Page 88 - Intro to Tensor Calculus

83

Epsilon Permutation Symbol

Associated with the e−permutation symbols there are the epsilon permutation symbols defined by the
relations
√ ijk 1 ijk
ijk = ge ijk and = √ e (1.3.30)
g
where g is the determinant of the metrices g ij .
It can be demonstrated that the e ijk permutation symbol is a relative tensor of weight −1 whereas the
ijk permutation symbol is an absolute tensor. Similarly, the e ijk permutation symbol is a relative tensor of
weight +1 and the corresponding ijk permutation symbol is an absolute tensor.
EXAMPLE 1.3-10. ( permutation symbol)
Show that e ijk is a relative tensor of weight −1 and the corresponding ijk permutation symbol is an
absolute tensor.
Solution: Examine the Jacobian 1 1 1
∂x ∂x ∂x
3
x 2 2 2
∂x 1 ∂x 2 ∂x
J = ∂x 1 ∂x 2 ∂x 3
x ∂x 3 ∂x 3 ∂x
3
∂x ∂x ∂x
∂x 1 ∂x 2 ∂x 3
and make the substitution
∂x i
i
a = j ,i, j =1, 2, 3.
j
∂x
From the definition of a determinant we may write
x
k
i
j
e ijk a a a = J( )e mnp . (1.3.31)
m n p
x
By definition, e mnp = e mnp in all coordinate systems and hence equation (1.3.31) can be expressed in the
form
x ∂x ∂x ∂x
h i −1 i j k
J( ) e ijk m n p = e mnp (1.3.32)
x ∂x ∂x ∂x
which demonstrates that e ijk transforms as a relative tensor of weight −1.
We have previously shown the metric tensor g ij is a second order covariant tensor and transforms
m
∂x ∂x n
according to the rule g ij = g mn . Taking the determinant of this result we find
i
∂x ∂x j
m 2 h i 2

∂x x
g = |g | = |g mn| = g J( ) (1.3.33)
ij
∂x i x

where g is the determinant of (g ij )and g is the determinant of (g ). This result demonstrates that g is a
ij
scalar invariant of weight +2. Taking the square root of this result we find that
p √ x
g = gJ( ). (1.3.34)
x
√
Consequently, we call g a scalar invariant of weight +1. Now multiply both sides of equation (1.3.32) by
√
g and use (1.3.34) to verify the relation
i
j
√ ∂x ∂x ∂x k p
ge ijk m n p = g e mnp . (1.3.35)
∂x ∂x ∂x
√
This equation demonstrates that the quantity ijk = ge ijk transforms like an absolute tensor.