Page 119 - Intro to Tensor Calculus

114

we simplify equation (1.4.26) to the form

" σ #
i β i
∂A i m ∂A σ α ∂x ∂x
+ A = + A σ . (1.4.27)
j
∂x j mj ∂x β αβ ∂x ∂x
Define the quantity
∂A i i m
i
A = + A (1.4.28)
,j j
∂x mj
i
as the covariant derivative of the contravariant tensor A . The equation (1.4.27) demonstrates that a covariant
derivative of a contravariant tensor will transform like a mixed second order tensor and
β
σ ∂x ∂x i
i
A ,j = A ,β σ . (1.4.29)
j
∂x ∂x
∂A i
i
Again it should be observed that for the condition where g ij are constants we have A ,j = and the
∂x j
covariant derivative of a contravariant tensor reduces to an ordinary derivative in this special case.
In a similar manner the covariant derivative of second rank tensors can be derived. We find these
derivatives have the forms:

∂A ij σ σ
A ij,k = − A σj − A iσ
∂x k ik jk
∂A i j i σ
A i = + A σ − A i (1.4.30)
j,k k j σ
∂x σk jk
∂A ij i j
ij σj iσ
A = + A + A .
,k k
∂x σk σk
In general, the covariant derivative of a mixed tensor
ij...k
A
lm...p
of rank n has the form

ij...k
∂A i j k
ij...k lm...p σj...k iσ...k ij...σ
A = + A + A + ··· + A
lm...p,q q lm...p lm...p lm...p
∂x σq σq σq
(1.4.31)

σ ij...k σ ij...k σ
ij...k
− A − A − ··· − A
σm...p lσ...p lm...σ
lq mq pq
and this derivative is a tensor of rank n +1. Note the pattern of the + signs for the contravariant indices
and the − signs for the covariant indices.
Observe that the covariant derivative of an nth order tensor produces an n+1st order tensor, the indices
of these higher order tensors can also be raised and lowered by multiplication by the metric or conjugate
metric tensor. For example we can write
jk
jk i
g im A jk | m = A jk | i and g im A | m = A |