convert routines of the DifferentialGeometry package - Maple Help

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convert routines of the DifferentialGeometry package

Calling Sequence

convert(A, DGArray)

convert(B, DGvector)

convert(C, DGform)

convert(C, DGspinor)

convert(E, DGtensor)

convert(F, DGjet)

convert(F, DGdiff)

convert(G, DGbiform, bidegree)

Parameters

A

-

a tensor

B

-

a Matrix, a tensor, or the infinitesimal symmetry generators for a system of differential equations, as calculated by PDEtools:-Infinitesimals

C

-

a differential form or a differential biform

E

-

an Array, a vector, a differential form, or a spinor-tensor

F

-

a Maple expression

G

-

a differential form

Description

• 

convert(A, DGArray) will convert a rank r tensor A to an r-dimensional array.  For example, if A= a * dx1 &t dx2 &t dx3 and T = convert(A, DGArray), then T[1, 2, 3] = a  and all other entries of T are 0.

• 

If B is a square matrix, then convert(B, DGvector) will return a linear vector field.

• 

If B is a a rank 1 contravariant tensor, then convert(B, DGvector) will return the corresponding vector field.

• 

If B is a list of infinitesimal symmetry generators for a system of differential equations, as calculated by PDEtools:-Infinitesimals, then convert(B, DGvector) will return a list of vector fields.

• 

If C is a rank 1 covariant tensor, then convert(B, DGform) will return a differential 1-form.

• 

If C is a differential biform, that is, a form expressed in terms of the contact forms on a jet space, then convert(C, DGform) will return the differential form on jet space obtained by replacing the contact forms by their coordinate formulas.

• 

If E is a spin-tensor, then convert(E, DGspinor, sigma, indexlist) will convert tensorial indices of E to pairs of spinorial indices.

• 

If E is an Array, then convert(E, DGtensor, indextype) will convert E to an r-dimensional array.

• 

If E is a vector field, then convert(E, DGtensor) will convert E to a rank 1 contravariant tensor.

• 

If E is a differential p-form, then convert(E, DGtensor) will convert E to a rank p contravariant tensor.

• 

If E is a spinor-tensor, convert(E, DGtensor, sigma, indexlist) will convert pairs of spinorial indices of E to tensorial indices.

• 

convert(F, DGjet) will convert a Maple expression F involving the Maple command diff, applied to functions u(x, y, z, ...), v(x, y, z, ...), ... to the standard indexed notation for derivatives, for example, diff(u(x, y, z), x) -> u[1], diff(u(x, y, z), x, y, z, z) -> u[1, 2, 3, 3] etc.

• 

convert(F, DGdiff) performs the inverse to the conversion command convert/DGjet, it will convert expressions F involving the indexed notation for derivatives to expressions containing the Maple  diff command.

• 

convert(G, DGbiform) will convert a differential p-form G, defined on a jet space J^k(M, N) , into a list of (p + 1)-biforms, Theta = [theta_0, theta_1, theta_2, .. theta_p], where theta_k has contact degree k, and G = theta_0 + theta_1 + theta_2 + ... + theta_p.

Examples