Page 31 - Intro to Tensor Calculus

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These cofactors are then combined into the single equation

1 ijk s t
i
A = δ rst j k (1.1.25)
a a
r
2!
r
which represents the cofactor of a . When the elements from any row (or column) are multiplied by their
i
corresponding cofactors, and the results summed, we obtain the value of the determinant. Whenever the
elements from any row (or column) are multiplied by the cofactor elements from a different row (or column),
and the results summed, we get zero. This can be illustrated by considering the summation
1 ijk s t m 1 ijk m s t
m
i
a a a
a A m = δ mst j k r = e e mst a a a
r
j k
r
2! 2!
1 ijk 1 ijk i
= e e rjk |A| = δ rjk |A| = δ |A|
r
2! 2!
Here we have used the e − δ identity to obtain
i
i k
i k
i
δ ijk = e ijk e rjk = e jik e jrk = δ δ − δ δ =3δ − δ =2δ i
rjk r k k r r r r
which was used to simplify the above result.
As an exercise one can show that an alternate form of the above summation of elements by its cofactors
is
r
r
a A m = |A|δ .
m i i
EXAMPLE 1.1-26. In N-dimensions the quantity δ j 1 j 2 ...j N is called a generalized Kronecker delta. It
k 1 k 2 ...k N
can be defined in terms of permutation symbols as
e j 1 j 2 ...j N e k 1 k 2 ...k N = δ j 1 j 2 ...j N (1.1.26)
k 1 k 2 ...k N
Observe that
δ j 1 j 2 ...j N e k 1 k 2 ...k N =(N!) e j 1 j 2 ...j N
k 1 k 2 ...k N
This follows because e k 1 k 2 ...k N is skew-symmetric in all pairs of its superscripts. The left-hand side denotes
a summation of N! terms. The first term in the summation has superscripts j 1 j 2 ...j N and all other terms
have superscripts which are some permutation of this ordering with minus signs associated with those terms
having an odd permutation. Because e j 1 j 2 ...j N is completely skew-symmetric we find that all terms in the
summation have the value +e j 1 j 2 ...j N . We thus obtain N! of these terms.