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DifferentialGeometry

  

DualBasis

  

calculate the dual basis to a given basis of vectors or 1-forms

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DualBasis(S, T)

Parameters

S

-

a list of independent vectors or 1-forms

T

-

(optional) a list of independent 1-forms if S is a list of vectors; a list of independent vectors if S is a list of 1-forms

Description

• 

Let S = [X_1, X_2, ..., X_n] be a list of vectors, defined on a manifold M, which define a basis for the tangent space at a point p.  Then the dual basis for the cotangent space at p is the list of 1-forms B = [alpha_1, alpha_2, ..., alpha_n] such that alpha_i(X_j) = delta_ij = {0 if i <> j and 1 if i = j}. The command DualBasis(S) will return the list of 1-forms B.

• 

Let S = [alpha_1, alpha_2, ..., alpha_n] be a list of 1-forms, defined on a manifold M, which define a basis for the cotangent space at a point p.  Then the dual basis for the tangent space at p is the list of vectors B = [X_1, X_2, ..., X_n] such that alpha_i(X_j) = delta_ij. The command DualBasis(S) will return the list of 1-forms B.

• 

More generally, let S = [X_1, X_2, ..., X_k] be a list of independent vectors defined on a manifold M and let T = [theta_1, theta_2, ..., theta_k] be a list of independent 1-forms which are transverse to S in the sense that the k x k matrix A_ij = alpha_i(X_j) is non-singular.  In this case DualBasis(S, T) returns a list of 1-forms B = [alpha_1, alpha_2, ..., alpha_k] such that span(B) = span(T) and alpha_i(X_j)  = delta_ij.

• 

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

Examples

withDifferentialGeometry&colon;

Initialize a 3-dimensional manifold M with coordinates [x, y, z].

DGsetupx&comma;y&comma;z&comma;M&colon;

 

Example 1.

S1D_x&comma;D_y&comma;D_z

S1D_x&comma;D_y&comma;D_z

(1)

B1DualBasisS1

B1dx&comma;dy&comma;dz

(2)

 

Example 2.

S2evalDGD_x&plus;D_yD_z&comma;D_yD_z&comma;D_x&plus;D_y&plus;3D_z

S2D_x+D_yD_z&comma;D_yD_z&comma;D_x+D_y+3D_z

(3)

B2DualBasisS2

B2dxdy4dz4&comma;dx+dy&comma;dy4+dz4

(4)

We check the answer by computing the interior products of S2[i] with B2[j].

Matrix3&comma;3&comma;i&comma;j&rarr;HookS2i&comma;B2j

100010001

(5)

 

Example 3.

The dual basis for the forms B2 from Example 2 are the vectors S2.

B3DualBasisB2

B3D_x+D_yD_z&comma;D_yD_z&comma;D_x+D_y+3D_z

(6)

 

Example 4.

Calculate the dual basis to the vectors S3 relative to the subspace of 1-forms T3.

S4evalDGD_x&plus;D_y&plus;D_z&comma;D_x2D_y&plus;D_z

S4D_x+D_y+D_z&comma;D_x2D_y+D_z

(7)

T4evalDGdx&plus;dy&plus;dz&comma;dx2dy&plus;3dz

T4dx+dy+dz&comma;dx2dy+3dz

(8)

B4DualBasisS4&comma;T4

B4dx3+dy3+dz3&comma;dx24dy3+7dz24

(9)

Matrix2&comma;2&comma;i&comma;j&rarr;HookS4i&comma;B4j

1001

(10)
M > 

See Also

DifferentialGeometry

Annihilator

ComplementaryBasis

DGbasis

CanonicalBasis

 


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