Vector Analysis - An Introduction to Dynamic Meteorology - page 500

(b) Cylindrical coordinates: (r, λ, z)

Coordinate Symbol Velocity component Unit vector

Radial r u i

Azimuthal λ v j

Upward z w k

i ∂ j 1

r

∂
∂λ +

k ∂
∂z

∇ =

∂r +

1
r

∂ (ru)
∂r

1
r

∂v
∂λ

∇· V = +

1
r

∂ (rv)
∂r

1
r

∂u
∂λ

k · ( ∇ × V ) = −

r ∂
∂r

∂ 2
∂λ 2

1
r

∂
∂r

1
r 2

2

∇

h

= +

(c) Spherical coordinates: (λ, φ, r)

Coordinate Symbol Velocity component Unit vector

Longitude λ u i

Latitude φ v j

Height r w k

i
r cos φ

∂
∂λ +

j 1
r

∂
∂φ +

k ∂
∂r

∇ =

∂u
∂λ +

1
r cos φ

∂ (v cos φ)
∂φ

∇· V =

∂v
∂λ −

1
r cos φ

∂ (u cos φ)
∂φ

k · ( ∇ × V ) =

∂ 2
∂λ 2

cos φ ∂
∂φ

1
r 2 cos 2 φ

∂
∂φ

2

∇

h

= + cos φ