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Calling Sequence
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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)
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Parameters
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A
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a tensor
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B
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a Matrix, a tensor, or the infinitesimal symmetry generators for a system of differential equations, as calculated by PDEtools:-Infinitesimals
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C
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a differential form or a differential biform
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E
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an Array, a vector, a differential form, or a spinor-tensor
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F
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a Maple expression
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G
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a differential form
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Description
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convert(A, DGArray) converts 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.
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If B is a a rank 1 contravariant tensor, then convert(B, DGvector) returns the corresponding vector field.
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If C is a rank 1 covariant tensor, then convert(B, DGform) returns a differential 1-form.
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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) returns the differential form on jet space obtained by replacing the contact forms by their coordinate formulas.
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If E is a differential p-form, then convert(E, DGtensor) converts E to a rank p contravariant tensor.
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convert(F, DGjet) converts 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.
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convert(F, DGdiff) performs the inverse to the conversion command convert/DGjet: it converts expressions F involving the indexed notation for derivatives to expressions containing the Maple diff command.
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convert(G, DGbiform) converts 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.
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Examples
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convert/DGArray
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Example 1.
We define a manifold M with coordinates [x, y, z]. On M we define two tensors T1 and T2. We convert these tensors to a Maple Array
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convert/DGvector
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Example 1.
We define a manifold M with coordinates [x, y, z]. We convert a matrix to a linear vector field.
Example 2.
We define a manifold M with coordinates [x, y, z]. On M we define two tensors T1 and T2 and contract the 1st and 3rd indices of T1 against the 1st and second indices of T2. The result is a contravariant rank 1 tensor S on M. We then convert S to a vector field X.
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_DG([["tensor", M, [["con_bas"], []]], [[[1], 2], [[3], 1]]])
_DG([["vector", M, []], [[[1], 2], [[3], 1]]])
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Example 3.
We use the PDEtools:-Infinitesimals program to find the infinitesimals symmetries of a system of differential equations. We convert the output of this program, which lists of the components of the infinitesimal symmetries, to a list of vector fields on the manifold of independent and dependent variables.
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convert/DGform
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Example 1.
We define a manifold M with coordinates [x, y, z]. On M we define a tensor T and contract the 1st and 3rd indices of T. The result is a covariant rank 1 tensor S on M. We then convert S to a differential 1-form alpha.
The tensor S and the 1 form alpha both have the same external displays but are have different internal representations. We can see this using DGinfo or lprint.
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_DG([["tensor", M, [["cov_bas"], []]], [[[1], 1], [[2], 2]]])
_DG([["form", M, 1], [[[1], 1], [[2], 2]]])
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Example 2.
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convert/DGspinor
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The solder form sigma is the fundamental spin-tensor in spinor algebra. It provides for a 1-1 correspondence between vectors and rank 2 Hermitian spinors. The convert/DGspinor uses a user-specified solder form to convert tensor indices of a given spinor-tensor to pairs of spinor indices.
To illustrate this convert procedure we first construct a solder form. Define a vector bundle over M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2].
Define a spacetime metric g on M. Our spinor conventions assume the metric has signature [1, -1, -1, -1].
Define an orthonormal frame on M with respect to the metric g.
Calculate the solder form sigma from the frame F. See Spinor, SolderForm.
Example 1.
Define a vector X and convert it to a contravariant rank 2 spinor psi. The convert/DGspinor command requires the solder form as a 3rd argument and a list of tensorial indices to be converted to spinorial indices as a 4th argument.
Example 2.
Define a rank 3 tensor and convert the 1st and 3rd indices to pairs of spinor indices to obtain a rank 5 spin-tensor chi. Note that the spinorial indices are moved to the right.
Example 3.
With the convert/DGspinor command, the spinor psi. can be converted back to the vector X and the spin-tensor chi back to the tensor T.
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convert/DGtensor
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We define a manifold M with coordinates [x, y, z].
Example 1.
We define a Matrix' A1 and convert it to a rank 2 tensor in 4 different ways:
Example 2.
We define a four-dimensional array and convert it to a rank 4 covariant tensor.
Example 3.
We define a vector field and convert it to a rank 1 contravariant tensor field.
The tensor T and vector X both have the same external displays but have different internal representations. We can see this using DGinfo or lprint.
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_DG([["vector", M, []], [[[2], x^2], [[3], -y^2]]])
_DG([["tensor", M, [["con_bas"], []]], [[[2], x^2], [[3], -y^2]]])
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Example 4.
We define a differential 2-form and convert it to a rank 2 covariant tensor field.
Example 5.
A spinor or spinor-tensor can be converted to a tensor using a solder form sigma. To work with spinors, define a vector bundle over M with base coordinates [x, y, z, t] and fiber coordinates [z1, z2, w1, w2].
Define a spacetime metric g on M. Our spinor convention require that the metric has signature [1, -1,-1,-1].
Define an orthonormal frame on M with respect to the metric g.
Calculate the solder form sigma from the frame F. See Spinor, SolderForm.
Define a contravariant rank 2 spinor psi and convert it to a rank 1 contravariant tensor X. In this situation, convert/DGtensor command requires the solder form as a 3rd argument and a list of pairs of spinorial indices to be converted to tensorial indices as a 4th argument.
Example 6.
Define a covariant rank 2 spinor psi and convert it to a rank 1 covariant tensor omega.
Example 7.
Define a covariant rank 4 spinor psi and convert it to a rank 2 covariant tensor G.
Example 8.
Define a mixed contravariant rank 5 spin-tensor psi and convert it to a rank 3 contravariant tensor in 2 different ways. (The tensors T1 and T2 are complex because psi is not Hermitian.)
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convert/DGjet, convert/DGdiff
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Example 1.
We define the 2nd order jet space J^2(R^2, R) for one function u of two independent variables [x, y].
We convert the Laplace of a function, defined using the Maple diff command into a standard index notation for coordinates on jet spaces.
Example 2.
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We retrieve a 4th order ODE from Kamke and rewrite it in standard index notation for coordinates on jet spaces.
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Example 3.
Define a 2-form on J^2(R^2, R). We convert the KdV equation, given in jet notation, to ordinary derivative notation.
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convert/DGbiform
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Define a 2-form on J^2(R^2, R). We define the 2nd order jet space J^2(R^2, R) for one function u of two independent variables [x, y].
Define a 2-form on J^2(R^2, R).
Define a 2-form on J^2(R^2, R). To obtain the representation of omega in terms of the contact forms Cu[1] = du[1] - u[1, 1]*dx - u[1, 2]*dy and Cu[2] = du[2] - u[1, 2]*dx - u[2, 2]*dy (*) we solve the equations (*) for du[1] and du[2], substitute for du[1] and du[2] into omega, and then grade the terms according to their contact degree -- [2, 0] : not contact forms; [1,1] : linear in contact forms; [0, 2] : quadratic in contact forms.
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