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DifferentialGeometry

  

ComplementaryBasis

  

extend a basis for a subspace to a basis for a larger subspace

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

ComplementaryBasis(S, T)

Parameters

S, T

-

lists of vectors, differential p-forms, or tensors (of the same type)

Description

• 

The procedure ComplementaryBasis(S, T) returns a list C of vectors, differential p-forms or tensors such that the span of [S, C] equals the span of the vectors, differential p-forms or tensors  defined by T.

• 

This command is part of the DifferentialGeometry package, and so can be used in the form ComplementaryBasis(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-ComplementaryBasis.

Examples

withDifferentialGeometry:

Initialize a 5-dimensional manifold M with coordinates [x, y, z, u, v].

DGsetupx,y,z,u,v,M:

 

Example 1.

S1D_x,D_y

S1D_x,D_y

(1)

T1D_x,D_y,D_z

T1D_x,D_y,D_z

(2)

C1ComplementaryBasisS1,T1

C1D_z

(3)

 

Example 2.

Note that a basis for S2 is [D_x, D_y] and a basis for T2 is [D_x, D_y, D_x + D_z, D_u].

S2D_x,D_y,D_x+D_y

S2D_x,D_y,D_x+D_y

(4)

T2evalDGD_x,D_y,D_x+D_z,D_xD_y,D_z,D_u

T2D_x,D_y,D_x+D_z,D_xD_y,D_z,D_u

(5)

C2ComplementaryBasisS2,T2

C2D_x+D_z,D_u

(6)

 

Example 3.

In most applications the subspace spanned by the first argument S will be a subspace of the span of the second argument T.  However, the procedure works in the more general context described above.

S3D_x,D_y

S3D_x,D_y

(7)

T3D_x,D_u,D_v

T3D_x,D_u,D_v

(8)

C3ComplementaryBasisS3,T3

C3D_u,D_v

(9)

 

Example 4.

The command ComplementaryBasis works with differential forms.

S4evalDGdx &w dy,dx &w dz

S4dxdy,dxdz

(10)

T4evalDGdx &w dy,dx &w dz,dx &w du,dy &w dv

T4dxdy,dxdz,dxdu,dydv

(11)

C4ComplementaryBasisS4,T4

C4dxdu,dydv

(12)

 

Example 5.

The command ComplementaryBasis works with tensors.

S5evalDGdx &t D_y,dx &t D_z

S5dxD_y,dxD_z

(13)

T5evalDGdx &t D_y,dx &t D_z,dx &t D_u,dy &t D_v

T5dxD_y,dxD_z,dxD_u,dyD_v

(14)

C5ComplementaryBasisS5,T5

C5dxD_u,dyD_v

(15)
M > 

See Also

DifferentialGeometry

Tools

CanonicalBasis

DGbasis

DualBasis

 


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