**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 ).