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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

Parameters

Description

Examples

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

withDifferentialGeometry:

DGsetupx,y,z,w,M:

 

Example 1.

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

S1dx,dy

S1dx,dy

(1)

T1D_x,D_y,D_z

T1D_x,D_y,D_z

(2)

T2D_x,D_y

T2D_x,D_y

(3)

AnnihilatorS1,T1

D_z

(4)

AnnihilatorS1,T2

(5)

AnnihilatorS1

D_w,D_z

(6)

 

Example 2.

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

S2D_y

S2D_y

(7)

AnnihilatorS2

dw,dz,dx

(8)

S3evalDGD_x2D_y+D_z,D_y+3D_zD_w

S3D_x2D_y+D_z,D_y+3D_zD_w

(9)

A3AnnihilatorS3

A32dx+dy+dw,7dx3dy+dz

(10)

Let us check this result.

Matrix2,2,i,j→HookS3i,A3j

0000

(11)
M > 

See Also

DifferentialGeometry

DualBasis

Hook

 


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