Page 82 - Intro to Tensor Calculus

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Employing the chain rule we write

∂x m ∂x m ∂x i ∂x m ∂x k i
δ
= n = n k (1.3.14)
n
i
i
∂x ∂x ∂x ∂x ∂x
m m
∂x
i
By hypothesis, the x ,i =1,... ,N are independent coordinates and therefore we have n = δ and (1.3.14)
∂x n
simplifies to
m i ∂x m ∂x k
δ = δ .
n k i n
∂x ∂x
Therefore, the Kronecker delta transforms as a mixed second order tensor.
Conjugate Metric Tensor
Let g denote the determinant of the matrix having the metric tensor g ij ,i,j =1,...,N as its elements.
In our study of cofactor elements of a matrix we have shown that

j
cof(g 1j )g 1k + cof(g 2j )g 2k + ... + cof(g Nj )g Nk = gδ . (1.3.15)
k
We can use this fact to find the elements in the inverse matrix associated with the matrix having the
components g ij . The elements of this inverse matrix are

1
ij
g = cof(g ij ) (1.3.16)
g
ij
and are called the conjugate metric components. We examine the summation g g ik and find:
ij
1j
2j
g g ik = g g 1k + g g 2k + ... + g Nj g Nk
1
= [cof(g 1j )g 1k + cof(g 2j )g 2k + ... + cof(g Nj )g Nk ]
g
1 h j i j
= gδ k = δ k
g
The equation
ij
g g ik = δ j (1.3.17)
k
is an example where we can use the quotient law to show g ij is a second order contravariant tensor. Because
of the symmetry of g ij and g ij the equation (1.3.17) can be represented in other forms.

i
EXAMPLE 1.3-5. Let A i and A denote respectively the covariant and contravariant components of a
~
vector A. Show these components are related by the equations

A i = g ij A j (1.3.18)
k jk
A = g A j (1.3.19)
where g ij and g ij are the metric and conjugate metric components of the space.