8/17/2019 Spherical Coord
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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 CoordinatesIn cylindrical coordinates,
x = r cos φ , y = r sin φ ,
z = z ,
we have
∇f = r∂f ∂r
+ φ1r
∂f
∂φ + z∂f
∂z ,
∇ · u = 1
r
∂ (rur)
∂r +
1
r
∂uφ
∂φ +
∂uz
∂z ,
∇× u = r1r ∂uz∂φ − ∂ uφ∂z
+ φ∂ur∂z − ∂uz∂r + z1r
∂ (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
+ θ1
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 = r 1r sin θ
∂
∂θ(uφ sin θ) −
∂uθ
∂φ
+ θ1
r
1
sin θ
∂ur
∂φ −
∂
∂r(ruφ)
+ φ1
r
∂
∂r(ruθ) −
∂ur
∂θ
.
