atlas[`&.`] - tensor product operator

Calling Sequence:

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

Parameters:

        T1, T2, ..., Tn - tensors.

Description:

•  The &. procedure calculates 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);

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

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]);

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

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

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

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

See Also:

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