extend a basis for a subspace to a basis for a larger subspace - Maple Help

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DifferentialGeometry[ComplementaryBasis] - extend a basis for a subspace to a basis for a larger subspace

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.

S1:=D_x,D_y

S1:=`*`D_x,`*`D_y

(1)

T1:=D_x,D_y,D_z

T1:=`*`D_x,`*`D_y,`*`D_z

(2)

C1:=ComplementaryBasisS1,T1

C1:=`*`D_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].

S2:=D_x,D_y,D_x+D_y

S2:=`*`D_x,`*`D_y,`*`D_x+`*`D_y

(4)

T2:=evalDGD_x,D_y,D_x+D_z,D_xD_y,D_z,D_u

T2:=`*`D_x,`*`D_y,`*`D_x+`*`D_z,`*`D_x`*`D_y,`*`D_z,`*`D_u

(5)

C2:=ComplementaryBasisS2,T2

C2:=`*`D_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.

S3:=D_x,D_y

S3:=`*`D_x,`*`D_y

(7)

T3:=D_x,D_u,D_v

T3:=`*`D_x,`*`D_u,`*`D_v

(8)

C3:=ComplementaryBasisS3,T3

C3:=`*`D_u,`*`D_v

(9)

 

Example 4.

The command ComplementaryBasis works with differential forms.

S4:=evalDGdx &w dy,dx &w dz

S4:=`*`dxdy,`*`dxdz

(10)

T4:=evalDGdx &w dy,dx &w dz,dx &w du,dy &w dv

T4:=`*`dxdy,`*`dxdz,`*`dxdu,`*`dydv

(11)

C4:=ComplementaryBasisS4,T4

C4:=`*`dxdu,`*`dydv

(12)

 

Example 5.

The command ComplementaryBasis works with tensors.

S5:=evalDGdx &t D_y,dx &t D_z

S5:=`*`dxD_y,`*`dxD_z

(13)

T5:=evalDGdx &t D_y,dx &t D_z,dx &t D_u,dy &t D_v

T5:=`*`dxD_y,`*`dxD_z,`*`dxD_u,`*`dyD_v

(14)

C5:=ComplementaryBasisS5,T5

C5:=`*`dxD_u,`*`dyD_v

(15)
M > 

See Also

DifferentialGeometry, Tools, CanonicalBasis, DGbasis, DualBasis


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