Page 13 - Intro to Tensor Calculus

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EXAMPLE 1.1-6. Some examples of the e−permutation symbol and Kronecker delta are:

1
e 123 = e 123 =+1 δ =1 δ 12 =0
1
1
e 213 = e 213 = −1 δ =0 δ 22 =1
2
1
e 112 = e 112 =0 δ =0 δ 32 =0.
3
EXAMPLE 1.1-7. When an index of the Kronecker delta δ ij is involved in the summation convention,
the effect is that of replacing one index with a different index. For example, let a ij denote the elements of an
N × N matrix. Here i and j are allowed to range over the integer values 1, 2,...,N. Consider the product

a ij δ ik

where the range of i, j, k is 1, 2,...,N. The index i is repeated and therefore it is understood to represent
a summation over the range. The index i is called a summation index. The other indices j and k are free
indices. They are free to be assigned any values from the range of the indices. They are not involved in any
summations and their values, whatever you choose to assign them, are fixed. Let us assign a value of j and
k to the values of j and k. The underscore is to remind you that these values for j and k are fixed and not
to be summed. When we perform the summation over the summation index i we assign values to i from the
range and then sum over these values. Performing the indicated summation we obtain

a ij δ ik = a 1j δ 1k + a 2j δ 2k + ··· + a kj δ kk + ··· + a Nj δ Nk .

In this summation the Kronecker delta is zero everywhere the subscripts are different and equals one where
the subscripts are the same. There is only one term in this summation which is nonzero. It is that term
where the summation index i was equal to the fixed value k This gives the result

a kj δ kk = a kj

where the underscore is to remind you that the quantities have fixed values and are not to be summed.
Dropping the underscores we write
a ij δ ik = a kj

Here we have substituted the index i by k and so when the Kronecker delta is used in a summation process
it is known as a substitution operator. This substitution property of the Kronecker delta can be used to
simplify a variety of expressions involving the index notation. Some examples are:

B ij δ js = B is

δ jk δ km = δ jm
e ijk δ im δ jn δ kp = e mnp .

Some texts adopt the notation that if indices are capital letters, then no summation is to be performed.
For example,

a KJ δ KK = a KJ