Page 10 - Intro to Tensor Calculus

6

Addition, Multiplication and Contraction
The algebraic operation of addition or subtraction applies to systems of the same type and order. That
is we can add or subtract like components in systems. For example, the sum of A i jk and B i jk is again a
i
i
system of the same type and is denoted by C jk = A i jk + B , where like components are added.
jk
The product of two systems is obtained by multiplying each component of the first system with each
component of the second system. Such a product is called an outer product. The order of the resulting
product system is the sum of the orders of the two systems involved in forming the product. For example,
i
if A is a second order system and B mnl is a third order system, with all indices having the range 1 to N,
j
i
then the product system is fifth order and is denoted C j imnl = A B mnl . The product system represents N 5
j
i mnl
terms constructed from all possible products of the components from A with the components from B .
j
The operation of contraction occurs when a lower index is set equal to an upper index and the summation
convention is invoked. For example, if we have a fifth order system C imnl and we set i = j and sum, then
j
we form the system
C mnl = C jmnl = C 1mnl + C 2mnl + ··· + C Nmnl .
j 1 2 N
Here the symbol C mnl is used to represent the third order system that results when the contraction is
performed. Whenever a contraction is performed, the resulting system is always of order 2 less than the
original system. Under certain special conditions it is permissible to perform a contraction on two lower case
indices. These special conditions will be considered later in the section.
The above operations will be more formally defined after we have explained what tensors are.

The e-permutation symbol and Kronecker delta

Two symbols that are used quite frequently with the indicial notation are the e-permutation symbol
and the Kronecker delta. The e-permutation symbol is sometimes referred to as the alternating tensor. The
e-permutation symbol, as the name suggests, deals with permutations. A permutation is an arrangement of
things. When the order of the arrangement is changed, a new permutation results. A transposition is an
interchange of two consecutive terms in an arrangement. As an example, let us change the digits 1 2 3 to
3 2 1 by making a sequence of transpositions. Starting with the digits in the order 1 2 3 we interchange 2 and
3 (first transposition) to obtain 1 3 2. Next, interchange the digits 1 and 3 ( second transposition) to obtain
312. Finally, interchange the digits 1 and 2 (third transposition) to achieve 3 2 1. Here the total number
of transpositions of 1 2 3 to 3 2 1 is three, an odd number. Other transpositions of 1 2 3 to 3 2 1 can also be
written. However, these are also an odd number of transpositions.