Page 118 - Intro to Tensor Calculus

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Employing the equation (1.4.17), with α replaced by σ, the equation (1.4.19) is expressible in the form

j k j k
∂A α σ ∂A j ∂x ∂x i ∂x ∂x
− A σ = α − A i α (1.4.20)
k
∂x β αβ ∂x ∂x ∂x β jk ∂x ∂x β
or alternatively
" #
j k
∂A α σ ∂A j i ∂x ∂x
− A σ = k − A i α β . (1.4.21)
β
∂x αβ ∂x jk ∂x ∂x
Define the quantity

∂A j i
A j,k = − A i (1.4.22)
∂x k jk
k
as the covariant derivative of A j with respect to x . The equation (1.4.21) demonstrates that the covariant
derivative of a covariant tensor produces a second order tensor which satisfies the transformation law
j
∂x ∂x k
A α,β = A j,k α β . (1.4.23)
∂x ∂x
Other notations frequently used to denote the covariant derivative are:

A j,k = A j;k = A j/k = ∇ k A j = A j | k . (1.4.24)

In the special case where g ij are constants the Christoffel symbols of the second kind are zero, and conse-
∂A j
quently the covariant derivative reduces to A j,k = . That is, under the special circumstances where the
∂x k
Christoffel symbols of the second kind are zero, the covariant derivative reduces to an ordinary derivative.
Covariant Derivative of Contravariant Tensor

i
i
A contravariant tensor A obeys the transformation law A = A α ∂x i which can be expressed in the
∂x α
form
α ∂x i
i
A = A α (1.4.24)
∂x
by interchanging the barred and unbarred quantities. We write the transformation law in the form of equation
(1.4.24) in order to make use of the second derivative relation from the previously derived equation (1.4.7).
j
Differentiate equation (1.4.24) with respect to x to obtain the relation
α
β
2 i
∂A i α ∂ x ∂x β ∂A ∂x ∂x i
= A + α . (1.4.25)
α
β
j
β
∂x j ∂x ∂x ∂x j ∂x ∂x ∂x
Changing the indices in equation (1.4.25) and substituting for the second derivative term, using the relation
from equation (1.4.7), produces the equation
" i m k # β α β i
i
∂A α σ ∂x i ∂x ∂x ∂x ∂A ∂x ∂x
= A − + . (1.4.26)
α
∂x j αβ ∂x σ mk ∂x ∂x β ∂x j ∂x ∂x ∂x α
β
j
Applying the relation found in equation (1.4.24), with i replaced by m, together with the relation
β
∂x ∂x k k
= δ ,
j
j
∂x ∂x β