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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

_DGvector,M,,1,1,_DGvector,M,,2,1

(1)

T1D_x,D_y,D_z

_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,3,1

(2)

C1ComplementaryBasisS1,T1

_DGvector,M,,3,1

(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

_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,1,1+_DGvector,M,,2,1

(4)

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

_DGvector,M,,1,1,_DGvector,M,,2,1,_DGvector,M,,1,1,3,1,_DGvector,M,,1,1,2,1,_DGvector,M,,3,1,_DGvector,M,,4,1

(5)

C2ComplementaryBasisS2,T2

_DGvector,M,,1,1,3,1,_DGvector,M,,4,1

(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

_DGvector,M,,1,1,_DGvector,M,,2,1

(7)

T3D_x,D_u,D_v

_DGvector,M,,1,1,_DGvector,M,,4,1,_DGvector,M,,5,1

(8)

C3ComplementaryBasisS3,T3

_DGvector,M,,4,1,_DGvector,M,,5,1

(9)

 

Example 4.

The command ComplementaryBasis works with differential forms.

S4evalDGdx&wdy,dx&wdz

_DGform,M,2,1,2,1,_DGform,M,2,1,3,1

(10)

T4evalDGdx&wdy,dx&wdz,dx&wdu,dy&wdv

_DGform,M,2,1,2,1,_DGform,M,2,1,3,1,_DGform,M,2,1,4,1,_DGform,M,2,2,5,1

(11)

C4ComplementaryBasisS4,T4

_DGform,M,2,1,4,1,_DGform,M,2,2,5,1

(12)

 

Example 5.

The command ComplementaryBasis works with tensors.

S5evalDGdx&tD_y,dx&tD_z

_DGtensor,M,cov_bas,con_bas,,1,2,1,_DGtensor,M,cov_bas,con_bas,,1,3,1

(13)

T5evalDGdx&tD_y,dx&tD_z,dx&tD_u,dy&tD_v

_DGtensor,M,cov_bas,con_bas,,1,2,1,_DGtensor,M,cov_bas,con_bas,,1,3,1,_DGtensor,M,cov_bas,con_bas,,1,4,1,_DGtensor,M,cov_bas,con_bas,,2,5,1

(14)

C5ComplementaryBasisS5,T5

_DGtensor,M,cov_bas,con_bas,,1,4,1,_DGtensor,M,cov_bas,con_bas,,2,5,1

(15)
M > 

See Also

DifferentialGeometry

Tools

CanonicalBasis

DGbasis

DualBasis