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MAS251/PHY202/MAS651 Handout 5
Grad, Div and Curl in Cylindrical and Spherical Coordinates
In applications, we often use coordinates other than Cartesian coordinates. It is important to rememberthat expressions for the operations of vector analysis are different in different coordinates. Here wegive explicit formulae for cylindrical and spherical coordinates.
1 Cylindrical Coordinates
In cylindrical coordinates,x = r cos φ , y = r sin φ , z = z ,
we have
∇f = r∂f
∂r+ φ
1
r
∂f
∂φ+ z
∂f
∂z,
∇ · u =1
r
∂(rur)
∂r+
1
r
∂uφ
∂φ+
∂uz
∂z,
∇× u = r
(1
r
∂uz
∂φ− ∂uφ
∂z
)+ φ
(∂ur
∂z− ∂uz
∂r
)+ z
1
r
[∂(ruφ)
∂r− ∂ur
∂φ
].
2 Spherical Coordinates
In spherical coordinates,
x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ ,
we have
∇f = r∂f
∂r+ θ
1
r
∂f
∂θ+ φ
1
r sin θ
∂f
∂φ,
∇ · u =1
r2
∂(r2ur)
∂r+
1
r sin θ
∂(uθ sin θ)
∂θ+
1
r sin θ
∂uφ
∂φ,
∇× u = r1
r sin θ
[∂
∂θ(uφ sin θ)− ∂uθ
∂φ
]+ θ
1
r
[1
sin θ
∂ur
∂φ− ∂
∂r(ruφ)
]+ φ
1
r
[∂
∂r(ruθ)−
∂ur
∂θ
].
