Using Tensors

 

Elementary Operations with Tensors

Tensor operations are operations on tensors that result in quantities that are still tensors.  A simple way of establishing whether or not a quantity is a tensor, is to see how it transforms under a coordinate transformation.  For example, we can deduce directly from the transformation law that two tensors of the same type can be added together to give a tensor of the same type, e.g.

(2.1.1)

                                      

The same holds true for subtraction and scalar multiplication.

A covariant tensor of rank 2 is said to be symmetric if

 

(2.1.2)

                                                 ,

 

in which case it has only  independently components (check this by establishing how many independent components there are of a symmetric matrix of order n).  A similar definition holds for a contravariant tensor .  The tensor  is said to be anti-symmetric or skew symmetric if

(2.1.3)

                                              ,

which has only  independently components; this is again a tensorial property.  A notation frequently used to denote the symmetric part of a tensor is

(2.1.4)

                     

 

and the anti-symmetric part is

(2.1.5)

                    

 

In general the symmetrization of a tensor relative to its covariant indices can be written:

(2.1.6)

In general the antisymmetrization of a tensor relative to its covariant indices can be written:(2.1.7)

 

For example, consider the covariant rank 3 antisymmetric tensor

(2.1.8) .

 

(A way to remember the above expression is to note that the positive terms are obtained by cycling the indices to the right and the corresponding negative terms by flipping the last two indices).  A totally symmetric tensor is defined to be one equal to its symmetric part, and a totally anti-symmetric tensor is one equal to its anti-symmetric part.

We can multiply two tensors of type  and  together and obtain a tensor of type , e.g.

(2.1.9)                                

 

In particular, a tensor of type  when multiplied by a scalar field  is again a tensor of type .  Given a tensor of mixed type , we can form a tensor of type  by the process of contraction, which simply involves setting a raised and lowered index equal.  For example,

(2.1.10)

 

i.e. A tensor of type  has become a tensor of type .  Notice that we can contract a tensor by multiplying by the Kronecker-Delta tensor , e.g.

(2.1.11)                           

 

 

In effect, multiplying by  turns the index  into  (or equivalently the index  into  ).