26.18 DERIVATIVES OF BASIS VECTORS AND CHRISTOFFEL SYMBOLS
where we have used the definition (26.75). By cyclically permuting the free indices
i, j, kin (26.79), we obtain two further equivalent relations,
∂gjk
∂ui=Γljiglk+Γlkigjl (26.80)and
∂gki
∂uj
=Γlkjgli+Γlijgkl. (26.81)If we now add (26.80) and (26.81) together and subtract (26.79) from the result,we find
∂gjk
∂ui+∂gki
∂uj−∂gij
∂uk=Γljiglk+Γlkigjl+Γlkjgli+Γlijgkl−Γlikglj−Γljkgil=2Γlijgkl,where we have used the symmetry properties of both Γlijandgij. Contracting
both sides withgmkleads to the required expression for the Christoffel symbol in
terms of the metric tensor and its derivatives, namely
Γmij=^12 gmk(
∂gjk
∂ui+∂gki
∂uj−∂gij
∂uk). (26.82)
Calculate the Christoffel symbolsΓmijfor cylindrical polar coordinates.We may use either (26.75) or (26.82) to calculate the Γmijfor this simple coordinate system.
In cylindrical polar coordinates (u^1 ,u^2 ,u^3 )=(ρ, φ, z), the basis vectorseiare given by
(26.59). It is straightforward to show that the only derivatives of these vectors with respect
to the coordinates that are non-zero are
∂eρ
∂φ=
1
ρeφ,∂eφ
∂ρ=
1
ρeφ,∂eφ
∂φ=−ρeρ.Thus, from (26.75), we have immediately that
Γ^212 =Γ^221 =
1
ρand Γ^122 =−ρ. (26.83)Alternatively, using (26.82) and the fact thatg 11 =1,g 22 =ρ^2 ,g 33 = 1 and the other
components are zero, we see that the only three non-zero Christoffel symbols are indeed
Γ^212 =Γ^221 and Γ^122 .Thesearegivenby
Γ^212 =Γ^221 =
1
2 g 22∂g 22
∂u^1=
1
2 ρ^2∂
∂ρ(ρ^2 )=1
ρ,
Γ^122 =−
1
2 g 11∂g 22
∂u^1=−
1
2
∂
∂ρ(ρ^2 )=−ρ,which agree with the expressions found directly from (26.75) and given in (26.83 ).