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

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 3 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.

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.

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.

Example 2.

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 uer 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.

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 an 4 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.

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.)

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.

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.

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.