Page 361 - Intro to Tensor Calculus

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APPENDIX B
CHRISTOFFEL SYMBOLS OF SECOND KIND

2
1
3
1. Cylindrical coordinates (r, θ, z)= (x ,x ,x )
r ≥ 0 h 1 =1
x = r cos θ
y = r sin θ 0 ≤ θ ≤ 2π h 2 = r
−∞ <z < ∞ h 3 =1
z = z
The coordinate curves are formed by the intersection of the coordinate surfaces
2 2 2
x + y = r , Cylinders
Planes
y/x =tan θ
z = Constant Planes.

1 2 2 1

= =
22 12 21 r
= −r
1
2
3
2. Spherical coordinates (ρ, θ, φ)=(x ,x ,x )
ρ ≥ 0 h 1 =1
x = ρ sin θ cos φ
y = ρ sin θ sin φ 0 ≤ θ ≤ π h 2 = ρ
z = ρ cos θ 0 ≤ φ ≤ 2π h 3 = ρ sin θ
The coordinate curves are formed by the intersection of the coordinate surfaces

2 2 2 2
x + y + z = ρ Spheres
2 2 2
x + y =tan θz Cones

y = x tan φ Planes.

1 2 2 1
= =
= −ρ
22 12 21 ρ
1 2 3 3 1

= =
= −ρ sin θ
33 13 31 ρ

2 3 3
= =cot θ
= − sin θ cos θ
33 32 23