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Astro 497 Spring 2008
General Orthogonal Curvilinear Coordinates
and Differential Vector Operators
Generalized coordinates q 1 , q 2 , q 3 , with corresponding unit vectors ê 1 , ê 2 , ê 3 . The
scale factors relating arc length, l, to coordinate increments are h 1 , h 2 , h 3 , with dl i =
h i (q 1 , q 2 , q 3 ) dq i .
General Expressions for Differential Vector Operators
Acting on a scalar field, f, or a vector field, A = {A i }
Gradient:
Laplacian:
∇ 2 f =
Divergence
[ (
1 ∂ h2 h 3
h 1 h 2 h 3 ∂q 1 h 1
∇ · A =
∇f = ê1
h 1
∂f
∂q 1
+ ê2
h 2
∂f
∂q 2
+ ê3
h 3
∂f
∂q 3
)
∂f
+ ∂ (
h1 h 3
∂q 1 ∂q 2 h 2
)
∂f
+ ∂ (
h1 h 2
∂q 2 ∂q 3 h 3
[
1 ∂
(h 2 h 3 A 1 ) + ∂ (h 1 h 3 A 2 ) + ∂ ]
(h 1 h 2 A 3 )
h 1 h 2 h 3 ∂q 1 ∂q 2 ∂q 3
)]
∂f
∂q 3
Curl
∣ ê 1 ê 2 ê 3 ∣∣∣∣∣∣∣∣∣
h 2 h 3 h 1 h 3 h 1 h 2
∇ × A =
∂ ∂ ∂
∂q 1 ∂q2 ∂q3
∣
h 1 A 1 h 2 A 2 h 3 A 3
Expressions for q i and h i in Common Coordinate Systems
Coordinate System q 1 q 2 q 3 h 1 h 2 h 3
Cartesian x y z 1 1 1
Spherical Polar r ϑ ϕ 1 r r sin ϑ
Cylindrical ρ ϕ z 1 ρ 1

Astro 497 Problem Set Spring 2008
Specific Forms of Differential Vector Operators
Cartesian Coordinates (x, y, z)
Acting on a scalar field, f, or a vector field, A = {A i }
∇f = ˆx ∂f
∂x + ŷ ∂f
∂y + ẑ ∂f
∂z
∇ 2 f = ∂2 f
∂x + ∂2 f
2 ∂y + ∂2 f
2 ∂z 2
∣ ˆx ŷ ẑ
∇ · A = ∂A x
∂x + ∂A y
∂y + ∂A z
∇ × A =
∂z
ˆ
∣∣∣∣∣∣∣∣
∂ ∂ ∂
∂x ∂y ∂z
∣
A x A y A z
Spherical Coordinates (r, ϑ, ϕ)
∇f = ˆr ∂f
∂r + ˆÕr
∂f
∂ϑ + ∂f
r sin ϑ ∂ϕ
∇ 2 f = 1 (
∂
r 2 ∂f )
1 ∂
ˆÕ
(
+ sin ϑ ∂f )
1
+
r 2 ∂r ∂r r 2 sin ϑ ∂ϑ ∂ϑ r 2 sin 2 ϑ
∇ · A = 1 ∂ (
r 2 ) 1 ∂
r 2 A r +
∂r r sin ϑ ∂ϑ (sin ϑ A ϑ) + 1 ∂ 2 A ϕ
r sin ϑ ∂ϕ 2
∣ ˆr
∣∣∣∣∣∣∣∣∣
r 2 sin ϑ r sin ϑ
ˆr
∇ × A =
∂ ∂ ∂
∂r ∂ϑ ∂ϕ
∣
A r r A ϑ r sin ϑ A ϕ
Cylindrical Coordinates (ρ, ϕ, z)
∂ 2 f
∂ϕ 2
∇f = ˆÖ∂f
∂ρ + ˆρ
∇ · A = 1 ρ
∂f
∂ϕ + ẑ ∂f
∂z
∂
∂ρ (ρ A ρ) + 1 ρ
∂A ϕ
∂ϕ + ∂A z
∂z
∇ 2 f = 1 ρ
ˆ
(
∂
ρ ∂f )
+ 1 ∂ 2 f
∂ρ ∂ρ ρ 2 ∂ϕ + ∂2 f
2 ∂z 2
∣ ẑ ∣∣∣∣∣∣∣∣∣
ˆÖρ ρ
∇ × A =
∂ ∂ ∂
∂ρ ∂ϕ ∂z
∣
A ρ ρ A ϕ A z