form all possible symmetric tensors of a given rank (linearly independent over the real numbers) from a list of symmetric tensors - Maple Programming Help

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Tensor[SymmetricProductsOfKillingTensors] - form all possible symmetric tensors of a given rank (linearly independent over the real numbers) from a list of symmetric tensors

Calling Sequences

     SymmetricProducstOfKillingTensors(K, p, ptlist )

Parameters

     K         - a list of lists of Killing tensors K ={K1, K2, ...], where Kr is a list of Killing tensors of rank r on a manifold M

     p         - a positive integer.

     ptlist    - (optional) a list of points on the manifold M

 

Description

Examples

Description

• 

This program first computes all partitions p1,p2,...of the integer p, that is, all non-decreasing sequences p1,p2,...with p1+p2+=p. Then, for each partition, all possible rank p symmetric tensors of the form T1T2T3... ,where the tensor Ti is taken from the list K, are generated. From the totality of tensors so obtained a maximal set of linearly independent tensors (over the real numbers) is selected. Each symmetric tensor in the returned list is a Killing tensor if each of the Kiare.

• 

The independent tensors are generated by a call to the DifferentialGeometry command DGbasis. For tensors with coefficients which are not rational functions, the DGbasis program may work faster using a Wronskian approach which requires the specification of a list of points on the manifold M.

• 

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

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

DGsetupx,y,M

frame name: M

(2.1)
M > 

gevalDGdx&sdy

gdx2dy+dy2dx

(2.2)

 

Calculate the rank 1 Killing tensors.

M > 

K1KillingTensorsg,1

K1ydx+xdy,dy,dx

(2.3)

 

Calculate all the rank 2 and rank 3 Killing tensors generated by K1.

M > 

S2SymmetricProductsOfKillingTensorsK1,2

S2y2dxdxyxdxdyyxdydx+x2dydy,y2dxdyy2dydx+xdydy,ydxdx+x2dxdy+x2dydx,dydy,dx2dy+dy2dx,dxdx

(2.4)
M > 

nopsS2

6

(2.5)
M > 

S3SymmetricProductsOfKillingTensorsK1,3

S3y3dxdxdx+y2xdxdxdy+y2xdxdydxyx2dxdydy+y2xdydxdxyx2dydxdyyx2dydydx+x3dydydy,y23dxdxdy+y23dxdydx2yx3dxdydy+y23dydxdx2yx3dydxdy2yx3dydydx+x2dydydy,y2dxdxdx2yx3dxdxdy2yx3dxdydx+x23dxdydy2yx3dydxdx+x23dydxdy+x23dydydx,y3dxdydyy3dydxdyy3dydydx+xdydydy,y3dxdxdyy3dxdydx+x3dxdydyy3dydxdx+x3dydxdy+x3dydydx,ydxdxdx+x3dxdxdy+x3dxdydx+x3dydxdx,dydydy,dx3dydy+dy3dxdy+dy3dydx,dx3dxdy+dx3dydx+dy3dxdx,dxdxdx

(2.6)
M > 

nopsS3

10

(2.7)

 

Check that all the tensors in S2 are indeed Killing tensors using the CheckKillingTensor command.

M > 

map2CheckKillingTensor,g,S2

0dxdxdx,0dxdxdx,0dxdxdx,0dxdxdx,0dxdxdx,0dxdxdx

(2.8)

 

Calculate all the rank 3 Killing tensors generated by S3 and the metric g. We get the list S3 again because the metric is already a constant linear combination of the rank 1 Killing tensors.

M > 

S3aSymmetricProductsOfKillingTensorsK1,g,3:

M > 

nopsS3

10

(2.9)

See Also

DifferentialGeometry

Tensor

CheckKillingTensor

DGbasis

GenerateTensors

GetComponents

KillingTensors

SymmetrizeIndices