Annihilator - Maple Help

DifferentialGeometry

 Annihilator
 find the subspace of vectors (or 1-forms) whose interior product with a given list of 1-forms (or vectors) vanishes

 Calling Sequence Annihilator(S, T)

Parameters

 S - a list of vectors or a list of 1-forms T - (optional) a list of 1-forms if S is a list of vectors or a list of vectors if S is a list of 1-forms

Description

 • Let S be a list of 1-forms and T a list of vectors.  Then Annihilator(S, T) calculates the subspace of vectors X in the span of T such that alpha(X) = 0 for all alpha in S.
 • Let S be a list of vectors and T a list of 1-forms.  Then Annihilator(S, T) calculates the subspace of 1-forms alpha in the span of T such that alpha(X) = 0 for all X in S.
 • If the optional argument T is not given, then T is taken to be the standard basis for the tangent space or cotangent space for the manifold M on which the elements of S are defined.
 • This command is part of the DifferentialGeometry package, and so can be used in the form Annihilator(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-Annihilator.

Examples

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

Example 1.

Calculate the annihilator of the set of 1-forms S1 relative to subspaces T1, T2, and the full tangent space.

 > $\mathrm{S1}≔\left[\mathrm{dx},\mathrm{dy}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (1)
 > $\mathrm{T1}≔\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (2)
 > $\mathrm{T2}≔\left[\mathrm{D_x},\mathrm{D_y}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (3)
 > $\mathrm{Annihilator}\left(\mathrm{S1},\mathrm{T1}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (4)
 > $\mathrm{Annihilator}\left(\mathrm{S1},\mathrm{T2}\right)$
 $\left[\right]$ (5)
 > $\mathrm{Annihilator}\left(\mathrm{S1}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (6)

Example 2.

Calculate the annihilator of the set of vectors S2 and S3.

 > $\mathrm{S2}≔\left[\mathrm{D_y}\right]$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right)\right]$ (7)
 > $\mathrm{Annihilator}\left(\mathrm{S2}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right)\right]$ (8)
 > $\mathrm{S3}≔\mathrm{evalDG}\left(\left[\mathrm{D_x}-2\mathrm{D_y}+\mathrm{D_z},\mathrm{D_y}+3\mathrm{D_z}-\mathrm{D_w}\right]\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]{,}\left[\left[{2}\right]{,}{-}{2}\right]{,}\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[{}\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]{,}\left[\left[{3}\right]{,}{3}\right]{,}\left[\left[{4}\right]{,}{-}{1}\right]\right]\right]\right)\right]$ (9)
 > $\mathrm{A3}≔\mathrm{Annihilator}\left(\mathrm{S3}\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{2}\right]{,}\left[\left[{2}\right]{,}{1}\right]{,}\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{M}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{-}{7}\right]{,}\left[\left[{2}\right]{,}{-}{3}\right]{,}\left[\left[{3}\right]{,}{1}\right]\right]\right]\right)\right]$ (10)

Let us check this result.

 > $\mathrm{Matrix}\left(2,2,\left(i,j\right)↦\mathrm{Hook}\left(\mathrm{S3}\left[i\right],\mathrm{A3}\left[j\right]\right)\right)$
 $\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]$ (11)
 M >