find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : Tensor : DifferentialGeometry/Tensor/InvariantTensorsAtAPoint

Tensor[InvariantTensorsAtAPoint] - find tensors or differential forms which are invariant under the infinitesimal action of a set of matrices

Calling Sequences

InvariantTensorsAtAPoint(A, S, options)

Parameters

A        - a list of square matrices, with dimension equal to the dimension of the space on which the tensors S are defined

S        - a list of tensors or differential forms, each of the same index type

options  - the keyword argument output

Description

 • This command calculates the tensors in the span of the tensors in the list S which are invariant with respect to the infinitesimal action generated by the matrices in the list A. This is a pointwise calculation.
 • Let  be the coordinates in terms of which the tensors in the list S are defined. If and then If   , then If  and are tensors, then . Thus, the action of $P$ on a tensor  defined at a point coincides with the Lie derivative of (as a tensor with constant coefficients) with respect to the linear vector fieldthat is, . See Example 6 for examples of this action of matrices on tensors.
 • If and then InvariantTensorsAtAPoint(A, S) returns a basis for the vector space of all tensors  (constant) such that
 • If no invariant tensors exist, an empty list is returned.
 • With output = "list", a list of invariant tensors is returned. This is the default. With output = "general", a single tensor with arbitrary coefficients is returned. If the number of matrices in the list A is 1 and output = "action", then the action of the matrix in A on the tensors in S is returned.
 • In many cases, the list of tensors to be used by InvariantTensorsAtAPoint can be created with the commands GenerateTensors, GenerateSymmetricTensors, GenerateForms.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{GroupActions}\right):$

Example 1.

Define a list of matrices for the first argument of InvariantTensorsAtAPoint .

 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right)\right]$
 ${A}{:=}\left[\left[\begin{array}{rr}{1}& {0}\\ {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]\right]$ (2.1)

Define a 2-dimensional space on which the tensors $S$ for the second argument of InvariantTensorsAtAPoint will be defined.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.2)

We take for the space of all rank 2 covariant tensors on $M$.

 M > $S≔\mathrm{evalDG}\left(\left[\mathrm{dx}&t\mathrm{dx},\mathrm{dx}&t\mathrm{dy},\mathrm{dy}&t\mathrm{dx},\mathrm{dy}&t\mathrm{dy}\right]\right)$
 ${S}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{dx}}{,}\right]$