Page 117 - Intro to Tensor Calculus

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EXAMPLE 1.4-5. (Notation) In the case of cylindrical coordinates we can use the above relations and
find the nonzero Christoffel symbols of the second kind:

1 1 ∂g 22 1
= − = −x = −r
22 2g 11 ∂x 1
2 2 1 ∂g 22 1 1

= = = =
12 21 2g 22 ∂x 1 x 1 r
2
1
Note 1: The notation for the above Christoffel symbols are based upon the assumption that x = r, x = θ
3
and x = z. However, in tensor calculus the choice of the coordinates can be arbitrary. We could just as well
1
3
2
have defined x = z, x = r and x = θ. In this latter case, the numbering system of the Christoffel symbols
changes. To avoid confusion, an alternate method of writing the Christoffel symbols is to use coordinates in
place of the integers 1,2 and 3. For example, in cylindrical coordinates we can write

θ θ 1 r
= = and = −r.
rθ θr r θθ
3
1
2
If we define x = r, x = θ, x = z, then the nonzero Christoffel symbols are written as

2 2 1 1
= = and = −r.
12 21 r 22
3
1
2
In contrast, if we define x = z, x = r, x = θ, then the nonzero Christoffel symbols are written

3 3 1 2
= = and = −r.
23 32 r 33
da
Note 2: Some textbooks use the notation Γ a,bc for Christoffel symbols of the first kind and Γ d bc = g Γ a,bc for
Christoffel symbols of the second kind. This notation is not used in these notes since the notation suggests
that the Christoffel symbols are third order tensors, which is not true. The Christoffel symbols of the first
and second kind are not tensors. This fact is clearly illustrated by the transformation equations (1.4.3) and
(1.4.7).

Covariant Differentiation

Let A i denote a covariant tensor of rank 1 which obeys the transformation law

∂x i
A α = A i α . (1.4.17)
∂x
β
Differentiate this relation with respect to x and show
j
2 i
∂ x ∂A i ∂x ∂x i
∂A α
= A i + α . (1.4.18)
j
α
β
∂x β ∂x ∂x β ∂x ∂x ∂x
Now use the relation from equation (1.4.7) to eliminate the second derivative term from (1.4.18) and express
it in the form
" #
i j k j i
∂A α σ ∂x i ∂x ∂x ∂A i ∂x ∂x
= A i σ − α β + j β α . (1.4.19)
β
∂x αβ ∂x jk ∂x ∂x ∂x ∂x ∂x