Page 198 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 17: Tensor Fields in Curvilinear Coordinates

Notes

Task Do the same for spherical coordinates.
Exercise 3.1: Relying upon formula (1) and results of exercise 1.1. Calculate the Christoffel
symbols for cylindrical coordinates.
Exercise 4.1: Do the same for spherical coordinates.
Exercise 5.1: Remember formula from which you derive

Ei =  R . ...(2)
 y i
Substitute (2) into left hand side of the derivational formula (1) and relying on the properties of
mixed derivatives prove that the Christoffel symbols are symmetric with respect to their lower
k
indices:    k ji .
ij

k
Notes Christoffel symbols  form a three-dimensional array with one upper index
ij
and two lower indices. However, they do not represent a tensor. We shall not prove this
fact since it again leads deep into differential geometry.
17.6 Formula for Christoffel Symbols

Let’s take formula (3) and substitute it into both sides of (1). As a result we get the following

equality for Christoffel symbols  k ji :

3  S q 3 3
 i j e   k ij S e . ...(1)
q 
q
k q
q 1 y k 1 q 1



Cartesian basis vectors e do not depend on y ; therefore, they are not differentiated when we
j
q
substitute (3) into (1). Both sides of (1) are expansions in the base e , e , e of the auxiliary
2
3
1
Cartesian coordinate system. Due to the uniqueness of such expansions we have the following
equality derived from (1):
3
 S q i j   k S . ...(2)
q

 y k 1 ij k

Exercise 1.1: Using concept of the inverse matrix ( T = S ) derive the following formula for the
–1
Christoffel symbols  from (2):
k
ij
3 k S q
k
ij 
  T q i j . ...(3)
q 1  y

Due to this formula (3) can be transformed in the following way:
3 k S q 3 k  2 x q 3  S q
k
ij 
  T q i j  T q i j  T q k j i . ...(4)



q 1  y q 1  y y q 1  y



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