atlas[`&.`] - tensor product operator 

Calling Sequence: 

    `&.`(T1, T2, ..., Tn)
    T1 &. T2 

Parameters: 

       T1, T2, ..., Tn - tensors. 

Description: 

• The &. - procedure allows one to calculate the tensor product of given tensors. The main syntax is as follows: Omega &. T i.e. where and are tensors. To calculate tensor product for tensors use the following `&.`(Omega[1],Omega[2], ...Omega[k]) i.e.

 

• Tensor product is linear operation with respect to its arguments. Thus if are 0-forms then:

 

Examples: 

>	restart: with(atlas):

 

Declare Vectors  

>	Vectors(X,Y,Z);

 

(2.1)

 

Declare tensors:  

>	Tensors(T=[a,b],Omega=[0,1],Omega[1]=[p,n],Omega[2]=[q,n],Omega[3]=[l,m]);

 

(2.2)

 

Using &.- procedure: 

Tensor product is linear operation with respect to its arguments
'`&.`(Omega[1],alpha*Omega[2]+beta*Omega[3])' = `&.`(Omega[1],alpha*Omega[2]+beta*Omega[3]); 

(2.3)

 

Some more examples:
`&.`(Omega,T,Omega[3],Omega[2],Omega[1]); 

(2.4)

 

And with Lie derivative ( is 1-form see Tensors defenition above):
'L[X]'(Omega&.T)=L[X](Omega&.T); 

(2.5)

 

And again:
'L[X]'(Omega[1]&.Omega[2])=L[X](Omega[1]&.Omega[2]); 

(2.6)

 

And finally
'Omega[3]&.(L[Z](Omega)&.T)'=Omega[3]&.(L[Z](Omega)&^T); 

(2.7)

 

>

 

See Also:  

atlas, atlas[L], atlas[Tensors], atlas[`&^`], atlas[Forms].