generate a list of all product tensors from a list of lists of tensors - Maple Help

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Tensor[GenerateTensors] - generate a list of all product tensors from a list of lists of tensors

Calling Sequences

    GenerateTensors(Tlist)

Parameters

   Tlist    - a list of lists of tensor fields

Description

• 

With Tlist = [T1,T2,...,Tr] the command GenerateTensors(Tlist) will generate a list of tensors by forming all possible r-fold tensor products t1t2...tr , where the first tensor t1 belongs to the list T1, the second tensor t2 belongs to the list T2, and so on.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 3 dimensional manifold M. 

DGsetupx,y,z,M:

 

Create a list L1 of all type (1, 1) tensors on M.

M > 

T1:=Tools:-DGinfoFrameBaseVectors

T1:=D_x,D_y,D_z

(2.1)
M > 

T2:=Tools:-DGinfoFrameBaseForms

T2:=dx,dy,dz

(2.2)
M > 

L1:=GenerateTensorsT1,T2

L1:=D_xdx,D_xdy,D_xdz,D_ydx,D_ydy,D_ydz,D_zdx,D_zdy,D_zdz

(2.3)
M > 

nopsL1

9

(2.4)

 

Example 2.

Create a list L2 of all rank 3 covariant tensors which are symmetric in their first 2 indices.

M > 

S:=GenerateSymmetricTensorsT1,2

S:=D_xD_x,12D_xD_y+12D_yD_x,12D_xD_z+12D_zD_x,D_yD_y,12D_yD_z+12D_zD_y,D_zD_z

(2.5)
M > 

L2:=GenerateTensorsS,T2

L2:=D_xD_xdx,D_xD_xdy,D_xD_xdz,12D_xD_ydx+12D_yD_xdx,12D_xD_ydy+12D_yD_xdy,12D_xD_ydz+12D_yD_xdz,12D_xD_zdx+12D_zD_xdx,12D_xD_zdy+12D_zD_xdy,12D_xD_zdz+12D_zD_xdz,D_yD_ydx,D_yD_ydy,D_yD_ydz,12D_yD_zdx+12D_zD_ydx,12D_yD_zdy+12D_zD_ydy,12D_yD_zdz+12D_zD_ydz,D_zD_zdx,D_zD_zdy,D_zD_zdz

(2.6)
M > 

nopsL2

18

(2.7)

 

Example 3.

Create a list L3 of all rank 3 covariant tensors which are skew-symmetric in their first 2 indices.

M > 

W:=Tools:-GenerateFormsT2,2

W:=dxdy,dxdz,dydz

(2.8)
M > 

mapconvert,W,DGtensor

dxdydydx,dxdzdzdx,dydzdzdy

(2.9)
M > 

L3:=GenerateTensorsW,T2

L3:=dxdydxdydxdx,dxdydydydxdy,dxdydzdydxdz,dxdzdxdzdxdx,dxdzdydzdxdy,dxdzdzdzdxdz,dydzdxdzdydx,dydzdydzdydy,dydzdzdzdydz

(2.10)

See Also

DifferentialGeometry, Tensor, DGinfo, GenerateSymmetricTensors, GenerateForms


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