Page 18 - Intro to Tensor Calculus

14

Generalized Kronecker delta
The generalized Kronecker delta is defined by the (n × n) determinant

i i i
δ m δ n ··· δ
p
j
j δ n ··· δ j
δ
m
p
δ ij...k = . . .
mn...p . .
. . . .
. . . .
k k k
δ m δ n ··· δ p
For example, in three dimensions we can write
i i i
δ δ
m n δ
p
j ijk
ijk
j
δ mnp = δ δ n δ p = e e mnp .
j
m
k k k
δ m δ n δ p
Performing a contraction on the indices k and p we obtain the fourth order system
s
r
r s
δ rs = δ rsp = e rsp e mnp = e prs e pmn = δ δ − δ δ .
mn mnp m n n m
As an exercise one can verify that the definition of the e-permutation symbol can also be defined in terms
of the generalized Kronecker delta as
= δ 123 ··· N .
e j 1 j 2 j 3 ···j N
j 1 j 2 j 3 ···j N
Additional definitions and results employing the generalized Kronecker delta are found in the exercises.
In section 1.3 we shall show that the Kronecker delta and epsilon permutation symbol are numerical tensors
which have fixed components in every coordinate system.
Additional Applications of the Indicial Notation
The indicial notation, together with the e − δ identity, can be used to prove various vector identities.
~
~
~
~
EXAMPLE 1.1-14. Show, using the index notation, that A × B = −B × A
Solution: Let
~ ~ ~
C = A × B = C 1 b e 1 + C 2 b e 2 + C 3 b e 3 = C i b e i and let
~
~
~
D = B × A = D 1 b e 1 + D 2 b e 2 + D 3 b e 3 = D i b e i .
We have shown that the components of the cross products can be represented in the index notation by
and D i = e ijk B j A k .
C i = e ijk A j B k
We desire to show that D i = −C i for all values of i. Consider the following manipulations: Let B j = B s δ sj
and A k = A m δ mk and write
D i = e ijk B j A k = e ijk B s δ sj A m δ mk (1.1.6)

where all indices have the range 1, 2, 3. In the expression (1.1.6) note that no summation index appears
more than twice because if an index appeared more than twice the summation convention would become
meaningless. By rearranging terms in equation (1.1.6) we have

D i = e ijk δ sj δ mk B s A m = e ism B s A m .