Page 116 - Intro to Tensor Calculus

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EXAMPLE 1.4-4. (Christoffel symbols of the second kind)
Find formulas for the calculation of the Christoffel symbols of the second kind in a generalized orthogonal
coordinate system with metric coefficients

2
g ij =0 for i 6= j and g (i)(i) = h , i =1, 2, 3
(i)
where i is not summed.
Solution: By definition we have

i im i1 i2 i3

= g [jk, m]= g [jk, 1] + g [jk, 2] + g [jk, 3] (1.4.12)
jk
By hypothesis the coordinate system is orthogonal and so
1
ij
ii
g =0 for i 6= j and g = i not summed.
g ii
The only nonzero term in the equation (1.4.12) occurs when m = i and consequently

i ii [jk, i]
= g [jk, i]= no summation on i. (1.4.13)
jk g ii
We can now consider the four cases considered in the example 1.4-2.
CASE I Let j = k = i and show

i [ii, i] 1 ∂g ii 1 ∂
= = = ln g ii no summation on i. (1.4.14)
ii g ii 2g ii ∂x i 2 ∂x i
CASE II Let k = j 6= i and show

i [jj, i] −1 ∂g jj

= = no summation on i or j. (1.4.15)
jj g ii 2g ii ∂x i
CASE III Let i = j 6= k and verify that

j j [jk, j] 1 ∂g jj 1 ∂
= = = = ln g jj no summation on i or j. (1.4.16)
jk kj g jj 2g jj ∂x k 2 ∂x k
CASE IV For the case i 6= j 6= k we find

i [jk, i]
= =0, i 6= j 6= k no summation on i.
jk g ii
The above cases represent all 27 terms.