Page 30 - Intro to Tensor Calculus

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Other forms of this identity are

e ijk r s t rst and e ijk a ir a js a kt = |A|e rst . (1.1.19)
a a a = |A|e
i j k

Consider the representation of the determinant
1 1
a 1 a 2 a 1
3
|A| = a 2 1 a 2 2 a 2

3 3 3
a 1 a 2 a 3
3
by use of the indicial notation. By column expansions, this determinant can be represented
r s t
|A| = e rst a a a (1.1.20)
1 2 3
and if one uses row expansions the determinant can be expressed as

ijk 1 2 3
|A| = e a a a . (1.1.21)
i j k
Define A i m as the cofactor of the element a m in the determinant |A|. From the equation (1.1.20) the cofactor
i
r
of a is obtained by deleting this element and we find
1
s t
1
A = e rsta a . (1.1.22)
r 2 3
The result (1.1.20) can then be expressed in the form

1
1
r
1
1
1
3
2
|A| = a A = a A + a A + a A . (1.1.23)
1 r 1 1 1 2 1 3
That is, the determinant |A| is obtained by multiplying each element in the first column by its corresponding
cofactor and summing the result. Observe also that from the equation (1.1.20) we find the additional
cofactors
2
3
r s
r t
A = e rst a a and A = e rsta a . (1.1.24)
t
1 2
s
1 3
Hence, the equation (1.1.20) can also be expressed in one of the forms
2
3
1
2
s
2
2
|A| = a A = a A + a A + a A 2 3
2
2
2
2
1
2
s
3
3
1
3
2
t
3
|A| = a A = a A + a A + a A 3 3
2
3
3
3
1
3
t
The results from equations (1.1.22) and (1.1.24) can be written in a slightly different form with the indicial
notation. From the notation for a generalized Kronecker delta defined by
e ijk e lmn = δ ijk ,
lmn
the above cofactors can be written in the form
1 1jk s t 1 1jk s t
1
123
s t
A = e e rst a a = e e rst a a = δ a a
r 2 3 j k rst j k
2! 2!
1 2jk s t 1 2jk s t
123
s t
2
A = e e srt a a = e e rst a a = δ a a
r 1 3 j k rst j k
2! 2!
1 3jk s t 1 3jk s t
t s
123
3
A = e e tsr a a = e e rst a a = δ a a .
r 1 2 j k rst j k
2! 2!