Page 14 - Intro to Tensor Calculus

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as δ KK represents a single term because of the capital letters. Another notation which is used to denote no
summation of the indices is to put parenthesis about the indices which are not to be summed. For example,

a (k)j δ (k)(k) = a kj ,

since δ (k)(k) represents a single term and the parentheses indicate that no summation is to be performed.
At any time we may employ either the underscore notation, the capital letter notation or the parenthesis
notation to denote that no summation of the indices is to be performed. To avoid confusion altogether, one
can write out parenthetical expressions such as “(no summation on k)”.

i
EXAMPLE 1.1-8. In the Kronecker delta symbol δ we set j equal to i and perform a summation. This
j
i
operation is called a contraction. There results δ , which is to be summed over the range of the index i.
i
Utilizing the range 1, 2,... ,N we have
i
2
1
δ = δ + δ + ··· + δ N
i 1 2 N
i
δ =1 + 1 + ··· +1
i
i
δ = N.
i
i
In three dimension we have δ ,i, j =1, 2, 3and
j
1
2
3
k
δ = δ + δ + δ =3.
2
1
k
3
In certain circumstances the Kronecker delta can be written with only subscripts. For example,
δ ij , i, j =1, 2, 3. We shall find that these circumstances allow us to perform a contraction on the lower
indices so that δ ii =3.
EXAMPLE 1.1-9. The determinant of a matrix A =(a ij ) can be represented in the indicial notation.
Employing the e-permutation symbol the determinant of an N × N matrix is expressed

|A| = e ij...k a 1i a 2j ··· a Nk
where e ij...k is an Nth order system. In the special case of a 2 × 2 matrix we write

|A| = e ij a 1i a 2j

where the summation is over the range 1,2 and the e-permutation symbol is of order 2. In the special case
of a 3 × 3 matrix we have

a 11 a 12 a 13

|A| = a 21 a 22 a 23 = e ijk a i1 a j2 a k3 = e ijk a 1i a 2j a 3k

a 31 a 32 a 33
where i, j, k are the summation indices and the summation is over the range 1,2,3. Here e ijk denotes the
e-permutation symbol of order 3. Note that by interchanging the rows of the 3 × 3 matrix we can obtain