiVector(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-MultiVector. Examples

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

Example 1.

First create a 4 dimensional manifold $M$.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],M\right):$

Calculate the bi-vector of the two vector fields $\mathrm{X1}$ and $\mathrm{X2}$.

 M > $\mathrm{X1}≔\mathrm{D_x1}$
 ${\mathrm{X1}}{:=}{\mathrm{D_x1}}$ (2.1)
 M > $\mathrm{X2}≔\mathrm{D_x2}$
 ${\mathrm{X2}}{:=}{\mathrm{D_x2}}$ (2.2)
 M > $\mathrm{MultiVector}\left(\left[\mathrm{X1},\mathrm{X2}\right]\right)$
 ${\mathrm{D_x1}}{}{\mathrm{D_x2}}{-}{\mathrm{D_x2}}{}{\mathrm{D_x1}}$ (2.3)

Example 2.

Calculate the tri-vector of the three vector fields $\mathrm{Y1},\mathrm{Y2}$, and $\mathrm{Y3}$.

 M > $\mathrm{Y1}≔\mathrm{evalDG}\left(\mathrm{D_x1}-\mathrm{D_x2}\right)$
 ${\mathrm{Y1}}{:=}{\mathrm{D_x1}}{-}{\mathrm{D_x2}}$ (2.4)
 M > $\mathrm{Y2}≔\mathrm{evalDG}\left(\mathrm{D_x2}-\mathrm{D_x3}\right)$
 ${\mathrm{Y2}}{:=}{\mathrm{D_x2}}{-}{\mathrm{D_x3}}$ (2.5)
 M > $\mathrm{Y3}≔\mathrm{evalDG}\left(\mathrm{D_x3}-\mathrm{D_x4}\right)$
 ${\mathrm{Y3}}{:=}{\mathrm{D_x3}}{-}{\mathrm{D_x4}}$ (2.6)
 M > $\mathrm{MultiVector}\left(\left[\mathrm{Y1},\mathrm{Y2},\mathrm{Y3}\right]\right)$
 ${\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{-}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{-}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{+}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{+}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{-}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{-}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{+}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{+}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{-}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{-}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{+}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{+}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{-}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{-}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{+}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{+}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{-}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{-}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{+}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{+}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{-}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{-}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{+}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{}{\mathrm{D_x2}}$ (2.7)

Example 3.

Use the MultiVector command to determine when a vector field $Z$ lies in the span of the vector fields $\mathrm{Y1},\mathrm{Y2},\mathrm{Y3}$.

 M > $Z≔\mathrm{evalDG}\left(a\mathrm{D_x1}+b\mathrm{D_x2}+c\mathrm{D_x3}+d\mathrm{D_x4}\right)$
 ${Z}{:=}{a}{}{\mathrm{D_x1}}{+}{b}{}{\mathrm{D_x2}}{+}{c}{}{\mathrm{D_x3}}{+}{d}{}{\mathrm{D_x4}}$ (2.8)
 M > $T≔\mathrm{MultiVector}\left(\left[Z,\mathrm{Y1},\mathrm{Y2},\mathrm{Y3}\right]\right)$
 ${T}{:=}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x4}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x3}}{}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x4}}{}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x3}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x4}}{}{\mathrm{D_x2}}{}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{+}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{-}\left({c}{+}{d}{+}{a}{+}{b}\right){}{\mathrm{D_x4}}{}{\mathrm{D_x3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}$ (2.9)
 M > $\mathrm{Tools}:-\mathrm{DGinfo}\left(T,"CoefficientSet"\right)$
 $\left\{{c}{+}{d}{+}{a}{+}{b}{,}{-}{d}{-}{c}{-}{b}{-}{a}\right\}$ (2.10)

So $Z$ is a linear combination of $\mathrm{Y1},\mathrm{Y2},\mathrm{Y3}$ precisely when $a+b+c+d=0$.